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hp  48gll  graphing  calculator 


user's  guide 


invent 

Edition  4 

HP  part  number  F2226-90020 


Notice 


REGISTER  YOUR  PRODUCT  AT:  www.reqister.hp.com 

THIS  MANUAL  AND  ANY  EXAMPLES  CONTAINED  HEREIN  ARE 
PROVIDED  'AS  IS"  AND  ARE  SUBJECT  TO  CHANGE  WITHOUT 
NOTICE.  HEWLETT-PACKARD  COMPANY  MAKES  NO  WARRANTY 
OF  ANY  KIND  WITH  REGARD  TO  THIS  MANUAL,  INCLUDING,  BUT 
NOT  LIMITED  TO,  THE  IMPLIED  WARRANTIES  OF 
MERCHANTABILITY,  NON-INFRINGEMENT  AND  FITNESS  FOR  A 
PARTICULAR  PURPOSE. 

HEWLETT-PACKARD  CO.  SHALL  NOT  BE  LIABLE  FOR  ANY  ERRORS 
OR  FOR  INCIDENTAL  OR  CONSEQUENTIAL  DAMAGES  IN 
CONNECTION  WITH  THE  FURNISHING,  PERFORMANCE,  OR  USE 
OF  THIS  MANUAL  OR  THE  EXAMPLES  CONTAINED  HEREIN. 


©  Copyright  2003  Hewlett-Packard  Development  Company,  L.P. 
Reproduction,  adaptation,  or  translation  of  this  manual  is  prohibited  without 
prior  written  permission  of  Hewlett-Packard  Company,  except  as  allowed 
under  the  copyright  laws. 


Hewlett-Packard  Company 
4995  Murphy  Canyon  Rd, 
Suite  301 

San  Diego,CA  92123 


Printing  History 

Edition  4 


April  2004 


Preface 


You  have  in  your  hands  a  compact  symbolic  and  numerical  computer  that  will 
facilitate  calculation  and  mathematical  analysis  of  problems  in  a  variety  of 
disciplines,  from  elementary  mathematics  to  advanced  engineering  and 
science  subjects.    Although  referred  to  as  a  calculator,  because  of  its 
compact  format  resembling  typical  hand-held  calculating  devices,  the  hp  48gll 
should  be  thought  of  as  a  graphics/programmable  hand-held  computer. 

The  hp  48gll  can  be  operated  in  two  different  calculating  modes,  the  Reverse 
Polish  Notation  (RPN)  mode  and  the  Algebraic  (ALG)  mode  (see  page  1-11  in 
user's  guide  for  additional  details).  The  RPN  mode  was  incorporated  into 
calculators  to  make  calculations  more  efficient.  In  this  mode,  the  operands  in 
an  operation  (e.g.,  '2  and  '3'  in  the  operation  '2+3')  are  entered  into  the 
calculator  screen,  referred  to  as  the  stack,  and  then  the  operator  (e.g.,  '+'  in 
the  operation  '2+3')  is  entered  to  complete  the  operation.  The  ALG  mode,  on 
the  other  hand,  mimics  the  way  you  type  arithmetic  expressions  in  paper. 
Thus,  the  operation  '2+3' ,  in  ALG  mode,  will  be  entered  in  the  calculator  by 
pressing  the  keys  '2' ,  '+' ,  and  '3' ,  in  that  order.  To  complete  the  operation 
we  use  the  ENTER  key.  Examples  of  applications  of  the  different  functions 
and  operations  in  this  calculator  are  illustrated  in  this  user's  guide  in  both 
modes. 

This  guide  contains  examples  that  illustrate  the  use  of  the  basic  calculator 
functions  and  operations.  The  chapters  in  this  user's  guide  are  organized  by 
subject  in  order  of  difficulty.  Starting  with  the  setting  of  calculator  modes  and 
display  options,  and  continuing  with  real  and  complex  number  calculations, 
operations  with  lists,  vectors,  and  matrices,  detailed  examples  of  graph 
applications,  use  of  strings,  basic  programming,  graphics  programming, 
string  manipulation,  advanced  calculus  and  multivariate  calculus  applications, 
advanced  differential  equations  applications  (including  Laplace  transform, 
and  Fourier  series  and  transforms),  and  probability  and  statistic  applications. 

For  symbolic  operations  the  calculator  includes  a  powerful  Computer 
Algebraic  System  (CAS)  that  lets  you  select  different  modes  of  operation,  e.g., 
complex  numbers  vs.  real  numbers,  or  exact  (symbolic)  vs.  approximate 


(numerical)  mode.  The  display  can  be  adjusted  to  provide  textbook-type 
expressions,  which  can  be  useful  when  working  with  matrices,  vectors, 
fractions,  summations,  derivatives,  and  integrals.  The  high-speed  graphics  of 
the  calculator  are  very  convenient  for  producing  complex  figures  in  very  little 
time. 

Thanks  to  the  infrared  port  and  the  RS  232  cable  available  with  your 
calculator,  you  can  connect  your  calculator  with  other  calculators  or 
computers.  The  high-speed  connection  through  infrared  or  RS  232  allows  the 
fast  and  efficient  exchange  of  programs  and  data  with  other  calculators  or 
computers.  The  calculator  provides  a  flash  memory  card  port  to  facilitate 
storage  and  exchange  of  data  with  other  users. 

The  programming  capabilities  of  the  calculator  allow  you  or  other  users  to 
develop  efficient  applications  for  specific  purposes.  Whether  it  is  advanced 
mathematical  applications,  specific  problem  solution,  or  data  logging,  the 
programming  languages  available  in  your  calculator  make  it  into  a  very 
versatile  computing  device. 

We  hope  your  calculator  will  become  a  faithful  companion  for  your  school 
and  professional  applications. 


Table  of  contents 


A  note  about  screenshots  in  this  guide,  Note  1 
Chapter  1  -  Getting  started,  1-1 

Basic  Operations,  1-1 

Batteries,  1-1 

Turning  the  calculator  on  and  off,  1-2 
Adjusting  the  display  contrast,  1-2 
Contents  of  the  calculator's  display,  1-2 
Menus,  1-3 

SOFT  menus  vs.  CHOOSE  boxes,  1-3 

Selecting  SOFT  menus  or  CHOOSE  boxes,  1-4 

The  TOOL  menu,  1-6 

Setting  time  and  date,  1-7 
Introducing  the  calculator's  keyboard,  1-10 
Setting  calculator's  mode,  1-11 

Operating  mode,  1-12 

Number  format  and  decimal  dot  or  comma,  1  -1  6 

Angle  measure,  1-21 

Coordinate  system,  1-22 

Beep,  Key  Click,  and  Last  Stack,  1-23 
Selecting  CAS  settings,  1  -24 
Selecting  Display  modes,  1  -24 

Selecting  the  display  font,  1-25 

Selecting  properties  of  the  line  editor,  1-26 

Selecting  properties  of  the  Stack,  1-26 

Selecting  properties  of  the  Equation  Writer  (EQW),  1-27 

Selecting  the  size  of  the  header,  1-28 

Selecting  the  clock  display,  1-28 

Chapter  2  -  Introducing  the  calculator,  2-1 
Calculator  objects,  2-1 
Editing  expressions  in  the  screen,  2-3 

Creating  arithmetic  expressions,  2-3 
Editing  arithmetic  expressions,  2-6 


Page  TOC-1 


Creating  algebraic  expressions,  2-7 
Editing  algebraic  expressions,  2-8 
Using  the  Equation  Writer  (EQW)  to  create  expressions,  2-1 0 

Creating  arithmetic  expressions,  2-1  1 
Editing  arithmetic  expressions,  2-16 
Creating  algebraic  expressions,  2-19 
Editing  algebraic  expressions,  2-20 

Creating  and  editing  summations,  derivatives,  and  integrals,  2-28 
Organizing  data  in  the  calculator,  2-32 

Functions  for  manipulation  of  variables,  2-33 

The  HOME  directory,  2-34 

The  CASDIR  sub-directory,  2-35 

Typing  directory  and  variable  names,  2-37 

Creating  sub-directories,  2-38 

Moving  among  sub-directories,  2-42 

Deleting  sub-directories,  2-42 
Variables,  2  46 

Creating  variables,  2-46 

Checking  variable  contents,  2-51 

Replacing  the  contents  of  variables,  2-53 

Copying  variables,  2-54 

Reordering  variables  in  a  directory,  2-57 

Moving  variables  using  the  FILES  menu,  2-58 

Deleting  variables,  2-59 
UNDO  and  CMD  functions,  2  61 
Flags,  2  62 

Example  of  flag  setting:  general  solution  vs.  principal  value,  2-63 
Other  flags  of  interest,  2-64 
CHOOSE  boxes  vs.  SOFT  menus,  2-65 
Selected  CHOOSE  boxes,  2-66 

Chapter  3  -  Simple  calculations  with  real  numbers,  3-1 
Checking  calculator  settings,  3-1 
Checking  calculator  mode,  3-2 
Real  number  calculations,  3-2 

Changing  sing  of  a  number,  variable,  or  expression,  3-3 


Page  TOC-2 


The  inverse  function,  3-3 

Addition,  subtraction,  multiplication,  division,  3-3 

Using  parentheses,  3-4 

Absolute  value  function,  3-4 

Squares  and  square  roots,  3-4 

Powers  and  roots,  3-5 

Base-10  logarithms  and  powers  of  10,  3-5 

Using  powers  of  10  in  entering  data,  3-5 

Natural  logarithms  and  exponential  function,  3-6 

Trigonometric  functions,  3-6 

Inverse  trigonometric  functions,  3-6 

Differences  between  functions  and  operators,  3-7 
Real  number  functions  in  the  MTH  menu,  3-7 

Hyperbolic  functions  and  their  inverses,  3-9 

Real  number  functions,  3-1  1 

Special  functions,  3-14 
Calculator  constants,  3-16 
Operation  with  units,  3-1 7 

The  UNITS  menu,  3-17 

Available  units,  3-1  8 

Converting  to  base  units,  3-21 

Attaching  units  to  numbers,  3-22 

Operations  with  units,  3-25 

Units  manipulation  tools,  3-27 
Physical  constants  in  the  calculator,  3-28 
Special  physical  functions,  3-3 1 

Function  Z FACTOR,  3-32 

Function  F0?i,  3-32 

Function  SIDENS,  3-32 

Function  TDELTA,  3-33 

Function  TINC,  3-33 
Defining  and  using  functions,  3-33 
Functions  defined  by  more  than  one  expression,  3-35 

The  IFTE  function,  3-35 

Combined  IFTE  functions,  3-36 


Chapter  4  -  Calculations  with  complex  numbers,  4-1 

Definitions,  4-1 

Setting  the  calculator  to  COMPLEX  mode,  4  1 

Entering  complex  numbers,  4-2 

Polar  representation  of  a  complex  number,  4-3 
Simple  operations  with  complex  numbers,  4-4 

Changing  sign  of  a  complex  number,  4-4 

Entering  the  unit  imaginary  number,  4-5 
The  CMPLX  menus,  4  5 

CMPLX  menu  through  the  MTH  menu,  4-5 

CMPLX  menu  in  keyboard,  4-7 
Functions  applied  to  complex  numbers,  4-8 

Functions  from  the  MTH  menu,  4-8 
Function  DROITE:  equation  of  a  straight  line,  4-9 

Chapter  5  -  Algebraic  and  arithmetic  operations,  5-1 
Entering  algebraic  objects,  5-1 
Simple  operations  with  algebraic  objects,  5-2 
Functions  in  the  ALG  menu,  5-3 

COLLECT,  5-5 
EXPAND,  5-5 
FACTOR,  5-5 
LNCOLLECT,  5-5 
LIN,  5-5 
PARTFRAC,  5-5 
SOLVE,  5-5 
SUBST,  5-5 
TEXPAND,  5-5 

Other  forms  of  substitution  in  algebraic  expressions,  5-6 
Operations  with  transcendental  functions,  5-7 

Expansion  and  factoring  using  log-exp  functions,  5-8 

Expansion  and  factoring  using  trigonometric  functions,  5-8 
Functions  in  the  ARITHMETIC  menu,  5  9 

DIVIS,  5-10 


Page  TOC-4 


FACTORS,  5-10 

LGCD,  5-10 

PROPFRAC,  5-10 

SIMP2,  5-10 

INTEGER  menu,  5-10 

POLYNOMIAL  menu,  5-1  1 

MODULO  menu,  5-12 
Applications  of  the  ARITHMETIC  menu,  5  1  2 

Modular  arithmetic,  5-1  2 

Finite  arithmetic  in  the  calculator,  5-15 
Polynomials,  5-1  8 

Modular  arithmetic  with  polynomials,  5-19 

The  CHINREM  function,  5-19 

The  EGCD  function,  5-19 

The  GCD  function,  5-19 

The  HERMITE  function,  5-20 

The  HORNER  function,  5-20 

The  variable  VX,  5-20 

The  LAGRANGE  function,  5-21 

The  LCM  function,  5-21 

The  LEGENDRE  function,  5-22 

The  PCOEF  function,  5-22 

The  PROOT  function,  5-22 

The  PTAYL  function,  5-22 

The  QUOTIENT  and  REMAINDER  functions,  5-23 
The  EPSX0  function  and  the  CAS  variable  EPS,  5-23 
The  PEVAL  function,  5-23 
The  TCHEBYCHEFF  function,  5-24 
Fractions,  5-24 

The  SIMP2  function,  5-24 
The  PROPFRAC  function,  5-24 
The  PARTFRAC  function,  5-25 
The  FCOEF  function,  5-25 
The  FROOTS  function,  5-26 

Step-by-step  operations  with  polynomial  fractions,  5-26 
The  CONVERT  menu  and  algebraic  operations,  5-27 


UNITS  convert  menu,  5-27 
BASE  convert  menu,  5-28 
TRIGONOMETRIC  convert  menu,  5-28 
MATRICES  convert  menu,  5-28 
REWRITE  convert  menu,  5-28 

Chapter  6  -  Solution  to  single  equations,  6  1 
Symbolic  solution  of  algebraic  equations,  6-2 

Function  ISOL,  6-1 
Function  SOLVE,  6-2 
Function  SOLVEVX,  6-4 
Function  ZEROS,  6-4 
Numerical  solver  menu,  6-5 
Polynomial  equations,  6-6 
Financial  calculations,  6-9 

Solving  equations  with  one  unknown  through  NUM.SLV,  6-14 
The  SOLVE  soft  menu,  6  27 

The  ROOT  sub-menu,  6-27 
Function  ROOT,  6-27 
Variable  EQ,  6-28 
The  SOLVR  sub-menu,  6-28 
The  DIFFE  sub-menu,  6-30 
The  POLY  sub-menu,  6-30 
The  SYS  sub-menu,  6-31 
The  TVM  sub-menu,  6-31 

Chapter  7  -  Solving  multiple  equations,  7-1 
Rational  equation  systems,  7-1 

Example  1  -  Projectile  motion,  7-1 

Example  2  -  Stresses  in  a  thick  wall  cylinder,  7-2 

Example  3  -  System  of  polynomial  equations,  7-4 

Solution  to  simultaneous  equations  with  MSLV,  7-4 
Example  1  -  Example  from  the  help  facility,  7-5 
Example  2  -  Entrance  from  a  lake  into  an  open  channel,  7-6 

Using  the  Multiple  Equation  Solver  (MES),  7-1 0 
Application  1  -  Solution  of  triangles,  7-10 


Page  TOC-6 


Application  2  -  Velocity  and  acceleration  in  polar  coordinates,  7-1  8 


Chapter  8  -  Operations  with  Lists,  8  1 
Definitions,  8-1 
Creating  and  storing  lists,  8-1 
Composing  and  decomposing  lists,  8-2 
Operations  with  lists  of  numbers,  8-3 
Changing  sign,  8-3 

Addition,  subtraction,  multiplication,  division,  8-3 
Real  number  functions  from  the  keyboard,  8-5 
Real  number  functions  from  the  WITH  menu,  8-5 

Examples  of  functions  that  use  two  arguments,  8-6 
Lists  of  complex  numbers,  8-7 
Lists  of  algebraic  objects,  8-8 
The  MTH/LIST  menu,  8  8 
Manipulating  elements  of  a  list,  8-1 0 

List  size,  8-10 

Extracting  and  inserting  elements,  8-10 

Element  position  in  the  list,  8-1  1 

HEAD  and  TAIL  functions,  8-1 1 

The  SEQ  function,  8-1  1 

The  MAP  function,  8-12 
Defining  functions  that  use  lists,  8-1  3 
Applications  of  lists,  8-14 

Harmonic  mean  of  a  list,  8-1  5 

Geometric  mean  of  a  list,  8-16 

Weighted  average,  8-17 

Statistics  of  grouped  data,  8-1  8 

Chapter  9  -  Vectors,  9  1 
Definitions,  9-1 
Entering  vectors,  9-2 

Typing  vectors  in  the  stack,  9-2 
Storing  vectors  into  variables,  9-3 


Page  TOC-7 


Using  the  Matrix  Writer  (MTWR)  to  enter  vectors,  9-3 

Building  a  vector  with  -^ARRY,  9-6 
Identifying,  extracting,  and  inserting  vector  elements,  9-7 
Simple  operations  with  vectors,  9-9 

Changing  sign,  9-9 

Addition,  subtraction,  9-9 

Multiplication  and  division  by  a  scalar,  9-10 

Absolute  value  function,  9-10 
The  MTH/VECTOR  menu,  9  1 0 

Magnitude,  9-1  1 

Dot  product,  9-1  1 

Cross  product,  9-1  1 

Decomposing  a  vector,  9-1  2 

Building  a  two-dimensional  vector,  9-12 

Building  a  three-dimensional  vector,  9-13 

Changing  coordinate  system,  9-1  3 
Application  of  vector  operations,  9-1 6 

Resultant  of  forces,  9-1 6 

Angle  between  vectors,  9-16 

Moment  of  a  force,  9-1  7 

Equation  of  a  plane  in  space,  9-1  8 
Row  vectors,  column  vectors,  and  lists,  9-1 9 

Function  OBJ^,  9-20 

Function  -^LIST,  9-20 

Function  DROP,  9-21 

Transforming  a  row  vector  into  a  column  vector,  9-21 
Transforming  a  column  vector  into  a  row  vector,  9-22 
Transforming  a  list  into  a  vector,  9-24 
Transforming  a  vector  (or  matrix)  into  a  list,  9-25 

Chapter  10  -  Creating  and  manipulating  matrices,  10  1 

Definitions,  10-1 

Entering  matrices  in  the  stack,  1 0-2 

Using  the  Matrix  Writer,  1 0-2 
Typing  the  matrix  directly  into  the  stack,  1  0-3 
Creating  matrices  with  calculator  functions,  1 0-3 


Page  TOC-8 


Functions  GET  and  PUT,  10-6 

Functions  GETI  and  PUTI,  10-6 

Function  SIZE,  10-7 

Function  TRN,  10-8 

Function  CON,  10-8 

Function  IDN,  10-9 

Function  RDM,  10-10 

Function  RANM,  10-1 1 

Function  SUB,  10-1  1 

Function  REPL,  10-12 

Function  ->DIAG,  10-12 

Function  DIAG->,  10-13 

Function  VANDERMONDE,  10-14 

Function  HUBERT,  10-14 
A  program  to  build  a  matrix  out  of  a  number  of  lists,  10-15 

Lists  represent  columns  of  the  matrix,  1 0-1  5 

Lists  represent  rows  of  the  matrix,  10-17 
Manipulating  matrices  by  columns,  10-18 

Function  -^COL,  10-18 

Function  COL^,  10-19 

Function  COL+,  1 0-20 

Function  COL-,  10-20 

Function  CSWP,  10-21 
Manipulating  matrices  by  rows,  1 0-2 1 

Function  ^ROW,  10-22 

Function  ROW  10-23 

Function  ROW+,  1 0-24 

Function  ROW-,  10-24 

Function  RSWP,  10-25 

Function  RCI,  10-25 

Function  RCIJ,  10-26 

Chapter  1 1  -  Matrix  Operations  and  Linear  Algebra,  111 
Operations  with  matrices,  11-1 
Addition  and  subtraction,  1  1-2 
Multiplication,  1  1  -2 


Page  TOC-9 


Characterizing  a  matrix  (the  matrix  NORM  menu),  1 1  -6 

Function  ABS,  1  1-7 

Function  SNRM,  11-7 

Functions  RNRM  and  CNRM,  1  1-8 

Function  SRAD,  1  1-9 

Function  COND,  1  1-9 

Function  RANK,  11-10 

Function  DET,  11-11 

Function  TRACE,  11-13 

Function  TRAN,  11-14 
Additional  matrix  operations  (the  matrix  OPER  menu),  11-14 

Function  AXL,  11-15 

Function  AXM,  11-15 

Function  LCXM,  11-15 
Solution  of  linear  systems,  11-16 

Using  the  numerical  solver  for  linear  systems,  11-17 

Least-square  solution  (function  LSQ),  1  1-23 

Solution  with  the  inverse  matrix,  1  1-26 

Solution  by  "division"  of  matrices,  1  1-26 

Solving  multiple  set  of  equations  with  the  same  coefficient 
matrix,  1  1  -27 

Gaussian  and  Gauss-Jordan  elimination,  1  1-28 

Step-by-step  calculator  procedure  for  solving  linear  systems,  1  1-37 

Solution  to  linear  systems  using  calculator  functions,  1  1-46 

Residual  errors  in  linear  system  solutions  (function  RSD)  ,  1  1-43 
Eigenvalues  and  eigenvectors,  1 1  -44 

Function  PCAR,  1  1-44 

Function  EGVL,  11-45 

Function  EGV,  1 1-46 

Function  JORDAN,  11-47 

Function  MAD,  1  1-47 
Matrix  factorization,  1 1  -48 

Function  LU,  1  1-49 

Orthogonal  matrices  and  singular  value  decomposition,  1  1-49 
Function  SCHUR,  1 1-50 
Function  LQ,  11-51 


Page  TOG  10 


Function  QR,  11-51 
Matrix  Quadratic  Forms,  11-51 

The  QUADF  menu,  11-52 
Linear  Applications,  1 1  -54 

Function  IMAGE,  1 1-54 
Function  ISOM,  1 1-54 
Function  KER,  1 1-55 
Function  MKISOM,  11-55 

Chapter  12  -  Graphics,  12-1 

Graphs  options  in  the  calculator,  1  2-1 

Plotting  an  expression  of  the  form  y  =  f(x) ,  1  2-2 

Some  useful  PLOT  operations  for  FUNCTION  plots,  1  2-5 
Saving  a  graph  for  future  use,  1  2-8 
Graphics  of  transcendental  functions,  12-10 

Graph  of  ln(X) ,  1  2-8 

Graph  of  the  exponential  function,  1  2-1  0 
The  PPAR  variable,  12  11 
Inverse  functions  and  their  graphs,  12-12 
Summary  of  FUNCTION  plot  operation,  12-13 

Plots  of  trigonometric  and  hyperbolic  functions  and  their  inverses,  12-16 
Generating  a  table  of  values  for  a  function,  12-17 

The  TPAR  variable,  12-18 
Plots  in  polar  coordinates,  12-19 
Plotting  conic  curves,  1  2-2 1 
Parametric  plots,  1  2-23 

Generating  a  table  of  parametric  equations,  1  2-26 
Plotting  the  solution  to  simple  differential  equations,  1  2-26 
Truth  plots,  1  2  29 

Plotting  histograms,  bar  plots,  and  scatterplots,  1  2-30 

Bar  plots,  1  2-30 

Scatter  plots,  1  2-32 
Slope  fields,  1  2  34 
Fast  3D  plots,  1  2  35 
Wireframe  plots,  1  2-37 
Ps-Contour  plots,  12  39 


Page  TOC-1 1 


Y-Slice  plots,  12-41 
Gridmap  plots,  1 2-42 
Pr-Surface  plots,  1 2-43 

The  VPAR  variable,  1 2-44 
Interactive  drawing,  1 2-44 

DOT+  and  DOT-,  1 2-45 

MARK,  1 2-46 

LINE,  12-46 

TUNE,  12-46 

BOX,  12-47 

CIRCL,  12-47 

LABEL,  12-47 

DEL,  12-47 

ERASE,  12-48 

MENU,  12-48 

SUB,  1 2-48 

REPL,  1 2-48 

PICT^,  1 2-48 

X,Y^,  12-48 
Zooming  in  and  out  in  the  graphics  display,  1 2- 

ZFACT,  ZIN,  ZOUT,  and  ZLAST,  1 2-49 

BOXZ,  12-50 

ZDFLT,  ZAUTO,  1 2-50 

HZIN,  HZOUT,  VZIN,  and  VZOUT,  1 2-50 

CNTR,  12-50 

ZDECI,  12-50 

ZINTG,  12-51 

ZSQR,  12-51 

ZTRIG,  12-51 
The  SYMBOLIC  menu  and  graphs,  1 2-5 1 

The  SYMB/GRAPH  menu,  12-52 
Function  DRAW3DMATRIX,  1 2-54 


Chapter  13  -  Calculus  Applications,  13-1 
The  CALC  (Calculus)  menu,  13-1 
Limits  and  derivatives,  13-1 


Function  lim,  1  3-2 
Derivatives,  1  3-3 
Function  DERIV  and  DERVX,!  3-3 
The  DERIV&INTEG  menu,  13-3 
Calculating  derivatives  with  d,  1  3-4 
The  chain  rule,  1  3-6 
Derivatives  of  equations,  1  3-6 
Implicit  derivatives,  1  3-7 
Application  of  derivatives,  1  3-7 

Analyzing  graphics  of  functions,!  3-7 
Function  DOMAIN,  13-9 
Function  TABVAL,  1  3-9 
Function  SIGNTAB,  13-10 
Function  TABVAR,  13-10 

Using  derivatives  to  calculate  extreme  points,  1  3-1  2 

Higher-order  derivatives,  1  3-1  3 
Anti-derivatives  and  integrals,  ,13-14 

Functions  INT,  INTVX,  RISCH,  SIGMA,  and  SIGMAVX,1  3-14 

Definite  integrals,  13-15 
Step-by-step  evaluation  of  derivatives  and  integrals,  13-16 
Integrating  an  equation,  13-18 
Techniques  of  integration,  13-18 

Substitution  or  change  of  variables,  1  3-1  8 

Integration  by  parts  and  differentials,  1  3-1 9 

Integration  by  partial  fractions,  1  3-20 

Improper  integrals,  13-21 
Integration  with  units,  1  3-2 1 
Infinite  series,  1  3-23 

Taylor  and  Maclaurin's  series,  1  3-23 

Taylor  polynomial  and  remainder,!  3-23 

Functions  TAYLR,  TAYRLO,  and  SERIES,!  3-24 

Chapter  14  -  Multi-variate  Calculus  Applications,  14  1 
Multi-variate  functions,  1 4-1 
Partial  derivatives,  14-1 

Higher-order  derivatives,  14-3 


Page  TOG  13 


The  chain  rule  for  partial  derivatives,  14-4 
Total  differential  of  a  function  z  =  z(x,y)  ,  14-5 
Determining  extrema  in  functions  of  two  variables,  14-5 
Using  function  HESS  to  analyze  extrema,  14-6 
Multiple  integrals,  1 4-8 

Jacobian  of  coordinate  transformation,  14-9 
Double  integral  in  polar  coordinates,  14-9 

Chapter  15  -  Vector  Analysis  Applications,  15-1 

Definitions,  15-1 

Gradient  and  directional  derivative,  1 5-1 

A  program  to  calculate  the  gradient,  1 5-2 
Using  function  HESS  to  obtain  the  gradient,  15-2 

Potential  of  a  gradient,  1 5-3 

Divergence,  15-4 

Laplacian,  15-4 

Curl,  1 5-5 

Irrotational  fields  and  potential  function,  1  5-5 
Vector  potential,  1 5-6 

Chapter  16  -  Differential  Equations,  16  1 

Basic  operations  with  differential  equations,  1 6-1 

Entering  differential  equations,  16-1 

Checking  solutions  in  the  calculator,  16-2 

Slope  field  visualization  of  solutions,  1  6-3 
The  CALC/DIFF  menu,  1 6  4 
Solution  to  linear  and  non-linear  equations,  1 6-4 

Function  LDEC,  16-4 

Function  DESOLVE,  16-7 

The  variable  ODETYPE,  16-8 
Laplace  transforms,  16-10 

Definitions,  16-10 

Laplace  transform  and  inverses  in  the  calculator,  16-1  1 
Laplace  transform  theorems,  1 6-1  2 

Dirac's  delta  function  and  Heaviside's  step  function,  16-15 
Applications  of  Laplace  transform  in  the  solution  of  ODEs,  1  6-1 7 


Page  TOG  14 


Fourier  series,  16-27 

Function  FOURIER,  16-28 

Fourier  series  for  a  quadratic  function,  1  6-29 

Fourier  series  for  a  triangular  wave,  1  6-35 

Fourier  series  for  a  square  wave,  1 6-39 

Fourier  series  applications  in  differential  equations,  16-42 
Fourier  Transforms,  1 6-43 

Definition  of  Fourier  transforms,  1  6-46 

Properties  of  the  Fourier  transform,  1  6-48 
Fast  Fourier  Transform  (FFT) ,  1 6-49 

Examples  of  FFT  applications,  16-50 
Solution  to  specific  second-order  differential  equations,  1 6-53 

The  Cauchy  or  Euler  equation  ,  1 6-53 

Legendre's  equation,  1  6-54 

Bessel's  equation,  16-55 

Chebyshev  or  Tchebycheff  polynomials,  1  6-57 

Laguerre's  equation,  16-58 

Weber's  equation  and  Hermite  polynomials,  1  6-59 
Numerical  and  graphical  solutions  to  ODEs,  1 6-60 

Numerical  solution  of  first-order  ODE,  16-60 

Graphical  solution  of  first-order  ODE,  1  6-62 

Numerical  solution  of  second-order  ODE,  16-64 

Graphical  solution  for  a  second-order  ODE,  16-66 

Numerical  solution  for  stiff  first-order  ODE,  1  6-68 
Numerical  solution  to  ODEs  with  the  SOLVE/DIFF  menu,  1 6  69 

Function  RKF,  1 6-70 

Function  RRK,  16-71 

Function  RKFSTEP,  16-72 

Function  RRKSTEP,  16-73 

Function  RKFST,  16-72 

Function  RKFERR,  16-72 

Function  RSBERR,  16-73 


Chapter  17  -  Probability  Applications,  17-1 
The  MTH/PROBABILITY..  sub-menu  -  part  1,17  1 

Factorials,  combinations,  and  permutations,  17-1 


Page  TOG  15 


Random  numbers,  1  7-2 
Discrete  probability  distributions,  1 7-4 

Binomial  distribution,  17-4 

Poisson  distribution,  17-5 
Continuous  probability  distributions,  1  7-6 

The  gamma  distribution,  17-6 

The  exponential  distribution,  1  7-7 

The  beta  distribution,  17-7 

The  Weibull  distribution,  1  7-7 

Functions  for  continuous  distributions,  1  7-7 
Continuous  distributions  for  statistical  inference,  1 7-9 

Normal  distribution  pdf,  17-9 

Normal  distribution  cdf,  17-10 

The  Student-t  distribution,  1  7-1 0 

The  Chi-square  distribution,  1  7-1  1 

The  F  distribution,  17-12 
Inverse  cumulative  distribution  functions,  17-13 

Chapter  18  -  Statistical  Applications,  18  1 
Pre-programmed  statistical  features,  1  8-1 

Entering  data,  1  8-1 

Calculating  single-variable  statistics,  18-2 

Obtaining  frequency  distributions,  18-5 

Fitting  data  to  a  function  y  =  f(x)  ,  18-10 

Obtaining  additional  summary  statistics,  1  8-1  3 

Calculation  of  percentiles,  18-14 
The  STAT  soft  menu,  18  15 

The  DATA  sub-menu,  1  8-1  5 

The  2PAR  sub-menu,  18-16 

The  1VAR  sub-menu,  18-16 

The  PLOT  sub-menu,  18-17 

The  FIT  sub-menu,  1  8-1  8 

Example  of  STAT  menu  operations,  1  8-1  8 
Confidence  intervals,  1  8-22 

Estimation  of  Confidence  Intervals,  1  8-23 

Definitions,  1  8-23 


Confidence  intervals  for  the  population  mean  when  the 
population  variance  is  known,  1  8-23 

Confidence  intervals  for  the  population  mean  when  the 
population  variance  is  unknown,  1  8-24 

Confidence  interval  for  a  proportion,  1  8-24 

Sampling  distributions  of  differences  and  sums  of  statistics,  1  8-25 

Confidence  intervals  for  sums  and  differences  of  mean  values,  1  8-26 
Determining  confidence  intervals,  1  8-27 

Confidence  intervals  for  the  variance,  1  8-33 
Hypothesis  testing,  1  8-34 

Procedure  for  testing  hypotheses,  1  8-35 

Errors  in  hypothesis  testing,  1  8-35 

Inferences  concerning  one  mean,  18-36 

Inferences  concerning  two  means,  18-39 

Paired  sample  tests,  1  8-40 

Inferences  concerning  one  proportion,  1  8-41 

Testing  the  difference  between  two  proportions,  1  8-41 

Hypothesis  testing  using  pre-programmed  features,  18-43 

Inferences  concerning  one  variance,  1  8-47 

Inferences  concerning  two  variances,  1  8-48 
Additional  notes  on  linear  regression,  1  8-49 

The  method  of  least  squares,  1  8-49 

Additional  equations  for  linear  regression,  1  8-51 

Prediction  error,  18-51 

Confidence  intervals  and  hypothesis  testing  in 

linear  regression,  1  8-52 
Procedure  for  inference  statistics  for  linear  regression 
using  the  calculator,  1  8-53 
Multiple  linear  fitting,  1  8-56 
Polynomial  fitting,  1  8  58 

Selecting  the  best  fitting,  1  8-62 

Chapter  19  -  Numbers  in  Different  Bases,  19  1 
Definitions,  19  1 
The  BASE  menu,  19  1 

Functions  HEC,  DEC,  OCT  and  BIN,  19-2 


Page  TOG  17 


Conversion  between  number  systems,  1 9-3 
Wordsize,  19-4 

Operations  with  binary  integers,  19-4 
The  LOGIC  menu,  1 9  5 
The  BIT  menu,  1 9  6 
The  BYTE  menu,  1 9  6 

Hexadecimal  numbers  for  pixel  references,  1 9-7 

Chapter  20  -  Customizing  menus  and  keyboard,  20-1 
Customizing  menus,  20-1 

The  PRG/MODES/MENU,  20-1 

Menu  numbers  (RCLMENU  and  MENU  functions),  20-2 

Custom  menus  (MENU  and  TMENU  functions),  20-2 

Menu  specification  and  CST  variable,  20-4 
Customizing  the  keyboard,  20-5 

The  PRG/MODES/KEYS  sub-menu,  20-5 

Recall  current  user-defined  key  list,  20-6 

Assign  an  object  to  a  user-defined  key,  20-6 

Operating  user-defined  keys,  20-6 

Un-assigning  a  user-defined  key,  20-7 

Assigning  multiple  user-defined  key,  20-7 

Chapter  21  -  Programming  in  User  RPL  language,  211 

An  example  of  programming,  21-1 

Global  and  local  variables  and  subprograms,  21-2 

Global  Variable  Scope,  21-4 

Local  Variable  Scope,  21-5 
The  PRG  menu,  21-5 

Navigating  through  RPN  sub-menus,  21-6 

Functions  listed  by  sub-menu,  21-6 

Shortcuts  in  the  PRG  menu,  21-9 

Keystroke  sequence  for  commonly  used  commands,  21-10 
Programs  for  generating  lists  of  numbers,  21-13 
Examples  of  sequential  programming,  21-15 

Programs  generated  by  defining  a  function,  21-15 


Page  TOG  18 


Programs  that  simulate  a  sequence  of  stack  operations,  21-17 
Interactive  input  in  programs,  21-19 

Prompt  with  an  input  string,  21-21 

A  function  with  an  input  string,  21-22 

Input  string  for  two  or  three  input  values,  21-24 

Input  through  input  forms,  21-27 

Creating  a  choose  box,  21-31 
Identifying  output  in  programs,  21-33 

Tagging  a  numerical  result,  21-33 

Decomposing  a  tagged  numerical  result  into  number  and  tag,  21-33 

"De-tagging"  a  tagged  quantity,  21-33 

Examples  of  tagged  output,  21-34 

Using  a  message  box,  21-37 
Relational  and  logical  operators,  21-43 

Relational  operators,  21-43 

Logical  operators,  21-44 
Program  branching,  2 1  -46 

Branching  with  IF,  21-46 

The  CASE  construct,  21-51 
Program  loops,  21-53 

The  START  construct,  21-53 

The  FOR  construct,  21-59 

The  DO  construct,  21-61 

The  WHILE  construct,  21-62 
Errors  and  error  trapping,  2 1  -64 

DOERR,  21-64 

ERRN,  21-64 

ERRM,  21-65 

ERRO,  21-65 

LASTARG,  21-65 

Sub-menu  IFERR,  21-65 
User  RPL  programming  in  algebraic  mode,  2 1  -66 

Chapter  22  -  Programs  for  graphics  manipulation,  22-1 
The  PLOT  menu,  22-1 

User-defined  key  for  the  PLOT  menu,  22-1 


Page  TOG  19 


Description  of  the  PLOT  menu,  22-2 
Generating  plots  with  programs,  22-14 
Two-dimensional  graphics,  22-14 
Three-dimensional  graphics,  22-15 
The  variable  EQ,  22-15 

Examples  of  interactive  plots  using  the  PLOT  menu,  22-15 
Examples  of  program-generated  plots,  22-17 
Drawing  commands  for  use  in  programming,  22-19 
PICT,  22-20 
PDIM,  22-20 
LINE,  22-20 
TUNE,  22-20 
BOX,  22-21 
ARC,  22-21 

PIX?,  PIXON,  and  PIXOFF,  22-22 
PVIEW,  22-22 
PX^C,  22-22 
C^PX,  22-22 

Programming  examples  using  drawing  functions,  22-22 

Pixel  coordinates,  22-25 
Animating  graphics,  22-26 

Animating  a  collection  of  graphics,  22-27 

More  information  on  the  ANIMATE  function,  22-29 
Graphic  objects  (GROBs),  22  30 

The  GROB  menu,  22-31 
A  program  with  plotting  and  drawing  functions,  22-33 

Modular  programming,  22-36 

Running  the  program,  22-36 

A  program  to  calculate  principal  stresses,  22-38 

Ordering  the  variables  in  the  sub-directory,  22-39 

A  second  example  of  Mohr's  circle  calculations,  22-39 
An  input  form  for  the  Mohr's  circle  program,  22-40 

Chapter  23  -  Character  strings,  23-1 

String-related  functions  in  the  TYPE  sub-menu,  23-1 
String  concatenation,  23-2 


Page  TOC-20 


The  CHARS  menu,  23  2 
The  characters  list,  23-3 

Chapter  24  -  Calculator  objects  and  flags,  24-1 
Description  of  calculator  objects,  24-1 
Function  TYPE,  24  2 
Function  VTYPE,  24  2 
Calculator  flags,  24-3 
System  flags,  24-3 

Functions  for  setting  and  changing  flags,  24-3 
User  flags,  24-4 

Chapter  25  -  Date  and  Time  Functions,  25-1 
The  TIME  menu,  25-1 

Setting  an  alarm,  25-1 

Browsing  alarms,  25-2 

Setting  time  and  date,  25-2 

TIME  Tools,  25-2 
Calculations  with  dates,  25-3 
Calculations  with  times,  25-4 
Alarm  functions,  25-4 

Chapter  26  -  Managing  memory,  26-1 
Memory  Structure,  26-1 

The  HOME  directory,  26-2 

Port  memory,  26-2 
Checking  objects  in  memory,  26-2 
Backup  objects,  26-3 
Backing  up  objects  in  port  memory,  26-3 

Backing  up  and  restoring  HOME,  26-4 

Storing,  deleting,  and  restoring  backup  objects,  26-5 
Using  data  in  backup  objects,  26-5 
Using  libraries,  26-6 

Installing  and  attaching  a  library,  26-6 

Library  numbers,  26-7 

Deleting  a  library,  26-7 


Creating  libraries,  26-7 
Backup  battery,  26-7 

Appendices 

Appendix  A  -  Using  input  forms,  A -1 

Appendix  B  -  The  calculator's  keyboard,  B -1 

Appendix  C  -  CAS  settings,  C  I 

Appendix  D  -  Additional  character  set,  D -1 

Appendix  E  -  The  Selection  Tree  in  the  Equation  Writer, 

Appendix  F  -  The  Applications  (APPS)  menu,  F -1 

Appendix  G  -  Useful  shortcuts,  G-l 

Appendix  H  -  The  CAS  help  facility,  H  I 

Appendix  I  -  Command  catalog  list,  1-1 

Appendix  J  -  The  MATHS  menu,  J 1 

Appendix  K  -  The  MAIN  menu,  K -1 

Appendix  L  -  Line  editor  commands,  L  I 

Appendix  M  -  Index,  M  l 

Limited  Warranty  -  W-l 

Service,  W-2 

Regulatory  information,  W-4 


A  note  about  screenshots  in  this  guide 

A  screenshot  is  a  representation  of  the  calculator  screen.  For  example,  the 
first  time  the  calculator  is  turned  on  you  get  the  following  screen  (calculator 
screens  are  shown  with  a  thick  border  in  this  section): 


:;HD  KYI  HEM  fi=  1 K '  hLG 
CH0HE1  


EDIT  |  VIEH  |  RCL  |  STOfr  |PURGE|CLEflR 


The  top  two  lines  represent  the  screen  header  and  the  remaining  area  in  the 
screen  is  used  for  calculator  output. 

Most  screenshots  in  this  guide  were  generated  using  a  computer-based 
emulator  (a  program  that  simulates  the  operation  of  the  calculator  in  a 
computer),  and  are  missing  the  screen  header  lines.  Instead,  they  will  show 
additional  screen  output  area  in  the  location  of  the  header  lines,  as  shown 
below: 


This  additional  screen  output  area  in  many  screen  shots  in  this  guide  will  not 
show  when  you  try  those  guide's  example  in  your  calculator.  Thus,  while  in 
the  guide  you  may  see  a  screenshot  as  the  following: 


:SIN(2.5) 

t  =_59S472 144104 

:-J5.5+LN(2.5) 

2.53303982043 
:  2.  3+5-1-2.  31 

13. S 


EDIT  |  VIEH  |  RCL  |  STOT  |PURGE|CLEflR 


the  calculator  will  actually  show  the  following  screen: 


Page  Note-1 


fiAD  KVI  HEM  f;= 
CH0HE1  


ALG 


■J5.5+LNC2.5) 

2.53303982043 
2.3+5-1-2.31 

13. S 


EDIT     I E H    F;lL    STO*  PURGE  CLEAR 


Notice  that  the  header  lines  cover  the  top  first  and  a  half  lines  of  output  in  the 
calculator's  screen.  Nevertheless,  the  lines  of  output  not  visible  are  still 
available  for  you  to  use.  You  can  access  those  lines  in  your  calculator  by 
pressing  the  up-arrow  key  (^b),  which  will  allow  you  to  scroll  down  the 
screen  contents. 

Also,  as  you  perform  the  three  operations  listed  in  the  screenshot,  in  the  order 
shown,  your  screen  will  show  them  occupying  higher  levels  in  the  display  as 
shown  next: 


:;HD  KVI  HEM  f;= 

' K '              ALG  j 

RAD  KVI  HEK  fi=  ' K '               HLG  < 

[HOME]- 

[HOME]-  ! 

:SIN(2.5) 

!  .593472144104 

:SIN(2.5) 

:J5.5+LHC2.5) 

598472144104 

2.53303932043 

fa  IT 


STO*  PURGE  CLEAR 


STO*  PURGE  CLEAR 


The  keystrokes  required  to  complete  these  exercises  are  the  following: 
The  next  operation, 

will  force  the  lines  corresponding  to  the  operation  SIN(2.5)  to  move  upwards 
and  be  hidden  by  the  header  lines. 

Many  screenshots  in  this  guide  have  also  been  modified  to  show  only  the 
operation  of  interest.  For  example,  the  screenshot  for  the  operation  SIN(2.5), 
shown  above,  may  be  simplified  in  this  guide  to  look  as: 


Page  Note-2 


:SIN(2.5) 

■598472144104 

These  simplifications  of  the  screenshots  are  aimed  at  economizing  output 
space  in  the  guide. 

Be  aware  of  the  differences  between  the  guide's  screenshots  and  the  actual 
screen  display,  and  you  should  have  no  problem  reproducing  the  exercises  in 
this  guide. 


Page  Note-3 


Chapter  1 
Getting  started 


This  chapter  is  aimed  at  providing  basic  information  in  the  operation  of  your 
calculator.  The  exercises  are  aimed  at  familiarizing  yourself  with  the  basic 
operations  and  settings  before  actually  performing  a  calculation. 

Basic  Operations 

The  following  exercises  are  aimed  at  getting  you  acquainted  with  the 
hardware  of  your  calculator. 

Batteries 

The  calculator  uses  3  AAA  (LR03)  batteries  as  main  power  and  a  CR2032 
lithium  battery  for  memory  backup. 

Before  using  the  calculator,  please  install  the  batteries  according  to  the 
following  procedure. 

To  install  the  main  batteries 

a.  Make  sure  the  calculator  is  off.  Slide  up  the  battery  compartment 
cover  as  illustrated. 


b.  Insert  3  new  AAA  (LR03)  batteries  into  the  main  compartment.  Make  sure 
each  battery  is  inserted  in  the  indicated  direction. 

To  install  the  backup  battery 

a.  Make  sure  the  calculator  is  off.  Press  down  the  holder.  Push  the 
plate  to  the  shown  direction  and  lift  it. 


Page  1-1 


Holder 


b.  Insert  a  new  CR2032  lithium  battery.  Make  sure  its  positive  (+)  side  is 
facing  up. 

c.  Replace  the  plate  and  push  it  to  the  original  place. 

After  installing  the  batteries,  press  [ON]  to  turn  the  power  on. 
Warning:  When  the  low  battery  icon  is  displayed,  you  need  to  replace  the 
batteries  as  soon  as  possible.  However,  avoid  removing  the  backup  battery 
and  main  batteries  at  the  same  time  to  avoid  data  lost. 

Turning  the  calculator  on  and  off 

The  jQN J  key  is  located  at  the  lower  left  corner  of  the  keyboard.  Press  it  once 
to  turn  your  calculator  on.  To  turn  the  calculator  off,  press  the  red  right-shift 
key  IsD  (first  key  in  the  second  row  from  the  bottom  of  the  keyboard), 
followed  by  the  Q^D  key.  Notice  that  the  Q^D  key  has  a  red  OFF  label 
printed  in  the  upper  right  corner  as  a  reminder  of  the  OFF  command. 

Adjusting  the  display  contrast 

You  can  adjust  the  display  contrast  by  holding  the  Q^D  key  while  pressing  the 
L±J  or  CED  keys.  The  L^J  (hold)  L±J  key  combination  produces  a  darker 
display.  The  L^J  (hold)  CED  key  combination  produces  a  lighter  display 

Contents  of  the  calculator's  display 

Turn  your  calculator  on  once  more.  The  display  should  look  as  indicated 
below. 


Page  1-2 


At  the  top  of  the  display  you  will  have  two  lines  of  information  that  describe 
the  settings  of  the  calculator.   The  first  line  shows  the  characters: 

RFiD  XYZ  HEX  R=  5  X  5 

For  details  on  the  meaning  of  these  specifications  see  Chapter  2. 
The  second  line  shows  the  characters:  {  }  indicating  that  the  HOME 

directory  is  the  current  file  directory  in  the  calculator's  memory.  In  Chapter  2 
you  will  learn  that  you  can  save  data  in  your  calculator  by  storing  them  in  files 
or  variables.  Variables  can  be  organized  into  directories  and  sub-directories. 
Eventually,  you  may  create  a  branching  tree  of  file  directories,  similar  to  those 
in  a  computer  hard  drive.  You  can  then  navigate  through  the  file  directory 
tree  to  select  any  directory  of  interest.  As  you  navigate  through  the  file 
directory  the  second  line  of  the  display  will  change  to  reflect  the  proper  file 
directory  and  sub-directory. 

At  the  bottom  of  the  display  you  will  find  a  number  of  labels,  namely, 


associated  with  the  six  soft  menu  keys,  Fl  through  F6: 

The  six  labels  displayed  in  the  lower  part  of  the  screen  will  change  depending 
on  which  menu  is  displayed.  But  (jl)  will  always  be  associated  with  the  first 
displayed  label,  (JL)  with  the  second  displayed  label,  and  so  on. 

Menus 

The  six  labels  associated  with  the  keys  (JL)  through  (jT)  form  part  of  a  menu 
of  functions.  Since  the  calculator  has  only  six  soft  menu  keys,  it  only  display  6 
labels  at  any  point  in  time.  However,  a  menu  can  have  more  than  six  entries. 
Each  group  of  6  entries  is  called  a  Menu  page.  The  current  menu,  known  as 
the  TOOL  menu  (see  below),  has  eight  entries  arranged  in  two  pages.  The 
next  page,  containing  the  next  two  entries  of  the  menu  is  available  by 


Page  1-3 


pressing  the  jNxrj  (NeXT  menu)  key.  This  key  is  the  third  key  from  the  left  in 
the  third  row  of  keys  in  the  keyboard.  Press  jNxrj  once  more  to  return  to  the 
main  TOOL  menu,  or  press  the  [tool)  key  (third  key  in  second  row  of  keys  from 
the  top  of  the  keyboard). 

The  TOOL  menu  is  described  in  detain  in  the  next  section.  At  this  point  we 
will  illustrate  some  properties  of  menus  that  you  will  find  useful  while  using 
your  calculator. 

SOFT  menus  vs.  CHOOSE  boxes 

Menus,  or  SOFT  menus,  associate  labels  in  the  lower  part  of  the  screen  with 
the  six  soft  menu  keys  (C^O  through  C2D  )•  By  pressing  the  appropriate  soft 
menu  key,  the  function  shown  in  the  associated  label  gets  activated.  For 
example,  with  the  TOOL  menu  active,  pressing  the  MSMB  key  (C™~) )  activates 
function  CLEAR,  which  erases  (clears  up)  the  contents  of  the  screen.  To  see 
this  function  in  action,  type  a  number,  say  LU L2J LiJ iitHS) ,  and  then  press 
the  (jD  key. 

SOFT  menus  are  typically  used  to  select  from  among  a  number  of  related 
functions.  However,  SOFT  menus  are  not  the  only  way  to  access  collections 
of  related  functions  in  the  calculator.  The  alternative  way  will  be  referred  to 
as  CHOOSE  boxes.  To  see  an  example  of  a  choose  box,  activate  the  TOOL 
menu  (press  [tool J ),  and  then  press  the  keystroke  combination 
(j^j_msf  (associated  with  the  LU  key).  This  will  provide  the  following 
CHOOSE  box: 


EASE  HEnU 

™Ec"™  "^j 

3. OCT  *  1! 

H.&m  * 

This  CHOOSE  box  is  labeled  BASE  MENU  and  provides  a  list  of  numbered 
functions,  from  1 .  HEX  x  to  6.  B->R.  This  display  will  constitute  the  first  page 
of  this  CHOOSE  box  menu  showing  six  menu  functions.  You  can  navigate 
through  the  menu  by  using  the  up  and  down  arrow  keys,  located  in 

the  upper  right  side  of  the  keyboard,  right  under  the  C2D  and  CZD  soft  menu 
keys.  To  activate  any  given  function,  first,  highlight  the  function  name  by 


Page  1  -4 


using  the  up  and  down  arrow  keys,  (/aS)<^?  ,  or  by  pressing  the  number 
corresponding  to  the  function  in  the  CHOOSE  box.  After  the  function  name  is 
selected,  press  the         soft  menu  key  (QD  )•  Thus,  if  you  wanted  to  use 
function  R->B  (Real  to  Binary),  you  could  press  • 

If  you  want  to  move  to  the  top  of  the  current  menu  page  in  a  CHOOSE  box, 
use     J</^N> .  To  move  to  the  bottom  of  the  current  page,  use  LjnJ<\3?  •  To 
move  to  the  top  of  the  entire  menu,  use  Lr> .  To  move  to  the  bottom  of 
the  entire  menu,  use  IrU*^?  . 

Selecting  SOFT  menus  or  CHOOSE  boxes 

You  can  select  the  format  in  which  your  menus  will  be  displayed  by  changing 
a  setting  in  the  calculator  system  flags  (A  system  flag  is  a  calculator  variable 
that  controls  a  certain  calculator  operation  or  mode.  For  more  information 
about  flags,  see  Chapter  24).  System  flag  1  17  can  be  set  to  produce  either 
SOFT  menus  or  CHOOSE  boxes.  To  access  this  flag  use: 

Your  calculator  will  show  the  following  screen,  highlighting  the  line  starting 
with  the  number  1  1  7: 

^^»SVSTEH  FLAGS  fflfflfflim 

111  SiHp  non  rat.  t 

112  i  fiHplifiGd 

113  Linear  siHp  on 
11H  Dif p  1+x  ■+  x+1 
115  SGRT  fiHpUficd 

By  default,  the  line  will  look  as  shown  above.  The  highlighted  line  (117 
CHOOSE  boxes)  indicates  that  CHOOSE  boxes  are  the  current  menu  display 
setting.  If  you  prefer  to  use  SOFT  menu  keys,  press  the  IKSQli soft  menu  key 
(QD  ),  followed  by  llillll  ((JD  )•  Press  IB3BIII  ((jD  )  once  more  to  return  to 
normal  calculator  display. 

If  you  now  press  L  r>  J  base  ,  instead  of  the  CHOOSE  box  that  you  saw  earlier, 
the  display  will  now  show  six  soft  menu  labels  as  the  first  page  of  the  STACK 
menu: 


Page  1-5 


To  navigate  through  the  functions  of  this  menu,  press  the  jNxrj  key  to  move  to 
the  next  page,  or  Jp^v  (associated  with  the  jNxrj  key)  to  move  to  the 
previous  page.  The  following  figures  show  the  different  pages  of  the  BASE 
menu  accessed  by  pressing  the  [nxtJ  key  twice: 

Pressing  the  jNxrj  key  once  more  will  takes  us  back  to  the  first  menu  page. 


Note:  With  the  SOFT  menu  setting  for  system  flag  1  1 7,  the  keystroke 
combination  (j^J(hold)  <<z? ,  will  show  a  list  of  the  functions  in  the  current  soft 
menu.  For  example,  for  the  two  first  pages  in  the  BASE  menu,  you  will  get: 


REX" 

!>EC 

OCT 

BIN 

R+B 

B+R 


LOGIC 

3IT 

BYTE 

STUB 
RCNS 


To  revert  to  the  CHOOSE  boxes  setting,  use: 

(Ara)  flfcPfl  ^C^O  ^^IKEOIl  Hill  Ellllll!. 

Notes: 

1.  The  TOOL  menu,  obtained  by  pressing  [tool]  ,  will  always  produce  a 
SOFT  menu. 

2.  Most  of  the  examples  in  this  user's  manual  are  shown  using  both  SOFT 
menus  and  CHOOSE  boxes.  Programming  applications  (Chapters  21  and 
22)  use  exclusively  SOFT  menus. 

3.  Additional  information  on  SOFT  menus  vs.  CHOOSE  boxes  is  presented 
in  Chapter  2  o  f  this  guide. 


The  TOOL  menu 

The  soft  menu  keys  for  the  menu  currently  displayed,  known  as  the  TOOL 
menu,  are  associated  with  operations  related  to  manipulation  of  variables  (see 
pages  for  more  information  on  variables): 

CjD      EDIT  the  contents  of  a  variable  (see  Chapter  2  and  Appendix 
L  for  more  information  on  editing) 


Page  1-6 


VIEW  the  contents  of  a  variable 

1:1: 

iiii  j  jlS  JS  iiii 

dD 

ReCaLl  the  contents  of  a  variable 

I1IS33S 

STOre  the  contents  of  a  variable 

!  .11!.!.!.!:!=!:!.::! 

C«D 

PURGE  a  variable 

!":"i:F"s!":"!: 

CLEAR  the  display  or  stack 

The  calculator  has  only  six  soft  menu  keys,  and  can  only  display  6  labels  at 
any  point  in  time.  However,  a  menu  can  have  more  than  six  entries.  Each 
group  of  6  entries  is  called  a  Menu  page.  The  TOOL  menu  has  eight  entries 
arranged  in  two  pages.  The  next  page,  containing  the  next  two  entries  of  the 
menu  are  available  by  pressing  the  jNxrj  (NeXT  menu)  key.  This  key  is  the 
third  key  from  the  left  in  the  third  row  of  keys  in  the  keyboard. 
In  this  case,  only  the  first  two  soft  menu  keys  have  commands  associated  with 
them.  These  commands  are: 

BxBSlI    GD      CASCMD:  CAS  CoMmanD,  used  to  launch  a  command  from 
the  CAS  by  selecting  from  a  list 
C~^Q      HELP  facility  describing  the  commands  available 
Pressing  the  [nxt)    key  will  show  the  original  TOOL  menu.    Another  way  to 
recover  the  TOOL  menu  is  to  press  the  [tool)  key  (third  key  from  the  left  in  the 
second  row  of  keys  from  the  top  of  the  keyboard). 

Setting  time  and  date 

The  calculator  has  an  internal  real  time  clock.  This  clock  can  be  continuously 
displayed  on  the  screen  and  be  used  for  alarms  as  well  as  running  scheduled 
tasks.  This  section  will  show  not  only  how  to  set  time  and  date,  but  also  the 
basics  of  using  CHOOSE  boxes  and  entering  data  in  a  dialog  box.  Dialog 
boxes  on  your  calculator  are  similar  to  a  computer  dialog  box. 

To  set  time  and  date  we  use  the  TIME  choose  box  available  as  an  alternative 
function  for  the  l_9j  key.  By  combining  the  red  right-shift  button,  (j^J,  with 
the  L  9  j  key  the  TIME  choose  box  is  activated.  This  operation  can  also  be 
represented  as  L  r>  j  time  .  The  TIME  choose  box  is  shown  in  the  figure  below: 


Page  1-7 


a. s«t  ourH.. 

3. Set  tiH«^  date.. 
H. Tools.. 


As  indicated  above,  the  TIME  menu  provides  four  different  options,  numbered 
1  through  4.  Of  interest  to  us  as  this  point  is  option  3.  Set  time,  dote...  Using 
the  down  arrow  key,  <\j?,  highlight  this  option  and  press  the  IIE3II  C™~)  soft 
menu  key.  The  following  input  form  (see  Appendix  1-A)  for  adjusting  time 
and  date  is  shown: 


mm 

sis  SET  TIHE  AM  DATE  ssssssssssssss 

TiH-2  : 

E:01:34  PM 

Date : 

3 '30 '03  IVD^Y 

Enter 

hour 

Setting  the  time  of  the  day 

Using  the  number  keys,  CD  CD  CD  CD  CD  CD  CD  CjDCDCD,  start  by 
adjusting  the  hour  of  the  day.  Suppose  that  we  change  the  hour  to  11,  by 
pressing  CDCD  as  the  hour  field  in  the  SET  TIME  AND  DATE  input  form  is 
highlighted.  This  results  in  the  number  7 7  being  entered  in  the  lower  line  of 
the  input  form: 


mmm 

ijiSET  TIHE  AM  DATE  ssssssssssssss 

TiH-2  : 

B:01=34  PM 

Date : 

3 '30 '03  IVD^Y 

11 

Press  the  1111331 C™~)  soft  menu  key  to  effect  the  change.  The  value  of  1  7  is 
now  shown  in  the  hour  field,  and  the  minute  field  is  automatically  highlighted: 


TiH-2  : 

Date  = 


SET  TIHE  HTl D  DATE 

11  :EH=34  PM 

3 '30 '03  IVD^Y 


Enter  Hinutc 


Page  1-8 


Let's  change  the  minute  field  to  25,  by  pressing:  L2J UU  IIE3I .  The 
seconds  field  is  now  highlighted.  Suppose  that  you  want  to  change  the 
seconds  field  to  45,  use:      GDCD  MM 


The  time  format  field  is  now  highlighted.    To  change  this  field  from  its  current 
setting  you  can  either  press  the  L +/- J    key  (the  second  key  from  the  left  in  the 
fifth  row  of  keys  from  the  bottom  of  the  keyboard),  or  press  the  Siilliiliii  soft 
menu  key  (  (jD  )• 

•     If  using  the  L +/- J    key,  the  setting  in  the  time  format  field  will  change  to 
either  of  the  following  options: 

o    AM  :  indicates  that  displayed  time  is  AM  time 
o    PM  :  indicates  that  displayed  time  is  PM  time 
o    24-hr  :  indicates  that  that  the  time  displayed  uses  a  24  hour 
format  where  1  8:00,  for  example,  represents  6pm 

The  last  selected  option  will  become  the  set  option  for  the  time  format  by 
using  this  procedure. 


If  using  the  [iilliiilill!  soft  menu  key,  the  following  options  are  available 


SET  TIHE  flnD  DATE§ 


AH 


Use  the  up  and  down  arrow  keys,^>  ^3? ,  to  select  among  these  three 
options  (AM,  PM,  24-hour  time).  Press  the  111111111111:11111  C^l  soft  menu  key  to 
make  the  selection. 


Setting  the  date 

After  setting  the  time  format  option,  the  SET  TIME  AND  DATE  input  form  will 
look  as  follows: 


Page  1-9 


TiHG 
Date 


SET  TIHE  flnD  DATE 
11  =25  =45  B 


3 '30 '03  IVD-^Y 

Choojg  flHj  FHj  or  ZH-hour  tim 


To  set  the  date,  first  set  the  date  format.  The  default  format  is  M/D/Y 
(month/day/year).  To  modify  this  format,  press  the  down  arrow  key.  This 
will  highlight  the  date  format  as  shown  below: 


TiHfl : 
Date: 


SET  TIHE  HTl D  DATE 

11  =£5  =45  RM 
3 '30 '03  Dfflj 


Se  the  iiJsjj:!:!:!:!::;! 


Choose  date  ■JiJpi.oM  forHJt 


soft  menu  key  (  B),  to  see  the  options  for  the  date  format: 

""lliPSET  TIHE  Alb  pHfET^  J 


Choose  dat«  display  format 

Highlight  your  choice  by  using  the  up  and  down  arrow  keys,^^  ^3? ,  and 
press  the  ItEiii CZD  soft  menu  key  to  make  the  selection. 


Introducing  the  calculator's  keyboard 

The  figure  below  shows  a  diagram  of  the  calculator's  keyboard  with  the 
numbering  of  its  rows  and  columns. 


Page  1-10 


Column:  1 
▼ 


Row 

1  ► 

2  ► 

3  ► 

4  ► 

5  ► 

6  ► 

7> 

8  ► 

9  ► 

10  ► 


2 

T 


3 

T 


4 
T 


5 
▼ 


6 

T 


7_j  LAJ  L?J  L*_ 

CALC  ALG  MATRICES  STAT  CONVERT  UNITS  ( )  — 


ARITH_CMPLX  DEF     LIB     #    BASE  {}«» 

1 


9  ANS-NUM 


ON  |    I  0 

CANCEL 


Column: 


▲ 

3 


A 

5 


The  figure  shows  1 0  rows  of  keys  combined  with  3,  5,  or  6  columns.    Row  1 
has  6  keys,  rows  2  and  3  have  3  keys  each,  and  rows  4  through  1  0  have  5 
keys  each.  There  are  4  arrow  keys  located  on  the  right-hand  side  of  the 
keyboard  in  the  space  occupied  by  rows  2  and  3. 

Each  key  has  three,  four,  or  five  functions.  The  main  key  function  correspond 
to  the  most  prominent  label  in  the  key.    Also,  the  green  left-shift  key,  key  (8, 7 J, 
the  red  right-shift  key,  key  (9, 7j,  and  the  blue  ALPHA  key,  key  [7, 7j,  can  be 


Page  1-11 


combined  with  some  of  the  other  keys  to  activate  the  alternative  functions 
shown  in  the  keyboard.    For  example,  the  [symb J  key,  key(4,4),  has  the 
following  six  functions  associated  with  it: 


Main  function,  to  activate  the  SYMBolic  menu 
Left-shift  function,  to  activate  the  MTH  (Math)  menu 
Right-shift  function,  to  activate  the  CATalog  function 
ALPHA  function,  to  enter  the  upper-case  letter  P 
ALPHA-Left-Shift  function,  to  enter  the  lower-case  letter  p 
ALPHA-Right-Shift  function,  to  enter  the  symbol  P 
Of  the  six  functions  associated  with  the  key  only  the  first  four  are  shown  in  the 
keyboard  itself.  This  is  the  way  that  the  key  looks  in  the  keyboard: 


[alpha]  (p) 


MTH 


CAT 


Notice  that  the  color  and  the  position  of  the  labels  in  the  key,  namely,  SYMB, 
MTH,  CA T  and  P,  indicate  which  is  the  main  function  (SYMB),  and  which  of 
the  other  three  functions  is  associated  with  the  left-shift  {JnJ(MTH),  right- 
shift  CgJ  (CAT)  ,  and  [alpha]  (P)  keys. 

For  detailed  information  on  the  calculator  keyboard  operation  referee  to 
Appendix  B  . 

Selecting  calculator  modes 

This  section  assumes  that  you  are  now  at  least  partially  familiar  with  the  use  of 
choose  and  dialog  boxes  (if  you  are  not,  please  refer  to  Chapter  2). 

Press  the  [mode)  button  (second  key  from  the  left  on  the  second  row  of  keys  from 
the  top)  to  show  the  following  CALCULATOR  MODES  input  form: 


Page  1-12 


§§§§§§§§§§§§§§§  CALCULATOR  HOPES  jjjjjjjjjjjjjjjj 
Operating  Ho»Jg..riIEFngEB 
nuHbcr  ForHat....Std  _FMj 
fln^U  Hcafurc... Radian* 

Coord  Syst«H  Rectangular 

^B**P    _K*y  CUch    ^Last  StacR 

Ihooig  coU uJ.ator  opgr atinj  HQjg 

Press  the  113.1:11 (T^H  soft  menu  key  to  return  to  normal  display.  Examples  of 
selecting  different  calculator  modes  are  shown  next. 

Operating  Mode 

The  calculator  offers  two  operating  modes:  the  Algebraic  mode,  and  the 
Reverse  Polish  Notation  (RPN)  mode.  The  default  mode  is  the  Algebraic 
mode  (as  indicated  in  the  figure  above),  however,  users  of  earlier  HP 
calculators  may  be  more  familiar  with  the  RPN  mode. 

To  select  an  operating  mode,  first  open  the  CALCULATOR  MODES  input  form 
by  pressing  the  LM0P£J  button.  The  Operating  Mode  field  will  be  highlighted. 
Select  the  Algebraic  or  RPN  operating  mode  by  either  using  the  L+/- J  key 
(second  from  left  in  the  fifth  row  from  the  keyboard  bottom),  or  pressing  the 
SID3III  soft  menu  key  (  ).  If  using  the  latter  approach,  use  up  and  down 
arrow  keys,  ^3?,  to  select  the  mode,  and  press  the  KQ1I  soft  menu  key 
to  complete  the  operation. 

To  illustrate  the  difference  between  these  two  operating  modes  we  will 
calculate  the  following  expression  in  both  modes: 


3. 

233 

To  enter  this  expression  in  the  calculator  we  will  first  use  the  equation  writer, 
I  r>  j  eqw  .  Please  identify  the  following  keys  in  the  keyboard,  besides  the 
numeric  keypad  keys: 


Page  1-13 


The  equation  writer  is  a  display  mode  in  which  you  can  build  mathematical 
expressions  using  explicit  mathematical  notation  including  fractions, 
derivatives,  integrals,  roots,  etc.  To  use  the  equation  writer  for  writing  the 
expression  shown  above,  use  the  following  keystrokes: 

^^^^^^^ 

After  pressing  t^jthe  calculator  displays  the  expression: 

V  (3*  (5-1/ (3*3)  ) /23A3+EXP  (2.5)  ) 

Pressing  [enter] again  will  provide  the  following  value.  Accept  Approx.  mode 
on,  if  asked,  by  pressing  II-IIIS.  [Note:  The  integer  values  used  above,  e.g., 
3,  5,  1,  represent  exact  values.  The  EXP(2.5),  however,  cannot  be  expressed 


as  an  exact  value,  therefore,  a  switch  to  Approx  mode  is  required] 


You  could  also  type  the  expression  directly  into  the  display  without  using  the 
equation  writer,  as  follows: 

to  obtain  the  same  result. 


Change  the  operating  mode  to  RPN  by  first  pressing  the  (mope)  button.  Select 
the  RPN  operating  mode  by  either  using  the  L^Jkey,  or  pressing  the  Mlmm 
soft  menu  key.    Press  the  1111111:111  C™~)    soft  menu  key  to  complete  the  operation. 
The  display,  for  the  RPN  mode  looks  as  follows:  


Page  1-14 


Notice  that  the  display  shows  several  levels  of  output  labeled,  from  bottom  to 
top,  as  1,  2,  3,  etc.  This  is  referred  to  as  the  stack  of  the  calculator.  The 
different  levels  are  referred  to  as  the  stack  levels,  i.e.,  stack  level  1,  stack  level 
2,  etc. 

Basically,  what  RPN  means  is  that,  instead  of  writing  an  operation  such  as  3 
+  2,  in  the  calculator  by  using  CZDC±DC2D^^),  we  write  first  the  operands, 
in  the  proper  order,  and  then  the  operator,  i.e.,  LU l&E) LU l&E) L±J  •  As 
you  enter  the  operands,  they  occupy  different  stack  levels.  Entering 
[  3  )[enter)  puts  the  number  3  in  stack  level  1.  Next,  entering  L  2  )[enter)  pushes 
the  3  upwards  to  occupy  stack  level  2.  Finally,  by  pressing  L  +  J,  we  are 
telling  the  calculator  to  apply  the  operator,  or  program,  L±J  to  the  objects 
occupying  levels  1  and  2.  The  result,  5,  is  then  placed  in  level  1.  A  simpler 
way  to  calculate  this  operation  is  by  using:  LUt^L)LUL±J  • 

Let's  try  some  other  simple  operations  before  trying  the  more  complicated 
expression  used  earlier  for  the  algebraic  operating  mode: 


Notice  the  position  of  the  y  and  the  x  in  the  last  two  operations.  The  base  in 
the  exponential  operation  is  y  (stack  level  2)  while  the  exponent  is  x  (stack 
level  1 )  before  the  key  CZD  is  pressed.  Similarly,  in  the  cubic  root  operation, 
y  (stack  level  2)  is  the  quantity  under  the  root  sign,  and  x  (stack  level  1)  is  the 
root. 

Try  the  following  exercise  involving  3  factors:  (5  +  3)  x  2 


123/32 
42 

3V27 


CU  CX3  CXI  ^  CU  CTD  LED 
LO^LUCzD 


[TJ(ENTER)[JJ[TJ 

LUCZD 


Calculates  (5  +3)  first. 
Completes  the  calculation. 


Let's  try  now  the  expression  proposed  earlier: 


Page  1-15 


L^CZ^t^D  Enter  3  in  level  1 

L  5  j L  i  J LfA/7"fflJ  Enter  5  in  level  1,  3  moves  to  y 

C3DCZD^^)  Enter  3  in  level  1,  5  moves  to  level  2,  3  to  level  3 

LiJLiJL><J  Place  3  and  multiply,  9  appears  in  level  1 

L  'fx  j  1/(3x3),  last  value  in  lev.  1;  5  in  level  2;  3  in  level  3 

CED  5  -  1/(3x3)  ,  occupies  level  1  now;  3  in  level  2 

L><J  3x  (5  -  1/(3x3)),  occupies  level  1  now. 

CX3CUCZD(^™D  Enter  23  in  level  1,  14.66666  moves  to  level  2. 

CBCZDCZD  Enter  3,  calculate  233  into  level  1 .  14.666  in  lev.  2. 

C±D  (3x  (5-l/(3x3)))/233  into  level  1 

CX3CZDCD  Enter  2.5  level  1 

L^lJ^   e25,  goes  into  level  1,  level  2  shows  previous  value. 

GD  (3x  (5  -  1  /(3x3)))/233  +  e2  5  =12.1  8369,  into  lev.  1 . 

O  V((3x  (5  -  l/(3x3)))/233  +  e25)  =  3.4905156,  into  1. 


Although  RPN  requires  a  little  bit  more  thought  than  the  algebraic  (ALG)  mode, 
there  are  multiple  advantages  in  using  RPN.  For  example,  in  RPN  mode  you 
can  see  the  equation  unfolding  step  by  step.  This  is  extremely  useful  to  detect 
a  possible  input  error.  Also,  as  you  become  more  efficient  in  this  mode  and 
learn  more  of  the  tricks,  you  will  be  able  to  calculate  expression  faster  and 
will  much  less  keystrokes.  Consider,  for  example  the  calculation  of  (4x6  - 
5)/(l  +4x6  -  5).  In  RPN  mode  you  can  write: 

obviously,  even  In  RPN  mode,  you  can  enter  an  expression  in  the  same  order 
as  the  algebraic  mode  by  using  the  Equation  writer.  For  example, 

  ^^^^^^^   

The  resulting  expression  is  shown  in  stack  level  1  as  follows: 


l: 

EDIT  |  VIEH  |  RCL  |  STOfr  |FUF;ijE|lLEHF; 

Notice  how  the  expression  is  placed  in  stack  level  1  after  pressing  [enter]  , 
Pressing  the  EVAL  key  at  this  point  will  evaluate  the  numerical  value  of  that 
expression  Note:  In  RPN  mode,  pressing  ENTER  when  there  is  no  command 


Page  1-16 


line  will  execute  the  DUP  function  which  copies  the  contents  of  stack  level  1  of 
the  stack  onto  level  2  (and  pushes  all  the  other  stack  levels  one  level  up).  This 
is  extremely  useful  as  showed  in  the  previous  example. 

To  select  between  the  ALG  vs.  RPN  operating  mode,  you  can  also  set/clear 
system  flag  95  through  the  following  keystroke  sequence: 

Qwx)  mm  ^  GT)  ^  GD  ^  GD  ^  IKIIilil  mm  mm 

Alternatively,  you  can  use  one  of  the  following  shortcuts: 

•  In  ALG  mode, 

CF(-95)  selects  RPN  mode 

•  In  RPN  mode, 

95  C+T)(enter)  SF  selects  ALG  mode 

For  more  information  on  calculator's  system  flags  see  Chapter  2. 

Number  Format  and  decimal  dot  or  comma 

Changing  the  number  format  allows  you  to  customize  the  way  real  numbers 
are  displayed  by  the  calculator.  You  will  find  this  feature  extremely  useful  in 
operations  with  powers  of  tens  or  to  limit  the  number  of  decimals  in  a  result. 

To  select  a  number  format,  first  open  the  CALCULATOR  MODES  input  form  by 
pressing  the  [mode)  button.  Then,  use  the  down  arrow  key,  ^3?,  to  select  the 
option  Number  format.  The  default  value  is  Std,  or  Sfanc/ard  format.  In  the 
standard  format,  the  calculator  will  show  floating-point  numbers  with  the 
maximum  precision  allowed  by  the  calculator  (12  significant  digits).  To  learn 
more  about  reals,  see  Chapter  2.  To  illustrate  this  and  other  number  formats 
try  the  following  exercises: 

•     Standard  format: 

This  mode  is  the  most  used  mode  as  it  shows  numbers  in  the  most  familiar 
notation. 

Press  the  m3m  soft  menu  key,  with  the  Number  format  set  to  Std,  to 
return  to  the  calculator  display.  Enter  the  number  1  23.45678901  23456. 
Notice  that  this  number  has  1  6  significant  figures.  Press  the  [enter]  key. 


Page  1-17 


The  number  is  rounded  to  the  maximum  1 2  significant  figures,  and  is 
displayed  as  follows: 


123.456789012 

123.456789012 

laaaiBaMfflBBBiHnMHBna 

In  the  standard  format  of  decimal  display,  integer  numbers  are  shown 
with  no  decimal  zeros  whatsoever.  Numbers  with  different  decimal 
figures  will  be  adjusted  in  the  display  so  that  only  those  decimal  figures 
that  are  necessary  will  be  shown.  More  examples  of  numbers  in  standard 
format  are  shown  next: 


:  125. 

125. 

:25.69S 

25.693 

: 56.254379 

56.254379 

EDIT  |  VIEH  |  RCL 

I  STO*  |F  UF;ijE|CLEHF; 

Fixed  format  with  no  decimals:    Press  the  [mode]  button.    Next,  use  the 
down  arrow  key,  <s3?/  to  select  the  option  Number  format.    Press  the 
SID3I1I  soft  menu  key  (  (JL)  ),  and  select  the  option  Fixed  with  the  arrow 
down  key  ^3? . 


CALCULATOR  MODE:" 
Operating  Hodc.  A Ucbrgic 
nuHbcr  ForHat....Hfi   0  _FH.. 
An^U  Hcafurc... Radian* 

Coord  Syst«H  RcctansuUr 

^B**P    _K*y  CUch    ^Last  StacR 

^00 Tl^liJf  Sf  _  jj/i/^-L  f Qj" Hat  


Notice  that  the  Number  Format  mode  is  set  to  Fix  followed  by  a  zero  (0). 
This  number  indicates  the  number  of  decimals  to  be  shown  after  the 
decimal  point  in  the  calculator's  display.  Press  the  IEI9I  soft  menu  key 
to  return  to  the  calculator  display.    The  number  now  is  shown  as: 


123. 


123. 


EDIT     I E H    RCL    STOP-  PURGE  CLEAR 


This  setting  will  force  all  results  to  be  rounded  to  the  closest  integer  (0 
digit  displayed  after  the  comma).  However,  the  number  is  still  stored  by 
the  calculator  with  its  full  1  2  significant  digit  precision.  As  we  change  the 
number  of  decimals  to  be  displayed,  you  will  see  the  additional  digits 
being  shown  again. 


Page  1-18 


Fixed  format  with  decimals: 

This  mode  is  mainly  used  when  working  with  limited  precision.  For 
example,  if  you  are  doing  financial  calculation,  using  a  FIX  2  mode  is 
convenient  as  it  can  easily  represent  monetary  units  to  a  1/1  00  precision. 
Press  the  (mode)  button.  Next,  use  the  down  arrow  key,  ^3? ,  to  select  the 
option  Number  format.  Press  the  soft  menu  key  (  QD  )/  and  select 

the  option  Fixed  with  the  arrow  down  key  ' 


j:  CALCULATOR  MODEL" 
lopcratin^  Hod*..  AUcbraic 
lriiJHt"2f  FurHi]t....lffl    0  _FHj 
Any!.*  Hcasurc...  Radians 

Coord  Syft«H  RcctonsuLar 

|^B«P    _K«y  aicR    ^Loft  StacR 

Choose  nuHb cr  display  ForHO t 


Press  the  right  arrow  key,  CD,  to  highlight  the  zero  in  front  of  the  option 
Fix.    Press  the  !i!ii!I!!i!!!i!!§!  soft  menu  key  and,  using  the  up  and  down  arrow 
keys,  (^N) ^3? ,  select,  say,  3  decimals.  


Opcr 

nurib 

Hr.q  1. 

.FH, 

Coor 

H 

S 

tacR 

Choo 

C 

Press  the  KCE111  soft  menu  key  to  complete  the  selection: 


CALCULATOR  MODE:" 
Operating  HodcALycbraic 
nuHbcr  ForHat....Fix   9  _FMj 
AnyU  Hcafurc... Radian* 

Coord  Syst«H  RcctanyuUr 

^B**P    _K*y  CUch    ^Last  StacR 

Choojg  d^ci Hal  places  to  display 


Press  the  I11ECE111  soft  menu  key  return  to  the  calculator  display.  The 
number  now  is  shown  as: 


123.457 

  123.457 

Notice  how  the  number  is  rounded,  not  truncated.  Thus,  the  number 

1  23.45678901  23456,  for  this  setting,  is  displayed  as  1  23.457,  and  not 

as  1  23.456  because  the  digit  after  6  is  >  5 


Page  1-19 


Scientific  format 

The  scientific  format  is  mainly  used  when  solving  problems  in  the  physical 
sciences  where  numbers  are  usually  represented  as  a  number  with  limited 
precision  multiplied  by  a  power  of  ten. 

To  set  this  format,  start  by  pressing  the  LM0P£J  button.  Next,  use  the  down 
arrow  key,  <\j?/  to  select  the  option  Number  format.  Press  the  H3SEB  soft 
menu  key  (  C^~) ),  and  select  the  option  Scientific  with  the  arrow  down  key 
<\3? .  Keep  the  number  3  in  front  of  the  Sci.  (This  number  can  be 
changed  in  the  same  fashion  that  we  changed  the  Fixed  number  of 
decimals  in  the  example  above). 


S  CALCULATOR  MODEL" 
lOpcrQtxn^  Hod*.. flU-zbraic 
lriUHt'h!i-  ForHat....EHl    3  _FMj 

Coord  SystCH  RcctonsuLar 

|^B**P    _K*y  aicR    ^Last  StacR 

Choojg  nuHb ir  digp_l.au  Fotho t 


Press  the  II1ECE111  soft  menu  key  return  to  the  calculator  display, 
number  now  is  shown  as: 


The 


1 . 235E2 

1 ■ 235E2 


This  result,  1 .23E2,  is  the  calculator's  version  of  powers-of-ten  notation, 
i.e.,  1.235  x  102.  In  this,  so-called,  scientific  notation,  the  number  3  in 
front  of  the  Sci  number  format  (shown  earlier)  represents  the  number  of 
significant  figures  after  the  decimal  point.  Scientific  notation  always 
includes  one  integer  figure  as  shown  above.  For  this  case,  therefore,  the 
number  of  significant  figures  is  four. 


•     Engineering  format 

The  engineering  format  is  very  similar  to  the  scientific  format,  except  that 
the  powers  of  ten  are  multiples  of  three. 

To  set  this  format,  start  by  pressing  the  LM0P£J  button.  Next,  use  the  down 
arrow  key,  <\j?/  to  select  the  option  Number  format.  Press  the  llllilll!  soft 
menu  key  (  C^~) ),  and  select  the  option  Engineering  with  the  arrow  down 
key  .  Keep  the  number  3  in  front  of  the  Eng.  (This  number  can  be 
changed  in  the  same  fashion  that  we  changed  the  Fixed  number  of 
decimals  in  an  earlier  example). 


Page  1-20 


CALCULATOR  MODEL" 
Opcrotins  Hod*.,  ft  Hebraic 
ri ij h t- hi I"  ForHat....En3   9  _FH.. 
flnslfl  M-2']?ur-2.... R']'ji-]ri? 

Coord  SystCH  RcctonsuLar 

^B**P    _K*y  aich    ^Last  StocR 

Choojg  dgci Hal  pUKgj  to  ■3iJp'.-]M 


Press  the  KGlisoft  menu  key  return  to  the  calculator  display, 
number  now  is  shown  as: 


The 


123. SEE 


Because  this  number  has  three  figures  in  the  integer  part,  it  is  shown  with 
four  significative  figures  and  a  zero  power  of  ten,  while  using  the 
Engineering  format.  For  example,  the  number  0.00256,  will  be  shown  as: 


: 123. 5E0 
2.560E-3 


123. 5E0 


2.560E-3 
H-HimillliHlHHllii'liHaHBla 


Decimal  comma  vs.  decimal  point 

Decimal  points  in  floating-point  numbers  can  be  replaced  by  commas,  if 
the  user  is  more  familiar  with  such  notation.  To  replace  decimal  points  for 
commas,  change  the  FA/I  option  in  the  CALCULATOR  MODES  input  form 
to  commas,  as  follows  (Notice  that  we  have  changed  the  Number  Format 
to  Std): 

Press  the  [mode)  button.  Next,  use  the  down  arrow  key,  ^3?,  once,  and  the 

right  arrow  key,  CD,  highlighting  the  option  FM,.    To  select  commas, 

press  the  li^ililllllll  soft  menu  key  (i.e.,  the  CZD  key).  The  input  form  will 
look  as  follows: 


CALCULATOR  MODEL" 
Opcrotins  Hod*.,  ft  Hebraic 
nuHbflf  ForHat....Std  Hfh, 
fln^Lc  Hcafur4....Ra4i4nf 

Coord  SystCH  RcctonsuLar 

^B**P    _K*y  aicR    ^Last  StacR 

Ifi  coHHfl  a j_  f ration  Harh? 


Press  the  BUI  soft  menu  key  return  to  the  calculator  display, 
number  1  23.45678901  2,  entered  earlier,  now  is  shown  as: 


The 


123,456739012 

123,456739013 


EDIT   '.' I E H    RCL    £T0t  PURGE  CLEAR 


Page  1-21 


Angle  Measure 

Trigonometric  functions,  for  example,  require  arguments  representing  plane 
angles.  The  calculator  provides  three  different  Angle  Measure  modes  for 
working  with  angles,  namely: 

•  Degrees:  There  are  360  degrees  (360°)  in  a  complete  circumference,  or 
90  degrees  (90°)  in  a  right  angle.  This  representation  is  mainly  used 
when  doing  basic  geometry,  mechanical  or  structural  engineering,  and 
surveying. 

•  Radians:  There  are  2n  radians  (2;rr)  in  a  complete  circumference,  or  n/2 
radians  (n/2  r)  in  a  right  angle.  This  notation  is  mainly  used  when  solving 
mathematics  and  physics  problems.  This  is  the  default  mode  of  the 
calculator. 

•  Grades:  There  are  400  grades  (400  9)  in  a  complete  circumference,  or 
700  grades  (7  00  g)  in  a  right  angle.  This  notation  is  similar  to  the  degree 
mode,  and  was  introduced  in  order  to  "simplify"  the  degree  notation  but 
is  seldom  used  now. 

The  angle  measure  affects  the  trig  functions  like  SIN,  COS,  TAN  and 
associated  functions. 

To  change  the  angle  measure  mode,  use  the  following  procedure: 

•  Press  the  LM0P£J  button.  Next,  use  the  down  arrow  key,  ^3? ,  twice.  Select 
the  Angle  Measure  mode  by  either  using  the  L^jkey  (second  from  left  in 
the  fifth  row  from  the  keyboard  bottom),  or  pressing  the  soft  menu 
key  (  QD  )•  If  using  the  latter  approach,  use  up  and  down  arrow 
keys,^}  ^z? ,  to  select  the  preferred  mode,  and  press  the  MB  f^Q  soft 
menu  key  to  complete  the  operation.  For  example,  in  the  following 
screen,  the  Radians  mode  is  selected: 

^^CflLCULflTOR  H0DES^^« 
Opcrotins  Hodc.  ft  Hebraic 
nuHbcr  ForHJt....  Std  _FHj 

Coord  SyftCH  RcctonsuLar 

^B«P    _K«y  aich    ^Loft  Stack 

Choose  gnjl g_  mq sure  

Coordinate  System 


Page  1-22 


The  coordinate  system  selection  affects  the  way  vectors  and  complex  numbers 
are  displayed  and  entered.  To  learn  more  about  complex  numbers  and 
vectors,  see  Chapters  4  and  9,  respectively. 

Two-  and  three-dimensional  vector  components  and  complex  numbers  can  be 
represented  in  any  of  3  coordinate  systems:  The  Cartesian  (2  dimensional)  or 
Rectangular  (3  dimensional),  Cylindrical  (3  dimensional)  or  Polar  (2 
dimensional),  and  Spherical  (only  3  dimensional).  In  a  Cartesian  or 
Rectangular  coordinate  system  a  point  P  will  have  three  linear  coordinates 
(x,y,z)  measured  from  the  origin  along  each  of  three  mutually  perpendicular 
axes  (in  2  d  mode,  z  is  assumed  to  be  0).  In  a  Cylindrical  or  Polar 
coordinate  system  the  coordinates  of  a  point  are  given  by  (r,9,z),  where  r  is  a 
radial  distance  measured  from  the  origin  on  the  xy  plane,  0  is  the  angle  that 
the  radial  distance  r  forms  with  the  positive  x  axis  -  measured  as  positive  in  a 
counterclockwise  direction  -,  and  z  is  the  same  as  the  z  coordinate  in  a 
Cartesian  system  (in  2  d  mode,  z  is  assumed  to  be  0).  The  Rectangular  and 
Polar  systems  are  related  by  the  following  quantities: 

r  =  Jx2+y2 


x  =  r-  cos(#) 
y-r-  sin(#) 


6  =  tan" 


z  —  z 


In  a  Spherical  coordinate  system  the  coordinates  of  a  point  are  given  by 
(p,6,4>)  where  p  is  a  radial  distance  measured  from  the  origin  of  a  Cartesian 
system,  6  is  an  angle  representing  the  angle  formed  by  the  projection  of  the 
linear  distance  p  onto  the  xy  axis  (same  as  6  in  Polar  coordinates),  and  0  is 
the  angle  from  the  positive  z  axis  to  the  radial  distance  p.  The  Rectangular 
and  Spherical  coordinate  systems  are  related  by  the  following  quantities: 

p  =  Jl 


x  =  p  •  sin(^)  •  cos(#) 
y  =  p-  sin(^)  •  sin(#)  0  =  tan" 

z  =  /}-cos(^)  (j)  -  tan" 


lx2  +y2  +z2 


(A 

\x) 

t 


To  change  the  coordinate  system  in  your  calculator,  follow  these  steps: 


Page  1-23 


•  Press  the  LM0P£J  button.  Next,  use  the  down  arrow  key,  ^3?,  three  times. 
Select  the  Angle  Measure  mode  by  either  using  the  L+/-  J  key  (second  from 
left  in  the  fifth  row  from  the  keyboard  bottom),  or  pressing  the  H3SEE9  soft 
menu  key  (  C™~)  )•  If  using  the  latter  approach,  use  up  and  down  arrow 
keys,^}  <\j/>,  to  select  the  preferred  mode,  and  press  the  1111111111  f^Q  soft 
menu  key  to  complete  the  operation.  For  example,  in  the  following 
screen,  the  Polar  coordinate  mode  is  selected: 

^^^CftLCULflTOR  H0DES^^« 
Operating  Hodc.  ft  Hebraic 
nuHb«r  ForHat....Std  _FHj 
fln^U  H-zojurh!.... F;-]-Ji-]riJ 

Coord  Syft«H  QIEI^^^H 

^E«flp    _K«y  CUch    ^Laft  Stack 

Ihi ■:■  ■:■  j £  coor dinqtg_JUJt «h  

Beep,  Key  Click,  and  Last  Stack 

The  last  line  of  the  CALCULATOR  MODES  input  form  include  the  options: 

_Beep  _Key  Click  _Lost  Stock 
By  choosing  the  check  mark  next  to  each  of  these  options,  the  corresponding 
option  is  activated.  These  options  are  described  next: 

_Beep   :  When  selected,  ,  the  calculator  beeper  is  active.  This  operation 

mainly  applies  to  error  messages,  but  also  some  user  functions  like 
BEEP. 

_Key  Click  :  When  selected,  each  keystroke  produces  a  "click"  sound. 
_Lost  Stock:  Keeps  the  contents  of  the  last  stack  entry  for  use  with  the  functions 
UNDO  and  ANS  (see  Chapter  2). 

The  _Beep  option  can  be  useful  to  advise  the  user  about  errors.  You  may 
want  to  deselect  this  option  if  using  your  calculator  in  a  classroom  or  library. 
The  _Key  Click  option  can  be  useful  as  an  audible  way  to  check  that  each 
keystroke  was  entered  as  intended. 

The  _Lost  Stock  option  is  very  useful  to  recover  the  last  operation  in  case  we 
need  it  for  a  new  calculation. 

To  select,  or  deselect,  any  of  these  three  options,  first  press  the  [mode)  button. 
Next, 


Page  1-24 


•  Use  the  down  arrow  key,  ~,  four  times  to  select  the  _Lost  Stock  option. 
Use  the  soft  menu  key  (i.e.,  the  CZD  key)  to  change  the  selection. 

•  Press  the  left  arrow  key  (T)   to  select  the  _Key  Click  option.  Use  the  11^333 
soft  menu  key  (i.e.,  the  (JL)  key)  to  change  the  selection. 

•  Press  the  left  arrow  key  CD  to  select  the  _Beep  option.  Use  the  jKilimjl  soft 
menu  key  (i.e.,  the  (JL)  key)  to  change  the  selection. 

Press  the  lllllLllllllll  C™~~)  soft  menu  key  to  complete  the  operation. 


CAS  stands  for  Computer  Algebraic  System.  This  is  the  mathematical  core  of 
the  calculator  where  the  symbolic  mathematical  operations  and  functions  are 
programmed  and  performed.  The  CAS  offers  a  number  of  settings  can  be 
adjusted  according  to  the  type  of  operation  of  interest.  These  are: 

•  The  default  independent  variable 

•  Numeric  vs.  symbolic  mode 

•  Approximate  vs.  Exact  mode 

•  Verbose  vs.  Non-verbose  mode 

•  Step-by-step  mode  for  operations 

•  Increasing  power  format  for  polynomials 

•  Rigorous  mode 

•  Simplification  of  non-rational  expressions 

Details  on  the  selection  of  CAS  settings  are  presented  in  Appendix  C. 

Selecting  Display  modes 

The  calculator  display  can  be  customized  to  your  preference  by  selecting 
different  display  modes.  To  see  the  optional  display  settings  use  the  following: 

•  First,  press  the  (km)  button  to  activate  the  CALCULATOR  MODES  input 
form.  Within  the  CALCULATOR  MODES  input  form,  press  the  EEB1  soft 
menu  key  ((JD  )  to  display  the  DISPLAY  MODES  input  form. 


Selecting  CAS  settings 


!Font:FtS_S:  SYSTEM  S 

lEdit:  g:'H-]i. i.  _FuU  Fas*  _Ind*nt 

!Stach:_SHaU  ^Textbook 

IEGH:    _ShoU  _ShoU  Stach  Difp 

fcdit  using  jhoU  font?  


Page  1-25 


•  To  navigate  through  the  many  options  in  the  DISPLAY  MODES  input  form, 
use  the  arrow  keys:  CD  CD  ^3?  {A^>  • 

•  To  select  or  deselect  any  of  the  settings  shown  above,  that  require  a  check 
mark,  select  the  underline  before  the  option  of  interest,  and  toggle  the 

soft  menu  key  until  the  right  setting  is  achieved.  When  an  option  is 
selected,  a  check  mark  will  be  shown  in  the  underline  (e.g.,  the  Textbook 
option  in  the  Stock:  line  above).  Unselected  options  will  show  no  check 
mark  in  the  underline  preceding  the  option  of  interest  (e.g.,  the  _Smoll, 
_Full  page,  and  Jndent  options  in  the  Edit:  line  above). 

•  To  select  the  Font  for  the  display,  highlight  the  field  in  front  of  the  Font: 
option  in  the  DISPLAY  MODES  input  form,  and  use  the  H3SSB  soft  menu 
key(CZD). 

•  After  having  selected  and  unselected  all  the  options  that  you  want  in  the 
DISPLAY  MODES  input  form,  press  the  Iliiilllllll  soft  menu  key.  This  will  take 
you  back  to  the  CALCULATOR  MODES  input  form.  To  return  to  normal 
calculator  display  at  this  point,  press  the  EH  soft  menu  key  once  more. 


Selecting  the  display  font 

Changing  the  font  display  allows  you  to  have  the  calculator  look  and  feel 
changed  to  your  own  liking.  By  using  a  6-pixel  font,  for  example,  you  can 
display  up  to  9  stack  levels!  Follow  these  instructions  to  select  your  display 
font: 

First,  press  the  (mqdF)  button  to  activate  the  CALCULATOR  MODES  input  form. 
Within  the  CALCULATOR  MODES  input  form,  press  the  EEB1  soft  menu  key 
(GD)  to  display  the  DISPLAY  MODES  input  form.  The  Font:  field  is 
highlighted,  and  the  option  Ft8_0: system  8  is  selected.  This  is  the  default 
value  of  the  display  font.    Pressing  the  illliiilili  soft  menu  key  (QD  ),  will 


provide  a  list  of  availab 


e  system  fonts,  as  shown  below: 


S  DISPLAY  MODEL" 


Font 
Edit 
Stac 


Syst«H 

Font  ? 

Syst«H 

Font  6 

dent 


Choos«  syst«H  Font 


The  options  available  are  three  standard  System  Fonts  (sizes  8/  7,  and  6)  and 
a  Browse.,  option.    The  latter  will  let  you  browse  the  calculator  memory  for 


Page  1-26 


additional  fonts  that  you  may  have  created  (see  Chapter  23)  or  downloaded 
into  the  calculator. 

Practice  changing  the  display  fonts  to  sizes  7  and  6.  Press  the  OK  soft  menu 
key  to  effect  the  selection.  When  done  with  a  font  selection,  press  the  112111 
soft  menu  key  to  go  back  to  the  CALCULATOR  MODES  input  form.  To  return 
to  normal  calculator  display  at  this  point,  press  the  IEQ5  soft  menu  key  once 
more  and  see  how  the  stack  display  change  to  accommodate  the  different  font. 

Selecting  properties  of  the  line  editor 

First,  press  the  [mode]  button  to  activate  the  CALCULATOR  MODES  input  form. 
Within  the  CALCULATOR  MODES  input  form,  press  the  BBisoft  menu  key 
(GD  )  to  display  the  DISPLAY  MODES  input  form.  Press  the  down  arrow  key, 
^▼7,  once,  to  get  to  the  Edit  line.  This  line  shows  three  properties  that  can  be 
modified.  When  these  properties  are  selected  (checked)  the  following  effects 
are  activated: 

_Small  Changes  font  size  to  small 

_Full  page        Allows  to  place  the  cursor  after  the  end  of  the  line 
Jndent  Auto  intend  cursor  when  entering  a  carriage  return 

Detailed  instructions  on  the  use  of  the  line  editor  are  presented  in  Chapter  2  in 
this  guide. 

Selecting  properties  of  the  Stack 

First,  press  the  [mode]  button  to  activate  the  CALCULATOR  MODES  input  form. 
Within  the  CALCULATOR  MODES  input  form,  press  the  I3S1  soft  menu  key 
(CZQ  )  to  display  the  DISPLAY  MODES  input  form.  Press  the  down  arrow  key, 
^▼7,  twice,  to  get  to  the  Stock  line.  This  line  shows  two  properties  that  can  be 
modified.  When  these  properties  are  selected  (checked)  the  following  effects 
are  activated: 

_Smoll  Changes  font  size  to  small.    This  maximized  the  amount  of 

information  displayed  on  the  screen.  Note,  this  selection 
overrides  the  font  selection  for  the  stack  display. 

_Textbook  Display  mathematical  expressions  in  graphical  mathematical 
notation 


Page  1-27 


To  illustrate  these  settings,  either  in  algebraic  or  RPN  mode,  use  the  equation 
writer  to  type  the  following  definite  integral: 

In  Algebraic  mode,  the  following  screen  shows  the  result  of  these  keystrokes 
with  neither  _Small  nor  _Textbook  are  selected: 

jagBHgSg33ME3H133MlEiaj 

With  the  _Smoll  option  selected  only,  the  display  looks  as  shown  below: 

With  the  _Textbook  option  selected  (default  value),  regardless  of  whether  the 
_Small  option  is  selected  or  not,  the  display  shows  the  following  result: 


:  e"XdX 

Je 

1 

EHMHEZEgEgliaaiEig] 

Selecting  properties  of  the  equation  writer  (EQW) 

First,  press  the  Qwx)  button  to  activate  the  CALCULATOR  MODES  input  form. 
Within  the  CALCULATOR  MODES  input  form,  press  the  I3S1  soft  menu  key 
(GQ  )  to  display  the  DISPLAY  MODES  input  form.  Press  the  down  arrow  key, 
<s3?/  three  times,  to  get  to  the  EQW  (Equation  Writer)  line.  This  line  shows 
two  properties  that  can  be  modified.  When  these  properties  are  selected 
(checked)  the  following  effects  are  activated: 

_Smoll  Changes  font  size  to  small  while  using  the  equation 

editor 

_Small  Stock  Disp  Shows  small  font  in  the  stack  for  textbook  style 

display 

Detailed  instructions  on  the  use  of  the  equation  editor  (EQW)  are  presented 
elsewhere  in  this  manual. 


Page  1-28 


f00  -X 

For  the  example  of  the  integral      e    dX ,  presented  above,  selecting  the 

_Small  Stack  Disp  in  the  EQW  line  of  the  DISPLAY  MODES  input  form 
produces  the  following  display: 


*"KdK 


Selecting  the  size  of  the  header 

First,  press  the  (km)  button  to  activate  the  CALCULATOR  MODES  input  form. 
Within  the  CALCULATOR  MODES  input  form,  press  the  EES1  soft  menu  key 
(GD  )  to  display  the  DISPLAY  MODES  input  form.  Press  the  down  arrow  key, 
<s3?/  four  times,  to  get  to  the  Header  line.  The  value  2  is  assigned  to  the 
Header  field  by  default.  This  means  that  the  top  part  of  the  display  will 
contain  two  lines,  one  showing  the  current  settings  of  the  calculator,  and  a 
second  one  showing  the  current  sub  directory  within  the  calculator's  memory 
(These  lines  were  described  earlier  in  the  manual).  The  user  can  select  to 
change  this  setting  to  1  or  0  to  reduce  the  number  of  header  lines  in  the 
display. 

Selecting  the  clock  display 

First,  press  the  (mode)  button  to  activate  the  CALCULATOR  MODES  input  form. 
Within  the  CALCULATOR  MODES  input  form,  press  the  BSl  soft  menu  key 
(GD  )  to  display  the  DISPLAY  MODES  input  form.  Press  the  down  arrow  key, 
^3? ,  four  times,  to  get  to  the  Header  line.  The  Header  field  will  be 
highlighted.  Use  the  right  arrow  key  (CD)  to  select  the  underline  in  front  of 
the  options  _Clock  or  _Analog.  Toggle  the  jK;jjj]31i  soft  menu  key  until  the 
desired  setting  is  achieved.  If  the  _Clock  option  is  selected,  the  time  of  the 
day  and  date  will  be  shown  in  the  upper  right  corner  of  the  display.  If  the 
_Analog  option  is  also  selected,  an  analog  clock,  rather  than  a  digital  clock, 
will  be  shown  in  the  upper  right  corner  of  the  display.  If  the  _Clock  option  is 
not  selected,  or  the  header  is  not  present,  or  too  small,  the  date  and  time  of 
day  will  not  be  shown  in  the  display. 


Page  1-29 


Chapter  2 

Introducing  the  calculator 

In  this  chapter  we  present  a  number  of  basic  operations  of  the  calculator 
including  the  use  of  the  Equation  Writer  and  the  manipulation  of  data  objects 
in  the  calculator.  Study  the  examples  in  this  chapter  to  get  a  good  grasp  of 
the  capabilities  of  the  calculator  for  future  applications. 

Calculator  objects 

Any  number,  expression,  character,  variable,  etc.,  that  can  be  created  and 
manipulated  in  the  calculator  is  referred  to  as  an  object.  Some  of  the  most 
useful  type  of  objects  are  listed  below. 

Real.  These  object  represents  a  number,  positive  or  negative,  with  12 
significant  digits  and  an  exponent  ranging  from  -499  to  +499.  example  of 
reals  are:  1 .,  -5.,  56.41 564  1 .5E45,  -555.74E-95 

When  entering  a  real  number,  you  can  use  the  Ieex j  key  to  enter  the  exponent 
and  the  L^J  key  to  change  the  sign  of  the  exponent  or  mantissa. 

Note  that  real  must  be  entered  with  a  decimal  point,  even  if  the  number  does 
not  have  a  fractional  part.  Otherwise  the  number  is  taken  as  an  integer 
number,  which  is  a  different  calculator  objects.  Reals  behave  as  you  would 
expect  a  number  to  when  used  in  a  mathematical  operation. 

Integers.  These  objects  represent  integer  numbers  (numbers  without  fractional 
part)  and  do  not  have  limits  (except  the  memory  of  the  calculator).  Example 
of  integers  are:  1,  564654112,  -413165467354646765465487.  Note 
how  these  numbers  do  not  have  a  decimal  point. 

Due  to  their  storage  format,  integer  numbers  are  always  maintain  full 
precision  in  their  calculation.  For  example,  an  operation  such  as  30/14,  with 
integer  numbers,  will  return  15/7  and  not  2.142....  To  force  a  real  (or 
floating-point)  result,  use  function  ->NUM  L  r>  j -~num  . 

Integers  are  used  frequently  in  CAS-based  functions  as  they  are  designed  to 
keep  full  precision  in  their  operation.. 


Page  2-1 


If  the  approximate  mode  (APPROX)  is  selected  in  the  CAS  (see  Appendix  C), 
integers  will  be  automatically  converted  to  reals.  If  you  are  not  planning  to  use 
the  CAS,  it  might  be  a  good  idea  to  switch  directly  into  approximate  mode. 
Refer  to  Appendix  C  for  more  details. 

Mixing  integers  and  reals  together  or  mistaking  an  integer  for  a  real  is  a 
common  occurrence.  The  calculator  will  detect  such  mixing  of  objects  and  ask 
you  if  you  want  to  switch  to  approximate  mode. 

Complex  numbers,  are  an  extension  of  real  numbers  that  include  the  unit 
imaginary  number,  /  2=  -1 .  A  complex  number,  e.g.,  3  +  2i,  is  written  as  (3, 
2)  in  the  calculator. 

Complex  numbers  can  be  displayed  in  either  Cartesian  or  polar  mode 
depending  on  the  setting  selected.  Note  that  complex  numbers  are  always 
stored  in  Cartesian  mode  and  that  only  the  display  is  affected.  This  allows  the 
calculator  to  keep  as  much  precision  as  possible  during  calculations. 

Most  mathematics  functions  work  on  complex  numbers.  There  is  no  need  to 
use  a  special  "complex  +"  function  to  add  complex  numbers,  you  can  use  the 
same  L  +  J  function  that  on  reals  or  integers. 

Vector  and  matrix  operations  utilize  objects  of  type  3,  real  arrays,  and,  if 
needed,  type  4,  complex  arrays.  Objects  type  2,  strings,  are  simply  lines  of 
text  (enclosed  between  quotes)  produced  with  the  alphanumeric  keyboard. 

A  list  is  just  a  collection  of  objects  enclosed  between  curly  brackets  and 
separated  by  spaces  in  RPN  mode  (the  space  key  is  labeled  jspc J )/  or  by 
commas  in  algebraic  mode.  Lists,  objects  of  type  5,  can  be  very  useful  when 
processing  collections  of  numbers.  For  example,  the  columns  of  a  table  can  be 
entered  as  lists.  If  preferred,  a  table  can  be  entered  as  a  matrix  or  array. 

Objects  type  8  are  programs  in  User  RPL  language.  These  are  simply  sets  of 
instructions  enclosed  between  the  symbols  «  ». 

Associated  with  programs  are  objects  types  6  and  7,  Global  and  Local 
Names,  respectively.  These  names,  or  variables,  are  used  to  store  any  type 
of  objects.  The  concept  of  global  or  local  names  is  related  to  the  scope  or 
reach  of  the  variable  in  a  given  program. 


Page  2-2 


An  algebraic  object,  or  simply,  an  algebraic  (object  of  type  9),  is  a  valid 
algebraic  expression  enclosed  between  apostrophes. 

Binary  integers,  objects  of  type  10,  are  used  in  some  computer  science 
applications. 

Graphics  objects,  objects  of  type  11,  store  graphics  produced  by  the 
calculator. 

Tagged  objects,  objects  of  type  1  2,  are  used  in  the  output  of  many  programs 
to  identify  results.  For  example,  in  the  tagged  object:  Mean:  23.2,  the  word 
Mean:  is  the  tag  used  to  identify  the  number  23.2  as  the  mean  of  a  sample,  for 
example. 

Unit  objects,  objects  of  type  1  3,  are  numerical  values  with  a  physical  unit 
attached  to  them. 

Directories,  objects  of  type  15,  are  memory  locations  used  to  organize  your 
variables  in  a  similar  fashion  as  folders  are  used  in  a  personal  computer. 

Libraries,  objects  of  type  16,  are  programs  residing  in  memory  ports  that  are 
accessible  within  any  directory  (or  sub-directory)  in  your  calculator.  They 
resemble  built-in  functions,  objects  of  type  18,  and  built-in  commands,  objects 
of  type  1 9,  in  the  way  they  are  used. 


Editing  expressions  in  the  screen 

In  this  section  we  present  examples  of  expression  editing  directly  into  the 
calculator  display  (algebraic  history  or  RPN  stack). 


Creating  arithmetic  expressions 

For  this  example,  we  select  the  Algebraic  operating  mode  and  select  a  Fix 
format  with  3  decimals  for  the  display.   We  are  going  to  enter  the  arithmetic 
expression: 

i.o+M 

5.0-  7-5 


V3.0-2.03 

To  enter  this  expression  use  the  following  keystrokes: 


Page  2-3 


The  resulting  expression  is:  5.*(1  .+1  ,/7.5)/(J"3.-2.A3). 
Press  (enter}  to  get  the  expression  in  the  display  as  follows: 


-.904070293597 

Notice  that,  if  your  CAS  is  set  to  EXACT  (see  Appendix  C)  and  you  enter  your 
expression  using  integer  numbers  for  integer  values,  the  result  is  a  symbolic 
quantity,  e.g., 

Before  producing  a  result,  you  will  be  asked  to  change  to  Approximate  mode. 
Accept  the  change  to  get  the  following  result  (shown  in  Fix  decimal  mode  with 
three  decimal  places  -  see  Chapter  1 ): 


5.000. 

fi  000+1-000! 

I1-000  7. 509 J 

J3. 009-2. 0003- 
MUHHIKgBIUIIIUBmilKMl 

In  this  case,  when  the  expression  is  entered  directly  into  the  stack,  as  soon  as 
you  press  [enter]  ,  the  calculator  will  attempt  to  calculate  a  value  for  the 
expression.  If  the  expression  is  entered  between  quotes,  however,  the 
calculator  will  reproduce  the  expression  as  entered.    In  the  following  example, 
we  enter  the  same  expression  as  above,  but  using  quotes.  For  this  case  we 
set  the  operating  mode  to  Algebraic,  the  CAS  mode  to  Exact  (deselect 
_Approx),  and  the  display  setting  to  Textbook.  The  keystrokes  to  enter  the 
expression  are  the  following: 


Page  2-4 


The  result  will  be  shown  as  follows: 


To  evaluate  the  expression  we  can  use  the  EVAL  function,  as  follows: 

[]W]r^lANS  [enter] 

As  in  the  previous  example,  you  will  be  asked  to  approve  changing  the  CAS 
setting  to  Approx.    Once  this  is  done,  you  will  get  the  same  result  as  before. 

An  alternative  way  to  evaluate  the  expression  entered  earlier  between  quotes 
is  by  using  the  option  LrLJ^M^  .  To  recover  the  expression  from  the  existing 
stack,  use  the  following  keystrokes:  CSCSCS^^ 

We  will  now  enter  the  expression  used  above  when  the  calculator  is  set  to  the 
RPN  operating  mode.  We  also  set  the  CAS  to  Exact  and  the  display  to 
Textbook.  The  keystrokes  to  enter  the  expression  between  quotes  are  the 
same  used  earlier,  i.e., 

Resulting  in  the  output 


Press  [enter)  once  more  to  keep  two  copies  of  the  expression  available  in  the 
stack  for  evaluation.  We  first  evaluate  the  expression  using  the  function  EVAL, 
and  next  using  the  function  ->NUM.  First,  evaluate  the  expression  using 
function  EVAL.  The  resulting  expression  is  semi-symbolic  in  the  sense  that 
there  are  floating-point  components  to  the  result,  as  well  as  a  V3.    Next,  we 
switch  stack  locations  and  evaluate  using  function  ->NUM: 

CD  Exchange  stack  levels  1  and  2  (the  SWAP  command) 

Ij^^num  Evaluate  using  function  ->NUM 


Page  2-5 


This  latter  result  is  purely  numerical,  so  that  the  two  results  in  the  stack, 
although  representing  the  same  expression,  seem  different.  To  verify  that  they 
are  not,  we  subtract  the  two  values  and  evaluate  this  difference  using  function 
EVAL: 

LnJ  Subtract  level  1  from  level  2 

[eval)  Evaluate  using  function  EVAL 

The  result  is  zero  (0.). 

Note:  Avoid  mixing  integer  and  real  data  to  avoid  conflicts  in  the 
calculations.  For  many  physical  science  and  engineering  applications, 
including  numerical  solution  of  equation,  statistics  applications,  etc.,  the 
APPROX  mode  (see  Appendix  C)  works  better.  For  mathematical 
applications,  e.g.,  calculus,  vector  analysis,  algebra,  etc.,  the  EXACT  mode  is 
preferred.  Become  acquainted  with  operations  in  both  modes  and  learn  how 
to  switch  from  one  to  the  other  for  different  types  of  operations  (see  Appendix 
Q.  

Editing  arithmetic  expressions 

Suppose  that  we  entered  the  following  expression,  between  quotes,  with  the 
calculator  in  RPN  mode  and  the  CAS  set  to  EXACT: 


18  5. 

I1  1.75J 

J5-23 

i™iH>ju:ir*««nc>]ai;iiiffl4Hi*ifl 

1  + 


rather  than  the  intended  expression:  5  ■ 
was  entered  by  using: 


7.5 


V3-23 


.  The  incorrect  expression 


To  enter  the  line  editor  use  C5D^3?  .  The  display  now  looks  as  follows: 


Page  2-6 


45*<  1-1/1.  75 V<-J"5-2A3 

The  editing  cursor  is  shown  as  a  blinking  left  arrow  over  the  first  character  in 
the  line  to  be  edited.    Since  the  editing  in  this  case  consists  of  removing  some 
characters  and  replacing  them  with  others,  we  will  use  the  right  and  left  arrow 
keys,  (3D  CD ,  to  move  the  cursor  to  the  appropriate  place  for  editing,  and  the 
delete  key,      J,  to  eliminate  characters. 

The  following  keystrokes  will  complete  the  editing  for  this  case: 

•  Press  the  right  arrow  key,  CD ,  until  the  cursor  is  immediately  to  the 
right  of  the  decimal  point  in  the  term  7.75 

•  Press  the  delete  key,  L4J,  twice  to  erase  the  characters  7. 

•  Press  the  right  arrow  key,  CD ,  once,  to  move  the  cursor  to  the  right  of 
the  7 

•  Type  a  decimal  point  with  Lj_J 

•  Press  the  right  arrow  key,  CD ,  until  the  cursor  is  immediately  to  the 
right  of  the  V5 

•  Press  the  delete  key,  L4J,  once  to  erase  the  character  5 

•  Type  a  3  with  LU 

•  Press  [enter)  to  return  to  the  stack 


The  edited  expression  is  now  available  in  the  stack. 


1- 


7.5 


43-2* 


EDIT   VIEH    RCL    STO*  PURGE  CLEAR 


Editing  of  a  line  of  input  when  the  calculator  is  in  Algebraic  operating  mode  is 
exactly  the  same  as  in  the  RPN  mode.  You  can  repeat  this  example  in 
Algebraic  mode  to  verify  this  assertion. 

Creating  algebraic  expressions 

Algebraic  expressions  include  not  only  numbers,  but  also  variable  names.  As 
an  example,  we  will  enter  the  following  algebraic  expression: 


Page  2-7 


R  +  y 


We  set  the  calculator  operating  mode  to  Algebraic,  the  CAS  to  Exact,  and  the 
display  to  Textbook.  To  enter  this  algebraic  expression  we  use  the  following 
keystrokes: 

ED  (J—  SSED^EDSC^EDCBCx^ 
Press  (enter}  to  get  the  following  result: 


Entering  this  expression  when  the  calculator  is  set  in  the  RPN  mode  is  exactly 
the  same  as  this  Algebraic  mode  exercise. 

Editing  algebraic  expressions 

Editing  of  an  algebraic  expression  with  the  line  editor  is  very  similar  to  that  of 
an  arithmetic  expression  (see  exercise  above).  Suppose  that  we  want  to 
modify  the  expression  entered  above  to  read 


To  edit  this  algebraic  expression  using  the  line  editor  use  l<n  j^r?  .  This 
activates  the  line  editor,  showing  the  expression  to  be  edited  as  follows: 


£*L*-T  <  1  +x^R  >  s  (.  R+y  >  +2* 


Page  2-8 


The  editing  cursor  is  shown  as  a  blinking  left  arrow  over  the  first  character  in 
the  line  to  be  edited.   As  in  an  earlier  exercise  on  line  editing,  we  will  use  the 
right  and  left  arrow  keys,  (3D  CD ,  to  move  the  cursor  to  the  appropriate 
place  for  editing,  and  the  delete  key,  L4J,  to  eliminate  characters. 

The  following  keystrokes  will  complete  the  editing  for  this  case: 

•  Press  the  right  arrow  key,  CD ,  until  the  cursor  is  to  the  right  of  the  x 

•  Type  LzULU  to  enter  the  power  2  for  the  x 

•  Press  the  right  arrow  key,  CD  /  until  the  cursor  is  to  the  right  of  the  y 

•  Press  the  delete  key,  L4J,  once  to  erase  the  characters  y. 

•  Type  (^)CSD(3  to  enter  an  x 

•  Press  the  right  arrow  key,  CD ,  4  times  to  move  the  cursor  to  the  right 
of  the  * 

•  Type        to  enter  a  square  root  symbol 

•  Type  QD^       to  enter  a  set  of  parentheses  (they  come  in  pairs) 

•  Press  the  right  arrow  key,  CD,  once,  and  the  delete  key,      J,  once, 
to  delete  the  right  parenthesis  of  the  set  inserted  above 

•  Press  the  right  arrow  key,  CD ,  4  times  to  move  the  cursor  to  the  right 
of  the  b 

•  Type  (JnJ^       to  enter  a  second  set  of  parentheses 

•  Press  the  delete  key,  i  +  J,  once,  to  delete  the  left  parenthesis  of  the 
set  inserted  above. 

•  Press  [enter)  to  return  to  normal  calculator  display. 

The  result  is  shown  next: 


2-L- 
"  R- 

[(2-R+ 

1+^ 
R  ,2'L 

1-  y  b 

2-  x3-JBH-IRI+2^ 

2  2 
R  +x 

SQCb-RMR+x) 
i™iH'JM:iH;««-ici]ai;iii;irU[M*ifl 

Notice  that  the  expression  has  been  expanded  to  include  terms  such  as 
|  R  | ,  the  absolute  value,  and  SQ(b-R),  the  square  of  b-R.  To  see  if  we  can 
simplify  this  result,  use  FACTOR(ANS(l ))  in  ALG  mode: 


Page  2-9 


U^-K+^XJ-JD-LAKI+^-JK  +X 


SQCb-RMR+x) 
FflCTORtflHSdH 


^■[[Jb-R+x-Jb]-IRI+J[R2+x2-R]^ 


2  2 
(R+x)-b  -R 


EaaiBaMfflMBBiianMHBna 

Press  LjnJ*^?  to  activate  the  line  editor  once  more.  The  result  is  now: 


4*  (.  (.  J"b*R+x*-J"b  >  *FlBS  (.  R  > 
+JX  <RA2+xA2*R)*L>*flBS 
(.  b    *FlBS  (.  R  >  *  (.  -J"L*flBS  (.  b 

>  V<<R+x>*<bA2*RA2>> 


♦skip  skiph  -kjel  del-*  del  l  ins  i 


•     Pressing  (enter}  once  more  to  return  to  normal  display. 


To  see  the  entire  expression  in  the  screen,  we  can  change  the  option 
_Smoll  Stock  Disp  in  the  DISPLAY  MODES  input  form  (see  Chapter  1 ). 
After  effecting  this  change,  the  display  will  look  as  follows: 


[ta.R+a.x)-JFT.iRi+a-jR3*x3.R.L.ibi].ib> 

SGtb-RMR+x) 
:FACTOR(AnS(iH 

(3'-Jb-R+3-x--Jb)-C-IH-R3+3--jR3+x3-R-L 


EDIT  |  '.'IEH  |  RCL  |  S TO*  |PURGE|CLERR 


Note:  To  use  Greek  letters  and  other  characters  in  algebraic  expressions  use 
the  CHARS  menu.  This  menu  is  activated  by  the  keystroke 
combination  L    j  chars  .  Details  are  presented  in  Appendix  D. 


Using  the  Equation  Writer  (EQW)  to  create  expressions 

The  equation  writer  is  an  extremely  powerful  tool  that  not  only  let  you  enter  or 
see  an  equation,  but  also  allows  you  to  modify  and  work/apply  functions  on 
all  or  part  of  the  equation.  The  equation  writer  (EQW),  therefore,  allows  you 
to  perform  complex  mathematical  operations,  directly,  or  in  a  step-by-step 
mode,  as  you  would  do  on  paper  when  solving,  for  example,  calculus 
problems. 


Page  2-10 


The  Equation  Writer  is  launched  by  pressing  the  keystroke  combination  Lr>  J 
lr>j  eqw  (the  third  key  in  the  fourth  row  from  the  top  in  the  keyboard).  The 
resulting  screen  is  the  following:  


EDIT  |  CURS  |  BIG  ■!  E ■■■  M L  |FflCT0|  SIHF 


The  six  soft  menu  keys  for  the  Equation  Writer  activate  the  following  functions: 

illlilllil:  lets  the  user  edit  an  entry  in  the  line  editor  (see  examples  above) 

illlllil:  highlights  expression  and  adds  a  graphics  cursor  to  it 

iiliili:  if  selected  (selection  shown  by  the  character  in  the  label)  the  font  used  in 

the  writer  is  the  system  font  8  (the  largest  font  available) 

fSEB:  lets  you  evaluate,  symbolically  or  numerically,  an  expression  highlighted 

in  the  equation  writer  screen  (similar  to  Lr>  JL^O) 

IIIiSQ:  lets  you  factor  an  expression  highlighted  in  the  equation  writer  screen 
(if  factoring  is  possible) 

ililill:  lets  you  simplify  an  expression  highlighted  in  the  equation  writer  screen 
(as  much  as  it  can  be  simplified  according  to  the  algebraic  rules  of  the  CAS) 

If  you  press  the  [nxt)  key,  two  more  soft  menu  options  show  up  as  shown 
below: 


The  six  soft  menu  keys  for  the  Equation  Writer  activate  the  following  functions: 

illlllil:  allows  access  to  the  collection  of  CAS  commands  listed  in  alphabetical 
order.  This  is  useful  to  insert  CAS  commands  in  an  expression  available  in  the 
Equation  Writer. 

illllSIII:  activates  the  calculator's  CAS  help  facility  to  provide  information  and 
examples  of  CAS  commands. 

Some  examples  for  the  use  of  the  Equation  Writer  are  shown  below. 

Creating  arithmetic  expressions 

Entering  arithmetic  expressions  in  the  Equation  Writer  is  very  similar  to 
entering  an  arithmetic  expression  in  the  stack  enclosed  in  quotes.  The  main 
difference  is  that  in  the  Equation  Writer  the  expressions  produced  are  written 


Page  2-1 1 


in  "textbook"  style  instead  of  a  line-entry  style.  Thus,  when  a  division  sign 
(i.e.,  GED)  is  entered  in  the  Equation  Writer,  a  fraction  is  generated  and  the 
cursor  placed  in  the  numerator.  To  move  to  the  denominator  you  must  use  the 
down  arrow  key.  For  example,  try  the  following  keystrokes  in  the  Equation 
Writer  screen:  CTDGDCSGDCD 
The  result  is  the  expression 

5+24 

kianEnaBglHEMBiBHBini 


The  cursor  is  shown  as  a  left-facing  key.  The  cursor  indicates  the  current 
edition  location.  Typing  a  character,  function  name,  or  operation  will  enter 
the  corresponding  character  or  characters  in  the  cursor  location.  For  example, 
for  the  cursor  in  the  location  indicated  above,  type  now: 

The  edited  expression  looks  as  follows: 


5 

|j.>41HII:MI:WrPI^J;TIJ;T4f.]i.^j:iJ 


Suppose  that  you  want  to  replace  the  quantity  between  parentheses  in  the 
denominator  (i.e.,  5+1/3)  with  (5+7i2/2).  First,  we  use  the  delete  key  (L4J) 
delete  the  current  1/3  expression,  and  then  we  replace  that  fraction  with  7i2/2, 
as  follows:   CrDCD 


When  we  hit  this  point  the  screen  looks  as  follows: 


5+2 


iaaaEnaBgiBaiiMHiHEBini 

In  order  to  insert  the  denominator  2  in  the  expression,  we  need  to  highlight 
the  entire  n2  expression.  We  do  this  by  pressing  the  right  arrow  key  (CD ) 
once.  At  that  point,  we  enter  the  following  keystrokes:  L+-  JL  2  J 


Page  2-12 


The  expression  now  looks  as  follows: 


EffllEnaBglHEBBiBHBini 

Suppose  that  now  you  want  to  add  the  fraction  1/3  to  this  entire  expression, 

i.e.,  you  want  to  enter  the  expression: 

5  1 

7T2  3 

5  +  2.(5  +  — ) 
2 

First,  we  need  to  highlight  the  entire  first  term  by  using  either  the  right  arrow 
(CD)  or  the  upper  arrow  (^N> )  keys,  repeatedly,  until  the  entire  expression  is 
highlighted,  i.e.,  seven  times,  producing:  


NOTE:  Alternatively,  from  the  original  position  of  the  cursor  (to  the  right  of  the 
2  in  the  denominator  of  n2/2),  we  can  use  the  keystroke  combination  C  r>  J(/^N) 
,  interpreted  as  (LcD  x  ). 


Once  the  expression  is  highlighted  as  shown  above,  type  CD CD CD CD  to 
add  the  fraction  1/3.  Resulting  in:  


*    2  J 


'34 


5+2 


Showing  the  expression  in  smaller-size 

To  show  the  expression  in  a  smaller-size  font  (which  could  be  useful  if  the 
expression  is  long  and  convoluted),  simply  press  the  liliiiiQD  soft  menu  key. 
For  this  case,  the  screen  looks  as  follows: 


Page  2-13 


To  recover  the  larger-font  display,  press  the  IIIiiiQD  soft  menu  key  once  more. 


Evaluating  the  expression 

To  evaluate  the  expression  (or  parts  of  the  expression)  within  the  Equation 
Writer,  highlight  the  part  that  you  want  to  evaluate  and  press  the  EEEOI  fTP) 
soft  menu  key. 

For  example,  to  evaluate  the  entire  expression  in  this  exercise,  first,  highlight 
the  entire  expression,  by  pressing  CTQ  ^  .  Then,  press  the  IlllJlillll  C^~)  soft 
menu  key.    If  your  calculator  is  set  to  Exact  CAS  mode  (i.e.,  the  _Approx 
CAS  mode  is  not  checked),  then  you  will  get  the  following  symbolic  result: 


If  you  want  to  recover  the  unevaluated  expression  at  this  time,  use  the  function 
UNDO,  i.e.,  CEtD  JJndo  (the  first  key  in  the  third  row  of  keys  from  the  top  of  the 
keyboard).  The  recovered  expression  is  shown  highlighted  as  before: 


5+2-  5+V 


If  you  want  a  floating-point  (numerical)  evaluation,  use  the  ->NUM  function 
(i.e.,  CS^SJ).  The  result  is  as  follows:  


S1967616 


Page  2-14 


Use  the  function  UNDO  (  [j^Jundo  )  once  more  to  recover  the  original 
expression: 


Evaluating  a  sub-expression 

Suppose  that  you  want  to  evaluate  only  the  expression  in  parentheses  in  the 

denominator  of  the  first  fraction  in  the  expression  above.  You  have  to  use  the 

arrow  keys  to  select  that  particular  sub-expression.  Here  is  a  way  to  do  it: 

^▼7    Highlights  only  the  first  fraction 

^▼7    Highlights  the  numerator  of  the  first  fraction 

CD    Highlights  denominator  of  the  first  fraction 

^▼7    Highlights  first  term  in  denominator  of  first  fraction 

CD    Highlights  second  term  in  denominator  of  first  fraction 

<^3?    Highlights  first  factor  in  second  term  in  denominator  of  first  fraction 

CD    Highlights  expression  in  parentheses  in  denominator  of  first  fraction 


Since  this  is  the  sub-expression  we  want  evaluated,  we  can  now  press  the 
OEEfl (jl.  soft  menu  key,  resulting  in:  


EaaEnaBgiBaiiMHiHEBini 

A  symbolic  evaluation  once  more.  Suppose  that,  at  this  point,  we  want  to 
evaluate  the  left-hand  side  fraction  only.  Press  the  upper  arrow  key  (<^g^ ) 
three  times  to  select  that  fraction,  resulting  in: 


Page  2-15 


iaaaEnaBgiBaiiMHiHEBina 

Then,  press  the  B!!I;I!!I  C^)  soft  menu  key  to  obtain: 


5 

4 

2 

TT  +15 

EaaEnaBgiBaiiMHiHEBini 

Let's  try  a  numerical  evaluation  of  this  term  at  this  point.  Use  L    j  *NUMs  to 
obtain:   

HflllWIMIfrfrCl^llflHfflBHl 

Let's  highlight  the  fraction  to  the  right,  and  obtain  a  numerical  evaluation  of 

that  term  too,  and  show  the  sum  of  these  two  decimal  values  in  small-font 
format  by  using:(T)  (j^J ^num  [IT)  f  we  get:  

^aaBnaBaBEaBiBEBini 

To  highlight  and  evaluate  the  expression  in  the  Equation  Writer  we  use:  ^> 
(jD  /  resulting  in: 

Editing  arithmetic  expressions 

We  will  show  some  of  the  editing  features  in  the  Equation  Writer  as  an 
exercise.  We  start  by  entering  the  following  expression  used  in  the  previous 
exercises: 


5 

-a 

f  £1 

5+2- 

5+^°- 

,     2  , 

H44iHN£ll4*EI4riai*HHfeH:il 

Page  2-16 


And  will  use  the  editing  features  of  the  Equation  Editor  to  transform  it  into  the 
following  expression: 


EaaEnaBgiBaiiMHiHEBini 

In  the  previous  exercises  we  used  the  arrow  keys  to  highlight  sub-expressions 
for  evaluation.  In  this  case,  we  will  use  them  to  trigger  a  special  editing  cursor. 
After  you  have  finished  entering  the  original  expression,  the  typing  cursor  (a 
left-pointing  arrow)  will  be  located  to  the  right  of  the  3  in  the  denominator  of 
the  second  fraction  as  shown  here: 


5         .  1 

5+2- 

f  21 
5  2 

EanEnaBglHEBBiBHBini 

Press  the  down  arrow  key        )  to  trigger  the  clear  editing  cursor.  The  screen 

now  looks  like  this: 


EffllEnaBglHEBBiBHBini 

By  using  the  left  arrow  key  (CD)  you  can  move  the  cursor  in  the  general  left 

direction,  but  stopping  at  each  individual  component  of  the  expression.  For 
example,  suppose  that  we  will  first  will  transform  the  expression  n2/2  into  the 
expression  LNl^/S)  .  With  the  clear  cursor  active,  as  shown  above,  press 
the  left-arrow  key  (CD)  twice  to  highlight  the  2  in  the  denominator  of  n2/2. 
Next,  press  the  delete  key  (HU)  once  to  change  the  cursor  into  the  insertion 
cursor.  Press  L  +  J  once  more  to  delete  the  2,  and  then  L  3  j  to  enter  a  3.  At 
this  point,  the  screen  looks  as  follows: 


-4 


5+2> 


5+^ 
34  J 


EBnBnaeaiiffliiMfliHHBHi 


Page  2-17 


Next,  press  the  down  arrow  key        )  to  trigger  the  clear  editing  cursor 
highlighting  the  3  in  the  denominator  of  n2/3.  Press  the  left  arrow  key  ((3D) 
once  to  highlight  the  exponent  2  in  the  expression  n2/3.  Next,  press  the 
delete  key  (L4J)  once  to  change  the  cursor  into  the  insertion  cursor.  Press 

J  once  more  to  delete  the  2,  and  then  L  5  J  to  enter  a  5.  Press  the  upper 
arrow  key         three  times  to  highlight  the  expression  n5/3.  Then,  type 

L  r>  J  m  to  apply  the  LN  function  to  this  expression.  The  screen  now  looks 

like  this: 


5 

5+2- 

-* 

H4tiHII«l4frEII4rifll*MM£H:il 

)  within  th 

e  parentheses  to  a  V2  by  using  these 

keystrokes :  CT)       CS  CD  CD  CD 

Next,  we  highlight  the  entire  expression  in  parentheses  an  insert  the  square 
root  symbol  by  using:  S+>^  fA^jix  j 

Next,  well  convert  the  2  in  front  of  the  parentheses  in  the  denominator  into  a 
2/3  by  using:  (S(3D^3^3^3(J3 
At  this  point  the  expression  looks  as  follows: 


EaaBnaBgnfflnMBiBEBini 

The  final  step  is  to  remove  the  1/3  in  the  right-hand  side  of  the  expression. 
This  is  accomplished  by  using:  ^^^^^CDCSCSCSCSCS 
The  final  version  will  be: 


In  summary,  to  edit  an  expression  in  the  Equation  Writer  you  should  use  the 
arrow  keys  (GO  CD  <^g^  ^7? )  to  highlight  expression  to  which  functions  will  be 
applied  (e.g.,  the  LN  and  square  root  cases  in  the  expression  above).  Use  the 


Page  2-18 


down  arrow  key  (<\3?)  in  any  location,  repeatedly,  to  trigger  the  clear  editing 
cursor.  In  this  mode,  use  the  left  or  right  arrow  keys  (CD  CD )  to  move  from 
term  to  term  in  an  expression.  When  you  reach  a  point  that  you  need  to  edit, 
use  the  delete  key  (L4J)  to  trigger  the  insertion  cursor  and  proceed  with  the 
edition  of  the  expression. 


Creating  algebraic  expressions 

An  algebraic  expression  is  very  similar  to  an  arithmetic  expression,  except 
that  English  and  Greek  letters  may  be  included.  The  process  of  creating  an 
algebraic  expression,  therefore,  follows  the  same  idea  as  that  of  creating  an 
arithmetic  expression,  except  that  use  of  the  alphabetic  keyboard  is  included. 


To  illustrate  the  use  of  the  Equation  Writer  to  enter  an  algebraic  equation  we 
will  use  the  following  example.  Suppose  that  we  want  to  enter  the  expression: 

Use  the  following  keystrokes: 

CZD  C±D  SO]  CDCD  CB  (^)CS(S1C±D  QD^_CB  ^)CB(S 
CDCD  CKD  O— ^  (^GD(3  GD  CD       (^CB®  CX3  C^CB© 

(^ED(3  CB  (^CB<3  CZDCDC±DCZD 

This  results  in  the  output: 


EanEnaBglHEBBiBHBini 

In  this  example  we  used  several  lower-case  English  letters,  e.g.,  x 

((ALPHA}[^)(x})f  several  Greek  letters,  e.g.,  X  (S™)CS(S1),  and  even  a 
combination  of  Greek  and  English  letters,  namely,  Ay  ({alpha) [  r>  J (c\ 
[alpha) l <n )(?)  ).    Keep  in  mind  that  to  enter  a  lower-case  English  letter,  you  need 
to  use  the  combination:  [alpha)[j^J  followed  by  the  letter  you  want  to  enter. 
Also,  you  can  always  copy  special  characters  by  using  the  CHARS  menu 
(I  r>  j  chars  )  if  you  don't  want  to  memorize  the  keystroke  combination  that 
produces  it.    A  listing  of  commonly  used  (alpha)  jr> )  keystroke  combinations 
was  listed  in  an  earlier  section. 


Page  2-19 


The  expression  tree 

The  expression  tree  is  a  diagram  showing  how  the  Equation  Writer  interprets 
an  expression.    See  Appendix  E  for  a  detailed  example. 


The  CURS  function 

The  CURS  function  (ililillllii!)  in  the  Equation  Writer  menu  (the  CZD  key)  converts 
the  display  into  a  graphical  display  and  produces  a  graphical  cursor  that  can 
be  controlled  with  the  arrow  keys  (GO QD      <\7? )  for  selecting  sub- 
expressions. The  sub-expression  selected  with  illEil!  will  be  shown  framed  in 
the  graphics  display.  After  selecting  a  sub-expression  you  can  press  [enter]  to 
show  the  selected  sub-expression  highlighted  in  the  Equation  writer.  The 
following  figures  show  different  sub-expressions  selected  with  and  the 


((y-3>x+5>(x2+4) 

fa-3>x+5)(x2+4) 

|SIHC4-x-2)| 

5IHi:4.x-2:iBMWl 

EDIT  |  CURS 

SIN(4'X-2) 


>(,2+4) 


SIHC4-X-2) 


C(ij-3>x+5)(Bh) 
SIHC4-X-2) 


((y-3>x+5)M+4) 
SIHC4-X-2) 


EDIT   CURS   RIG  -  E'.'HL  FACTO  SIHP 


Editing  algebraic  expressions 

The  editing  of  algebraic  equations  follows  the  same  rules  as  the  editing  of 
algebraic  equations.  Namely: 

•  Use  the  arrow  keys  (CT)  CD <^> ^3? )  to  highlight  expressions 

•  Use  the  down  arrow  key  (*\3? ),  repeatedly,  to  trigger  the  clear  editing 
cursor  .  In  this  mode,  use  the  left  or  right  arrow  keys  (ODCD )  to 
move  from  term  to  term  in  an  expression. 


Page  2-20 


•     At  an  editing  point,  use  the  delete  key  (L4J)  to  trigger  the  insertion 
cursor  and  proceed  with  the  edition  of  the  expression. 
To  see  the  clear  editing  cursor  in  action,  let's  start  with  the  algebraic 
expression  that  we  entered  in  the  exercise  above: 


8 


3* 


BaaEnaBgaBEaHiHEBini 

Press  the  down  arrow  key,  ^3?,  at  its  current  location  to  trigger  the  clear 
editing  cursor.   The  3  in  the  exponent  of  6^  will  be  highlighted.  Use  the  left 
arrow  key,  (3D ,  to  move  from  element  to  element  in  the  expression.  The  order 
of  selection  of  the  clear  editing  cursor  in  this  example  is  the  following  (press 
the  left  arrow  key,  CD,  repeatedly): 

1 .   The  1  in  the  1/3  exponent 

e 

Ay 


2. 
3. 
4. 
5. 
6. 
7. 


jli  in  the  exponential  function 


X 


9.  3  in  the  V3  term 

10.  the  2  in  the  2/V3  fraction 

At  any  point  we  can  change  the  clear  editing  cursor  into  the  insertion  cursor 
by  pressing  the  delete  key  (L4  J).  Let's  use  these  two  cursors  (the  clear  editing 
cursor  and  the  insertion  cursor)  to  change  the  current  expression  into  the 
following: 


EaaEnaBgiBaiiMHiHEBini 

If  you  followed  the  exercise  immediately  above,  you  should  have  the  clear 
editing  cursor  on  the  number  2  in  the  first  factor  in  the  expression.  Follow 
these  keystrokes  to  edit  the  expression: 


Page  2-21 


CD  l^^)LrLJL2j  Enters  the  factorial  for  the  3  in  the  square  root 

(entering  the  factorial  changes  the  cursor  to  the  selection  cursor) 
^3?*^?  CD  CD  Selects  the  ju  in  the  exponential  function 

L±JLiJL2lJ^™JLilJ(3    Modifies  exponential  function  argument 
CD  CD  CD  CD  Selects  Ay 

l^LJ                              Places  a  square  root  symbol  on  Ay 
(this  operation  also  changes  the  cursor  to  the  selection  cursor) 
^P^PCD^^  (JnJ  Select  e1/3  and  enter  the  SIN  function 
The  resulting  screen  is  the  following:  


SIH 

r  r 

nTiinTEairmranirawraH™ 

Evaluating  a  sub-expression 

Since  we  already  have  the  sub-expression  SIN(0l/3^  highlighted,  let's  press 


IsiNtsje1)! 

mnnimmDnnnixniOBini 

Some  algebraic  expressions  cannot  be  simplified  anymore.  Try  the  following 
keystrokes:  <^>(2D  •  You  will  notice  that  nothing  happens,  other  than  the 
highlighting  of  the  entire  argument  of  the  LN  function.  This  is  because  this 
expression  cannot  be  evaluated  (or  simplified)  any  more  according  to  the 
CAS  rules.  Trying  the  keystrokes:  {±^>CjD  again  does  not  produce  any 
changes  on  the  expression.  Another  sequence  of  ^A^fTT]  keystrokes, 
however,  modifies  the  expression  as  follows: 


BIG  "|  EYALlFflCToT^^ 


One  more  application  of  the  (j^>(jD  keystrokes  produces  more  changes: 


Page  2-22 


■LH 

_i±_ 

-    3-P  1 

This  expression  does  not  fit  in  the  Equation  Writer  screen.  We  can  see  the 
entire  expression  by  using  a  smaller-size  font.  Press  the  ElE  C~^~~]  soft  menu 
key  to  get: 


?>i_n 

M. 

EaaEnaBaBEaHiHEBini 

Even  with  the  larger-size  font,  it  is  possible  to  navigate  through  the  entire 
expression  by  using  the  clear  editing  cursor.  Try  the  following  keystroke 
sequence:  QD  ^3?  ^3?  ^3?  ^3? ,  to  set  the  clear  editing  cursor  atop  the  factor 
3  in  the  first  term  of  the  numerator.  Then,  press  the  right  arrow  key,  CD ,  to 
navigate  through  the  expression. 


Simplifying  an  expression 

Press  the  E33(jlD  soft  menu  key  to  get  the  screen  to  look  as  in  the  previous 
figure  (see  above).  Now,  press  the  lliiillll  (jD  soft  menu  key,  to  see  if  it  is 
possible  to  simplify  this  expression  as  it  is  shown  in  the  Equation  Writer.  The 
result  is  the  following  screen: 


EffllBIiaMHMHBMlgBHHini 

This  screen  shows  the  argument  of  the  SIN  function,  namely,  \J~6  , 

LN{6) 

transformed  into  e   3    .  This  may  not  seem  like  a  simplification,  but  it  is  in 
the  sense  that  the  cubic  root  function  has  been  replaced  by  the  inverse 
functions  exp-LN. 


Page  2-23 


Factoring  an  expression 

In  this  exercise  we  will  try  factoring  a  polynomial  expression.  To  continue  the 
previous  exercise,  press  the  (enter}  key.  Then,  launch  the  Equation  Writer  again 
by  pressing  the  IsD_eqw  key.  Type  the  equation: 

resulting  in 

I       2  2    2  2 

X  +2-X-Y+Y  -*  +p 

Let's  select  the  first  3  terms  in  the  expression  and  attempt  a  factoring  of  this 
sub-expression:  G^t^^s^CBCDCBCE)  •  This  produces: 

baaEnaBgiBaiiMHiHEEina 

Now,  press  the  EEBB  soft  menu  key,  to  get  

  ■^*iMiiMia<yci4irTi^mafcn:id 

Press  Ij^Jundo  to  recover  the  original  expression.  Next,  enter  the  following 
keystrokes:  ^^^CD(J^CDCD(J^(J^C^  to  select 

the  last  two  terms  in  the  expression,  i.e., 

^aaBnaBgiEEaaiBEBini 

press  the  llljllim  soft  menu  key,  to  get 


x2+2-Y-x+v2Hffl=kljHia 
  EanEnaBguaaHfliHHBini 

Press  [JfJuNDo  to  recover  the  original  expression.  Now,  let's  select  the  entire 
expression  by  pressing  the  upper  arrow  key  (<^g^ )  once.    And  press  the 
!IIi!IIS!l  soft  menu  key,  to  get 


Page  2-24 


;+y- 


Press  L  r>  j  UNDOs  to  recover  the  original  expression. 


Note:  Pressing  the  IDEE!!  or  the        soft  menu  keys,  while  the  entire  original 
expression  is  selected,  produces  the  following  simplification  of  the  expression: 

iron  nn^iOTE  iron  mHHBna 

Using  the  CMDS  menu  key 

With  the  original  polynomial  expression  used  in  the  previous  exercise  still 
selected,  press  the  [nxtJ  key  to  show  the  illiliii  and  lEIHIIII  soft  menu  keys. 
These  two  commands  belong  to  the  second  part  of  the  soft  menu  available 
with  the  Equation  Writer.    Let's  try  this  example  as  an  application  of  the  iulill 


ABCUY 

HDDTHuD 

hDDTOREhL 

HL'jE: 

Next,  select  the  command  DERVX  (the  derivative  with  respect  to  the  variable  X, 
the  current  CAS  independent  variable)  by  using:  ^™J(§]^3?^?^7  . 
Command  DERVX  will  now  be  selected: 


DEF  [ 
DEGREE  i 
DEF;  I'.' 

DESOLVE 

DIhGMhF-  [ 

Press  the  IIE33II  soft  menu  key  (C^Q  ),  to  get 


J,  roget. 


Page  2-25 


Next,  press  the  jNxrj  key  to  recover  the  original  Equation  Writer  menu,  and 
press  the  IIISE!!!!!  soft  menu  key  ((JL) )  to  evaluate  this  derivative.    The  result  is: 

Using  the  HELP  menu 

Press  the  (W)  key  to  show  the  BUS  and  lOHIill  soft  menu  keys.  Press  the  CUD 
soft  menu  key  to  get  the  list  of  CAS  commands.  Then,  press  {alpha}  @  ^v?  ^v? 
<\j?  to  select  the  command  DERVX.  Press  the  !!!!!:![!:!!!!!  soft  menu  key  (QD  ),  to  get 
information  on  the  command  DERVX: 

Returns  the  deriuatiue 
with  respect  to  the 
current  variable 

dervx<ln<<x+i>/<x-i>>> 

See:  DERIV  IHTVX 

Detailed  explanation  on  the  use  of  the  help  facility  for  the  CAS  is  presented  in 
Chapter  1 .  To  return  to  the  Equation  Writer,  press  the  IIES!  soft  menu  key. 
Press  the  (enter}  key  to  exit  the  Equation  Writer. 

Using  the  editing  functions  BEGIN,  END,  COPY,  CUT  and  PASTE 

To  facilitate  editing,  whether  with  the  Equation  Writer  or  on  the  stack,  the 
calculator  provides  five  editing  functions,  BEGIN,  END,  COPY,  CUT  and 
PASTE,  activated  by  combining  the  right-shift  key  (CB)  with  keys  (2,1),  (2,2), 
(3,1),  (3,2),  and  (3,3),  respectively.    These  keys  are  located  in  the  leftmost 
part  of  rows  2  and  3.  The  action  of  these  editing  functions  are  as  follows: 

BEGIN:  marks  the  beginning  of  a  string  of  characters  for  editing 
END:     marks  the  ending  of  a  string  of  characters  for  editing 
COPY:    copies  the  string  of  characters  selected  by  BEGIN  and  END 
CUT:     cuts  the  string  of  characters  selected  by  BEGIN  and  END 
PASTE:  pastes  a  string  of  characters,  previously  copied  or  cut,  into  the  current 
cursor  position 

To  see  and  example,  lets  start  the  Equation  Writer  and  enter  the  following 
expression  (used  in  an  earlier  exercise): 


Page  2-26 


CD  CD  CDCD  CD  CD  CD  (^CED®  CD  OfL-  CD  SC3(§ 
CDCD  CD  CO— «  (^™)CtD(3  CD  CD  CD  C^CrD®  CD  C^)CD(3 

CDCDS               CD  C™)CD(3  CDCDCOCD 
The  original  expression  is  the  following:  


8 


3* 


BaaEnaBgaBEiMHiHEBini 

We  want  to  remove  the  sub-expression  x+2-X-Ay  from  the  argument  of  the  LN 
function,  and  move  it  to  the  right  of  the  A  in  the  first  term.  Here  is  one 
possibility:  ^CDCSCS^^^CB^CDCSD^CXDCB^ 
The  modified  expression  looks  as  follows: 


EanEnaBglHEBBiBHBini 

Next,  we'll  copy  the  fraction  2/ V3  from  the  leftmost  factor  in  the  expression, 

and  place  it  in  the  numerator  of  the  argument  for  the  LN  function.  Try  the 
following  keystrokes: 

The  resulting  screen  is  as  follows:  


The  functions  BEGIN  and  END  are  not  necessary  when  operating  in  the 
Equation  Writer,  since  we  can  select  strings  of  characters  by  using  the  arrow 
keys.  Functions  BEGIN  and  END  are  more  useful  when  editing  an  expression 
with  the  line  editor.  For  example,  let's  select  the  expression  x+2-A-Ay  from  this 
expression,  but  using  the  line  editor  within  the  Equation  Writer,  as  follows: 

The  line  editor  screen  will  look  like  this  (quotes  shown  only  if  calculator  in  RPN 
mode): 


Page  2-27 


EXP<-ui>*LN<2^J"3^eA<l^ 
aiHiHIHCC'Hll'Hgi'HlUjjU 


To  select  the  sub-expression  of  interest,  use: 

CD  CD  CD  CD  CD  CD  CD  CD 
CD  CD  CD  CD  CD  CD  CD  CD  CD  CDO^S 
The  screen  shows  the  required  sub-expression  highlighted: 


EXP  <  -pl  >  *bl  VTTT^  T: <  I 


We  can  now  copy  this  expression  and  place  it  in  the  denominator  of  the  LN 
argument,  as  follows:C?L}_^:  CD  CD  •••  (27  times)  ...  CD 

C5DCS...  (9  times)  ... 
The  line  editor  now  looks  like  this: 


Pressing  [flvraJ  shows  the  expression  in  the  Equation  Writer  (in  small-font  format, 
press  the  ISBQlD  soft  menu  key): 


Press  (enter}  to  exit  the  Equation  Writer. 


Creating  and  editing  summations,  derivatives,  and  integrals 

Summations,  derivatives,  and  integrals  are  commonly  used  for  calculus, 
probability  and  statistics  applications.  In  this  section  we  show  some  examples 
of  such  operations  created  with  the  equation  writer.  Use  ALG  mode. 

Summations 

We  will  use  the  Equation  Writer  to  enter  the  following  summation: 


Page  2-28 


Press  C  r>  j  eqw  to  activate  the  Equation  Writer.  Then  press  L  r>  J  5  to  enter 

the  summation  sign.  Notice  that  the  sign,  when  entered  into  the  Equation 
Writer  screen,  provides  input  locations  for  the  index  of  the  summation  as  well 
as  for  the  quantity  being  summed.  To  fill  these  input  locations,  use  the 
following  keystrokes: 

^pha]      (k\  CD  CD  CD  CD  °°_  CD  CD  CD  CD  CD  (3  CD  CD 
The  resultinq  screen  is: 

i  * — ;  1 

k=lk 

^aaBnaBgiEEaaiBEBini   

To  see  the  corresponding  expression  in  the  line  editor,  press  C  J</^N>  and  the 
CD  soft  menu  key,  to  show:  


2t!k  =  l?o> 

cimi™i™nra3 

This  expression  shows  the  general  form  of  a  summation  typed  directly  in  the 
stack  or  line  editor: 

H(index  =  starting_value,  ending_value,  summation  expression) 
Press  [enter)  to  return  to  the  Equation  Writer.  The  resulting  screen  shows  the 
value  of  the  summation, 


EanEnaBgiHEBBiBHBini 

To  recover  the  unevaluated  summation  use  [j^Jundo  .  To  evaluate  the 
summation  again,  you  can  use  the  CD  soft  menu  key.  This  shows  again  that 

V  — 

You  can  use  the  Equation  Writer  to  prove  that 

00 

=  +CO  . 

k=\  k 

This  summation  (representing  an  infinite  series)  is  said  to  diverge. 
Double  summations  are  also  possible,  for  example: 


Page  2-29 


1 


n  m 
j=lk=lJ 


^■>41HII:MI:WrPI^J;TIJ;T4f.]i.^j:iJ 


Derivatives 

We  will  use  the  Equation  Writer  to  enter  the  following  derivative: 

— (a>t2+/3>t  +  S) 
dt 

Press  L  r>  j  eqw  to  activate  the  Equation  Writer.  Then  press  L  r>  J  §  to  enter 

the  (partial)  derivative  sign.  Notice  that  the  sign,  when  entered  into  the 
Equation  Writer  screen,  provides  input  locations  for  the  expression  being 
differentiated  and  the  variable  of  differentiation.  To  fill  these  input  locations, 
use  the  following  keystrokes: 

The  resulting  screen  is  the  following: 


6t 


laanEnaBgiHEBBiBHBini   

To  see  the  corresponding  expression  in  the  line  editor,  press  CB^}  and  the 
CjD  soft  menu  key,  to  show:  


This  indicates  that  the  general  expression  for  a  derivative  in  the  line  editor  or 
in  the  stack  is:  dvariable(function  of  variables) 


Press  [enter]  to  return  to  the  Equation  Writer.  The  resulting  screen  is  not  the 
derivative  we  entered,  however,  but  its  symbolic  value,  namely, 


ffl-2-t+P 


EaaBnaBgnfflnMBiBEBini 

To  recover  the  derivative  expression  use  L  r>  J  UNDOs  .  To  evaluate  the  derivative 
again,  you  can  use  the  (JL)  soft  menu  key.  This  shows  again  that 


Page  2-30 


—  (a-t2  -j3-t  +  S)  =  2a-t  +  J3 
dt 

Second  order  derivatives  are  possible,  for  example: 


4^) 


which  evaluates  to: 


EffllEnaBglHEnBiBHBini 


d 

Note:  The  notation  — (  )  is  proper  of  partial  derivatives.  The  proper 

dx 

notation  for  total  derivatives  (i.e.,  derivatives  of  one  variable)  is  — (  ).  The 

dx 

calculator,  however,  does  not  distinguish  between  partial  and  total 
derivatives. 


Definite  integrals 

We  will  use  the  Equation  Writer  to  enter  the  following  definite 

integral:  j  t  •  sin(7)  •  dt .  Press  (JfJ_EQw  to  activate  the  Equation  Writer. 

Then  press  IsD   {  to  enter  the  integral  sign.  Notice  that  the  sign,  when 

entered  into  the  Equation  Writer  screen,  provides  input  locations  for  the  limits 
of  integration,  the  integrand,  and  the  variable  of  integration.  To  fill  these 
input  locations,  use  the  following  keystrokes:CECK)(d^CB(SlCD(^^  C3D 


t-SIN(t)dt 

0 

To  see  the  corresponding  expression  in  the  line  editor,  press  <^^<^^  and  the 
(jD  soft  menu  key,  to  show: 


Page  2-31 


JX0,T,t*SIN<t>,t> 

This  indicates  that  the  general  expression  for  a  derivative  in  the  line  editor  or 
in  the  stack  is:  l(lower_limit,  upper Jimit/integrand,variable_of_integration) 


Press  [enter]  to  return  to  the  Equation  Writer.  The  resulting  screen  is  not  the 
definite  integral  we  entered,  however,  but  its  symbolic  value,  namely, 


EaaEnaBgiBaiiMHiHEBini 

To  recover  the  derivative  expression  use  L  r>  J  UNDOs  .  To  evaluate  the  derivative 
again,  you  can  use  the  GD  soft  menu  key.  This  shows  again  that 


Jo  t  •  sin(7)  •  dt  =  sin(r)  -  r  •  cos(r) 


Double  integrals  are  also  possible.  For  example, 


(x+y)dy  dx4 


-3 


-x 


laaaBnaBgiEEaaiBEBini 

which  evaluates  to  36.    Partial  evaluation  is  possible,  for  example: 


(x+y)dy 


^3 

Hdx 

-3 

H44iHN£ll4*EI4riai*HHfeH:il 

This  integral  evaluates  to  36. 


Organizing  data  in  the  calculator 

You  can  organize  data  in  your  calculator  by  storing  variables  in  a  directory 
tree.  To  understand  the  calculator's  memory,  we  first  take  a  look  at  the  file 
directory.    Press  the  keystroke  combination  [j^Jfiles_  (fjrst  key  in  second  row 
of  keys  from  the  top  of  the  keyboard)  to  get  the  calculator's  File  Manager 
screen: 


Page  2-32 


This  screen  gives  a  snapshot  of  the  calculator's  memory  and  of  the  directory 
tree.  The  screen  shows  that  the  calculator  has  three  memory  ports  (or  memory 
partitions),  port  0:IRAM,  port  1  :ERAM,  and  port  2:FLASH  .  Memory  ports  are 
used  to  store  third  party  application  or  libraries,  as  well  as  for  backups.  The 
size  of  the  three  different  ports  is  also  indicated.  The  fourth  and  subsequent 
lines  in  this  screen  show  the  calculator's  directory  tree.  The  top  directory 
(currently  highlighted)  is  the  Home  directory,  and  it  has  predefined  into  it  a 
sub-directory  called  CASDIR.  The  File  Manager  screen  has  three  functions 
associated  with  the  soft-menu  keys: 

HEH3  (GD  )•  Change  to  selected  directory 

BBSS  (CZD  )'  Cancel  action 

MM  (CjD  V-  Approve  a  selection 

For  example,  to  change  directory  to  the  CASDIR,  press  the  down-arrow  key, 
^▼7  ,  and  press  EEEH3  (CZD  )•  This  action  closes  the  File  Manager  window 
and  returns  us  to  normal  calculator  display.  You  will  notice  that  the  second 
line  from  the  top  in  the  display  now  starts  with  the  characters  {  HOME 
CASDIR  }  indicating  that  the  current  directory  is  CASDIR  within  the  HOME 
directory. 


Functions  for  manipulation  of  variables 

This  screen  includes  20  commands  associated  with  the  soft  menu  keys  that 
can  be  used  to  create,  edit,  and  manipulate  variables.  The  first  six  functions 
are  the  following: 

To  edit  a  highlighted  variable 

To  copy  a  highlighted  variable 

To  move  a  highlighted  variable 

To  recall  the  contents  of  a  highlighted  variable 

To  evaluate  a  highlighted  variable 

To  see  the  directory  tree  where  the  variable  is  contained 
If  you  press  the  [nxt)  key,  the  next  set  of  functions  is  made  available: 

To  purge,  or  delete,  a  variable 


Page  2-33 


SIi!!!i!!Ii         To  rename  a  variable 

To  create  a  new  variable 
To  order  a  set  of  variables  in  the  directory 
To  send  a  variable  to  another  calculator  or  computer 
To  receive  a  variable  from  another  calculator  or  computer 
If  you  press  the  jNxrj  key,  the  third  set  of  functions  is  made  available: 
To  return  to  the  stack  temporarily 
To  see  contents  of  a  variable 
GjEEO         To  edit  contents  of  a  binary  variable  (similar  to  llillii) 
11113333         To  show  the  directory  containing  the  variable  in  the  header 
Provides  a  list  of  variable  names  and  description 
To  sort  variables  according  to  a  sorting  criteria 
If  you  press  the  [nxt)  key,  the  last  set  of  functions  is  made  available: 
To  send  variable  with  X-modem  protocol 
1311111:1         To  change  directory 
To  move  between  the  different  soft  menu  commands,  you  can  use  not  only  the 
NEXT  key  (® ),  but  also  the  PREV  key  (C5D^  ). 

The  user  is  invited  to  try  these  functions  on  his  or  her  own.  Their  applications 
are  straightforward. 

The  HOME  directory 

The  HOME  directory,  as  pointed  out  earlier,  is  the  base  directory  for  memory 
operation  for  the  calculator.  To  get  to  the  HOME  directory,  you  can  press  the 
UPDIR  function  (f Jnj upm  )  --  repeat  as  needed  --  until  the  i  HOME:}  spec  is 
shown  in  the  second  line  of  the  display  header.  Alternatively,  you  can  use 
C3D(hold)  t™_  ,  press  (enter}  if  in  the  algebraic  mode.  For  this  example,  the 
HOME  directory  contains  nothing  but  the  CASDIR.  Pressing  Q^D  will  show 
the  variables  in  the  soft  menu  keys: 


Subdirectories 

To  store  your  data  in  a  well  organized  directory  tree  you  may  want  to  create 
subdirectories  under  the  HOME  directory,  and  more  subdirectories  within 


Page  2-34 


subdirectories,  in  a  hierarchy  of  directories  similar  to  folders  in  modern 
computers.  The  subdirectories  will  be  given  names  that  may  reflect  the 
contents  of  each  subdirectory,  or  any  arbitrary  name  that  you  can  think  of. 

The  CASDIR  sub-directory 

The  CASDIR  sub-directory  contains  a  number  of  variables  needed  by  the 
proper  operation  of  the  CAS  (Computer  Algebraic  System,  see  appendix  C). 
To  see  the  contents  of  the  directory,  we  can  use  the  keystroke  combination: 


(jT]fl££5_  which  opens  the  File  Manager  once  more: 


This  time  the  CASDIR  is  highlighted  in  the  screen.    To  see  the  contents  of  the 
directory  press  the  IIIEOII  (f^O  )  soft  menu  key  or  (enter),  to  get  the  following 
screen : 


mm 

\R  HODULO  inTG  E 

{  ]F;EhLhl"l"UME  LIST  2? 

EffPERIOD  RLG  12 

naHYK  GlmHE  H 

\R  EPS  REAL  10 

The  screen  shows  a  table  describing  the  variables  contained  in  the  CASDIR 
directory.  These  are  variables  pre-defined  in  the  calculator  memory  that 
establish  certain  parameters  for  CAS  operation  (see  appendix  C).    The  table 
above  contains  4  columns: 

•  The  first  column  indicate  the  type  of  variable  (e.g.,  'EG)'  means  an 
equation-type  variable,  |  R  indicates  a  real-value  variable,  {  }  means  a  list, 
nam  means  'a  global  name',  and  the  symbol       represents  a  graphic 
variable. 

•  The  second  column  represents  the  name  of  the  variables,  i.e.,  PRIMIT, 
CASINFO,  MODULO,  REALASSUME,  PERIOD,  VX,  and  EPS. 

•  Column  number  3  shows  another  specification  for  the  variable  type,  e.g., 
ALG  means  an  algebraic  expression,  GROB  stands  for  graphics  object, 
INTG  means  an  integer  numeric  variable,  LIST  means  a  list  of  data, 


Page  2-35 


GNAME  means  a  global  name,  and  REAL  means  a  real  (or  floating-point) 
numeric  variable. 

•     The  fourth  and  last  column  represents  the  size,  in  bytes,  of  the  variable 
truncated,  without  decimals  (i.e.,  nibbles).  Thus,  for  example,  variable 
PERIOD  takes  12.5  bytes,  while  variable  REALASSUME  takes  27.5  bytes 
(1  byte  =  8  bits,  1  bit  is  the  smallest  unit  of  memory  in  computers  and 
calculators). 

CASDIR  Variables  in  the  stack 

Pressing  the  jQN J  key  closes  the  previous  screen  and  returns  us  to  normal 
calculator  display.  By  default,  we  get  back  the  TOOL  menu: 


We  can  see  the  variables  contained  in  the  current  directory,  CASDIR,  by 
pressing  the  G^D  key  (first  key  in  the  second  row  from  the  top  of  the 
keyboard).  This  produces  the  following  screen:  

Pressing  the  jNxrj  key  shows  one  more  variable  stored  in  this  directory: 


•  To  see  the  contents  of  the  variable  EPS,  for  example,  use  LiUB33l.  This 
shows  the  value  of  EPS  to  be  .  0000000001 

•  To  see  the  value  of  a  numerical  variable,  we  need  to  press  only  the  soft 
menu  key  for  the  variable.  For  example,  pressing  93  followed  by  [enter]  f 
shows  the  same  value  of  the  variable  in  the  stack,  if  the  calculator  is  set  to 
Algebraic.  If  the  calculator  is  set  to  RPN  mode,  you  need  only  press  the 
soft  menu  key  for  (enter)  . 

•  To  see  the  full  name  of  a  variable,  press  the  apostrophe  first  L  1  J,  and 
then  the  soft  menu  key  corresponding  to  the  variable.  For  example,  for 
the  variable  listed  in  the  stack  as  PERIO,  we  use:  Ljjlilililli,  which 
produces  as  output  the  string:  5  PERIOD  5  .  This  procedure  applies  to 
both  the  Algebraic  and  RPN  calculator  operating  modes. 

Variables  in  CASDIR 

The  default  variables  contained  in  the  CASDIR  directory  are  the  following: 
PRIMIT  Latest  primitive  (anti-derivative)  calculated,  not  a  default 


Page  2-36 


variable,  but  one  created  by  a  previous  exercise 
CASINFO         a  graph  that  provides  CAS  information 
MODULO         Modulo  for  modular  arithmetic  (default  =  1  3) 
REALASSUME    List  of  variable  names  assumed  as  real  values 
PERIOD  Period  for  trigonometric  functions  (default  =  2n) 

VX  Name  of  default  independent  variable  (default  =  X) 

EPS  Value  of  small  increment  (epsilon),  (default  =  10"10) 

These  variables  are  used  for  the  operation  of  the  CAS. 

Typing  directory  and  variable  names 

To  name  subdirectories,  and  sometimes,  variables,  you  will  have  to  type 
strings  of  letters  at  once,  which  may  or  may  not  be  combined  with  numbers. 
Rather  than  pressing  the  {alpha}  f  [alpha}[j^J/  or     [alpha)[j^J  key  combinations  to 
type  each  letter,  you  can  hold  down  the  {alpha}  key  and  enter  the  various  letter. 
You  can  also  lock  the  alphabetic  keyboard  temporarily  and  enter  a  full  name 
before  unlocking  it  again.  The  following  combinations  of  keystrokes  will  lock 
the  alphabetic  keyboard: 

[alpha} [alpha}  locks  the  alphabetic  keyboard  in  upper  case.  When  locked  in  this 
fashion,  pressing  the  LjnJ  before  a  letter  key  produces  a  lower  case  letter, 
while  pressing  the  LjU  key  before  a  letter  key  produces  a  special  character. 
If  the  alphabetic  keyboard  is  already  locked  in  upper  case,  to  lock  it  in  lower 
case,  type,  {^J(alpha) 

[alpha] [alpha] I <i }  (alpha)  locks  the  alphabetic  keyboard  in  lower  case.  When  locked 
in  this  fashion,  pressing  the  LjnJ  before  a  letter  key  produces  an  upper  case 
letter.  To  unlock  lower  case,  press  {^J(alpha) 

To  unlock  the  upper-case  locked  keyboard,  press  [alpha] 

Let's  try  some  exercises  typing  directory/ variable  names  in  the  stack. 
Assuming  that  the  calculator  is  in  the  Algebraic  mode  of  operation  (although 
the  instructions  work  as  well  in  RPN  mode),  try  the  following  keystroke 
sequences.  With  these  commands  we  will  be  typing  the  words  'MATH', 
'Math',  and  'MatH' 


Page  2-37 


[  ■  )[alpha][alpha](m\(a]  (fj  (h\  [enter] 


The  calculator  display  will  show  the  following  (left-hand  side  is  Algebraic 
mode,  right-hand  side  is  RPN  mode): 


'MATH1 
'Math' 
'MatH' 


£: 

MATH  5= 
4: 

MatH  3: 


MatH 


1  MATH  1 
'Math' 
'MatH' 


Note:  if  system  flag  60  is  set,  you  can  lock  the  alphabetical  keyboard  by  just 
pressing  {alpha}.    See  Chapter  1  for  more  information  on  system  flags. 

Creating  subdirectories 

Subdirectories  can  be  created  by  using  the  FILES  environment  or  by  using  the 
command  CRDIR.  The  two  approaches  for  creating  sub-directories  are 
presented  next. 

Using  the  FILES  menu 

Regardless  of  the  mode  of  operation  of  the  calculator  (Algebraic  or  RPN),  we 
can  create  a  directory  tree,  based  on  the  HOME  directory,  by  using  the 
functions  activated  in  the  FILES  menu.  Press  [j^Jfiles_  to  activate  the  FILES 
menu.  If  the  HOME  directory  is  not  already  highlighted  in  the  screen,  i.e., 


1:IF;hM 
L :  ERflH 
3 : FLASH 


5  File  Hanger  3 


LCflSDIR 


use  the  up  and  down  arrow  keys  ((^aN)<^7  )  to  highlight  it.  Then,  press  the 
EDI!  (C™~)  )  soft  menu  key.  The  screen  may  look  like  this: 


HiHOfj:  3HHH53  I  gg^oH" 


EDIT  |  COPY  |  HOVE  |  RCL  |  E'.'HL  |  TREE 


Page  2-38 


showing  that  only  one  object  exists  currently  in  the  HOME  directory,  namely, 
the  CASDIR  sub-directory.    Let's  create  another  sub-directory  called  MANS 
(for  MANualS)  where  we  will  store  variables  developed  as  exercises  in  this 
manual.  To  create  this  sub-directory  first  enter:  [nxtJ  ((""^H  )  .  This  will 

produce  the  following  input  form: 


[■bject  = 
"MH-Z: 

_  Directory 

Enter  \\m  Object 


The  Object  input  field,  the  first  input  field  in  the  form,  is  highlighted  by  default. 
This  input  field  can  hold  the  contents  of  a  new  variable  that  is  being  created. 
Since  we  have  no  contents  for  the  new  sub-directory  at  this  point,  we  simply 
skip  this  input  field  by  pressing  the  down-arrow  key,  <s3?/  once.  The  Name 
input  field  is  now  highlighted: 


^  VfifilfiELE  §§§§§§§§§§§§§§§§§§§ 
Object  = 


_  Directory 

Enter  variable  naHe 


This  is  where  we  enter  the  name  of  the  new  sub-directory  (or  variable,  as  the 
case  may  be),  as  follows:  {^m\{^m\(^\®^\(s}{m^ 


The  cursor  moves  to  the  _Directory  check  field.  Press  the  iKSIOl  ((JL) )  soft  menu 
key  to  specify  that  you  are  creating  a  directory,  and  press  I31III  to  exit  the 
input  form.    The  variable  listing  for  the  HOME  directory  will  be  shown  in  the 
screen  as  follows: 

-ffeHory:  3HH5ii  I  Select: 


The  screen  indicates  that  there  is  a  new  directory  (MANS)  within  the  HOME 
directory. 


Page  2-39 


Next,  we  will  create  a  sub-directory  named  INTRO  (for  INTROduction),  within 
MANS,  to  hold  variables  created  as  exercise  in  subsequent  sections  of  this 
chapter.  Press  the  (jwj  key  to  return  to  normal  calculator  display  (the  TOOLS 
menu  will  be  shown).  Then,  press  L^iJ  to  show  the  HOME  directory  contents 
in  the  soft  menu  key  labels.  The  display  may  look  like  this  (if  you  have 
created  other  variables  in  the  HOME  directory  they  will  show  in  the  soft  menu 
key  labels  too): 


To  move  into  the  MANS  directory,  press  the  corresponding  soft  menu  key 
(C2lD  m  this  case),  and  [enter]  if  in  algebraic  mode.  The  directory  tree  will  be 
shown  in  the  second  line  of  the  display  as  i  MflHS}  .    However,  there 

will  be  no  labels  associated  with  the  soft  menu  keys,  as  shown  below, 
because  there  are  no  variables  defined  within  this  directory. 
Let's  create  the  sub-directory  INTRO  by  using: 

CjT]  FILE^_  Mm  {wo)  101  ^T?  {ALPHA}  {ALPHA}  Q(n\(t](r\(o\  (ENTER)         ;  ]  Tjjjjj  MM 

Press  the  (jwj  key,  followed  by  the  IjwJ  key,  to  see  the  contents  of  the  MANS 
directory  as  follows: 


Press  the  EEEEH  soft  menu  key  to  move  into  the  INTRO  sub-directory.  This  will 
show  an  empty  sub-directory.  Later  on,  we  will  do  some  exercises  in  creating 
variables. 

Using  the  command  CRDIR 

The  command  CRDIR  can  be  used  to  create  directories.  This  command  is 
available  through  the  command  catalog  key  (the  CgJ_c^  key,  second  key  in 
fourth  row  of  keys  from  the  top  of  the  keyboard),  through  the  programming 
menus  (the  C5D^—  key,  same  key  as  the  Lr±J_^z;  key),  or  by  simply  typing  it. 

•  Through  the  catalog  key 

Press  C3_cAr  {Aumj©  .  Use  the  up  and  down  arrow  keys  (fA^^r?)  to 
locate  the  CRDIR  command.  Press  the  IIIEII1  soft  menu  key  to  activate  the 
command. 

•  Through  the  programming  menus 

Press  CSD5?G_  .  This  will  produce  the  following  pull-down  menu  for 
programming: 


Page  2-40 


PROG  HERU 

a .  HEHOR  v..  |! 
3 .  BRRRCH.. 
H .  TEST.. 

5.  TYPE.. 

6.  LIST..  1 

Use  the  down  arrow  key  (<\3?)  to  select  the  option  2.  MEMORY...  ,  or  just 


press  L2J  •  Then,  press  I!!!!! 

!!!!!.  This  will  produce 

thef 

menu: 

HEHORY  HERD 

2.  HEH          *  { 

3 .  BYTES 

H.REHOE  1 
5.  DIRECTORY.. 
e.RRITHHETIC. 

Use  the  down  arrow  key  (<\3?)  to  select  the  5.  DIRECTORY  option,  or  just 
press  l_5J  •  Then,  press  131!!!.  This  will  produce  the  following  pull-down 
menu: 


DIRECTORY  HERU 

2 . RCL  I 

3.ST0 

H.PRTH 

S . CRDIR 

C.PGDIR  l 

Use  the  down  arrow  key  (<^3?)  to  select  the  5.  CRDIR  option,  and  press 


Command  CRDIR  in  Algebraic  mode 

Once  you  have  selected  the  CRDIR  through  one  of  the  means  shown  above, 
the  command  will  be  available  in  your  stack  as  follows: 


CRDIRO 


At  this  point,  you  need  to  type  a  directory  name,  say  chop!  : 

[alpha]  [alpha]  I  <i  j  [alpha)  g)  (h\  (3  (pjj  I  )  {alpha)  [enter] 

The  name  of  the  new  directory  will  be  shown  in  the  soft  menu  keys,  e.g., 

p  LKUlKL'chapl'J  1 

HOVflL 


Page  2-41 


Command  CRDIR  in  RPN  mode 

To  use  the  CRDIR  in  RPN  mode  you  need  to  have  the  name  of  the  directory 

already  available  in  the  stack  before  accessing  the  command.  For  example: 

[alpha)  [alpha)  [  <i  j  [alpha)  g)  (Rj  (a]  (p)  I  2  )  [alpha)  [enter) 

Then  access  the  CRDIR  command  by  either  of  the  means  shown  above,  e.g., 
through  the  Lr>J  cat  key: 


7: 
&: 
5: 
4: 
3: 
£: 
l: 

CATALOG :  F63  COHHftnDS 

C R OS "  fc^-^-OTT! 

CSHP 
CURL 

CVCLOTOHIC 
CVLin 

p2' 

Press  the  IliEII  soft  menu  key  to  activate  the  command,  to  create  the  sub- 
directory: 


Moving  among  subdirectories 

To  move  down  the  directory  tree,  you  need  to  press  the  soft  menu  key 
corresponding  to  the  sub-directory  you  want  to  move  to.  The  list  of  variables 
in  a  sub-directory  can  be  produced  by  pressing  the  G^D  (VARiables)  key.  To 
move  up  in  the  directory  tree,  use  the  function  UPDIR,  i.e.,  enter  LjnJ^™-  • 

Alternatively,  you  can  use  the  FILES  menu,  i.e.,  press  l<n  j  files   m  Use  the  up 
and  down  arrow  keys  (^^^7 )  to  select  the  sub-directory  you  want  to  move 
to,  and  then  press  the  I3II1I:!  (CHange  DIRectory)  orCZD  soft  menu  key.  This 
will  show  the  contents  of  the  sub-directory  you  moved  to  in  the  soft  menu  key 
labels. 

Deleting  subdirectories 

To  delete  a  sub-directory,  use  one  of  the  following  procedures: 
Using  the  FILES  menu 

Press  the  l<n )F,ILES    key  to  trigger  the  FILES  menu.  Select  the  directory 
containing  the  sub-directory  you  want  to  delete,  and  press  the  Elllllli!!  if  needed. 
This  will  close  the  FILES  menu  and  display  the  contents  of  the  directory  you 
selected.    In  this  case  you  will  need  to  press  [enter)  .    Press  the  I31III  soft  menu 


Page  2-42 


key  to  list  the  contents  of  the  directory  in  the  screen.  Select  the  sub-directory 
(or  variable)  that  you  want  to  delete.  Press  (j^IQMSI.  A  screen  similar  to  the 
following  will  be  shown: 


■S2" 

Are  You  Sure? 


The  'S2'  string  in  this  form  is  the  name  of  the  sub-directory  that  is  being 
deleted.    The  soft  menu  keys  provide  the  following  options: 

(CjD  )    Proceed  with  deleting  the  sub-directory  (or  variable) 
(GD  )    Proceed  with  deleting  all  sub-directories  (or  variables) 
EEI333    (CZD  )    Do  not  delete  sub-directory  (or  variable)  from  a  list 

(CfL) )    Do  not  delete  sub-directory  (or  variable) 
After  selecting  one  of  these  four  commands,  you  will  be  returned  to  the  screen 
listing  the  contents  of  the  sub-directory.  The  \SMM  command,  however,  will 
show  an  error  message: 


A  Interrupted 


and  you  will  have  to  press  lillli,  before  returning  to  the  variable  listing. 
Using  the  command  PGDIR 

The  command  PGDIR  can  be  used  to  purge  directories.  Like  the  command 
CRDIR,  the  PGDIR  command  is  available  through  the  (j^J_cat  or  through  the 
L*i  )P,RG    key,  or  it  can  simply  be  typed  in. 

•  Through  the  catalog  key 

Press  CgJ_^i;  (alpha} {alpha} (p](gJ  .  This  should  highlight  the  PGDIR  command. 
Press  the  liill!!!  soft  menu  key  to  activate  the  command. 

•  Through  the  programming  menus 

Press  (jnJPRG_  •  This  will  produce  the  following  pull-down  menu  for 
programming: 


Page  2-43 


PROG  HERU 

3.HEH0RY..  ! 

?■ .  BRRRCH.. 
H .  TEST.. 

5 .  TYPE.. 

6.  LIST..  L 

Use  the  down  arrow  key  )  to  select  the  option  2.  MEMORY...  Then, 
press  139 .  This  will  produce  the  following  pull-down  menu: 


HEHORY  HERU 


2.  MEM 

3.  BYTES 
H.REHOP- 

5. DIRECTORY.. 
e.ARITHHETIC. 


Use  the  down  arrow  key  to  select  the  5.  DIRECTORY  option.  Then, 
press  199.  This  will  produce  the  following  pull-down  menu: 


DIRECTORY  HERU 


a.RCL 
3.ST0 
H.PRTH 

5 .  CRDIR 

6.  PGDIR 


Use  the  down  arrow  key  (<\3?)  to  select  the  6.  PGDIR  option,  and  press 


Command  PGDIR  in  Algebraic  mode 

Once  you  have  selected  the  PGDIR  through  one  of  the  means  shown  above, 
the  command  will  be  available  in  your  stack  as  follows: 

:,GDIRO  ^ 


At  this  point,  you  need  to  type  the  name  of  an  existing  directory,  say  S4  : 

[alpha}  {s}  C4~3 [enter] 

As  a  result,  sub-directory  llllll  is  deleted:  


PGDIRCS41) 


HOVflL 


Instead  of  typing  the  name  of  the  directory,  you  can  simply  press  the 
corresponding  soft  menu  key  at  the  listing  of  the  PGDIR(  )  command,  e.g., 


Page  2-44 


CATALOG :  F63  COHHflnDS 

PERH 

PEVflL 

PICK 

PICKS 

PICT 

Vfll 

Press  iiiiiliEliiiiii,  to  get: 


PGDIRCS4') 
PGDIRO 


HOVflL 


Then,  press  lil!  to  enter  'S3'  as  the  argument  to  PGP 


:PGDIRCS4') 

HOVflL 

PGDIRCS34 

R. 


Press  Icnter)  to  delete  the  sub-directory: 


:PGDIRC'S4') 

HOVflL 

:PGDIRC'S3') 

HOVflL 

Command  PGDIR  in  RPN  mode 

To  use  the  PGDIR  in  RPN  mode  you  need  to  have  the  name  of  the  directory, 
between  quotes,  already  available  in  the  stack  before  accessing  the  command. 
For  example:  CZDS^tHCZDQw^ 

Then  access  the  PGDIR  command  by  either  of  the  means  shown  above,  e.g., 


directory: 


&: 
5: 

CATALOG :  F63  COHHflnDS 

PERH 
PEVflL 

4: 

3: 
£: 
l: 

PICK 

PICKS 

PICT 

R?  1 

<ey  to  activate  the  command  c 

Page  2-45 


Using  the  PURGE  command  from  the  TOOL  menu 

The  TOOL  menu  is  available  by  pressing  the  Itqql J  key  (Algebraic  and  RPN 
modes  shown): 


EDIT   VIEH  Il'ThlKI  RCL   PURGE  CLEAR 


EDIT   VIEH  STACK   RCL   PURGE  CLEAR 


The  PURGE  command  is  available  by  pressing  the  DQ333  soft  menu  key 
In  the  following  examples  we  want  to  delete  sub-directory  SI: 

•  Algebraic  mode:    Enter  03333  [W^mM^m) 

•  RPN  mode:  Enter  CWT  nniliilll  (enter) M n^ff^rW) 


:czd). 


Variables 

Variables  are  like  files  on  a  computer  hard  drive.  One  variable  can  store  one 
object  (numerical  values,  algebraic  expressions,  lists,  vectors,  matrices, 
programs,  etc).  Even  sub-directories  can  be  through  of  as  variables  (in  fact,  in 
the  calculator,  a  subdirectory  is  also  a  type  of  calculator  object). 

Variables  are  referred  to  by  their  names,  which  can  be  any  combination  of 
alphabetic  and  numerical  characters,  starting  with  a  letter  (either  English  or 
Greek).  Some  non-alphabetic  characters,  such  as  the  arrow  (-^)  can  be  used 
in  a  variable  name,  if  combined  with  an  alphabetical  character.  Thus,  '-»A' 
is  a  valid  variable  name,  but '— »'  is  not.  Valid  examples  of  variable  names 
are:  'A',  'B',  'a',  'b',  'a',  'p',  'AT,  'AB12',  '->A1 2,/Vel,/Z0,,,zl,/  etc. 

A  variable  can  not  have  the  same  name  than  a  function  of  the  calculator.  You 
can  not  have  a  SIN  variable  for  example  as  there  is  a  SIN  command  in  the 
calculator.  The  reserved  calculator  variable  names  are  the  following: 
ALRMDAT,  CST,  EQ,  EXPR,  I  ERR,  IOPAR,  MAXR,  MINR,  PICT,  PPAR,  PRTPAR, 
VPAR,  ZPAR,  der_,  e,  i,  nl ,n2,      si ,  s2,      ZDAT,  ZPAR,  n,  oo 

Variables  can  be  organized  into  sub-directories. 

Creating  variables 

To  create  a  variable,  we  can  use  the  FILES  menu,  along  the  lines  of  the 
examples  shown  above  for  creating  a  sub-directory.  For  example,  within  the 


Page  2-46 


sub-directory  i  HOME  MM  MS  INTRO},  created  in  an  earlier  example,  we 
want  to  store  the  following  variables  with  the  values  shown: 


Name 

Contents 

Type 

A 

12.5 

real 

a 

-0.25 

real 

A12 

3xl05 

real 

Q 

V(m+r)' 

algebraic 

R 

[3,2,1] 

vector 

zl 

3+5i 

complex 

Pi 

«  ->  r  '7i*rA2'  » 

program 

Using  the  FILES  menu 

We  will  use  the  FILES  menu  to  enter  the  variable  A.  We  assume  that  we  are 
in  the  sub-directory  {  HOME  MANS  INTRO}  .  To  get  to  this  sub-directory, 
use  the  following:  [j^Jfiles_  and  select  the  INTRO  sub-directory  as  shown  in 
this  screen: 


1:IF;hM 

L :  ERliM 

355KB 

2: FLASH 

hHflnS 

L!iiltli|] 
krtSDIR 

Press  113111  to  enter  the  directory.  You  will  get  a  files  listing  with  no  entries 
(the  INTRO  sub-directory  is  empty  at  this  point) 

[UgHory:  3HH653  I  S«Uct:  fl 
rio  Entries 


EDIT   COPY   HOVE    RCL    E'.'HL  TREE 


Press  the  [nxtJ  key  to  move  to  the  next  set  of  soft  menu  keys,  and  press  the 
I0III  soft  menu  key.    This  will  produce  the  NEW  VARIABLE  input  form: 


_  Directory 
Enter  ncu  Object 


Page  2-47 


To  enter  variable  A  (see  table  above),  we  first  enter  its  contents,  namely,  the 
number  1  2.5,  and  then  its  name,  A,  as  follows:  U_JLlJLiJL5j 
mm  (ALP^)(A]mm.  Resultinginjjie  following  screen 


HEW  YftRIftELE»W«^ 

&bj«t:  12.5 

"l']Hs::  fl 

Directory 
Create  a  ncH  directory? 


Press  Iliiilonce  more  to  create  the  variable.  The  new  variable  is  shown  in  the 
following  variable  listing: 


The  listing  indicates  a  real  variable  (|R),  whose  name  is  A,  and  that  occupies 
1 0.5  bytes  of  memory.    To  see  the  contents  of  the  variable  in  this  screen, 
press  (WlBEHI. 

•     Press  the  Ellilll!  (CjQ  )  soft  menu  key  to  see  the  contents  in  a  graphical 
format. 

1275 


•  Press  the  ESED  (CjD  )  soft  menu  key  to  see  the  contents  in  text  format. 

•  Press  199  to  return  to  the  variable  list 

•  Press  tjwj  once  more  to  return  to  normal  display.  Variable  A  should  now 
be  featured  in  the  soft  menu  key  labels: 


Using  the  STO  ►  command 

A  simpler  way  to  create  a  variable  is  by  using  the  STO  command  (i.e.,  the 
[sto>)  key).  We  provide  examples  in  both  the  Algebraic  and  RPN  modes,  by 
creating  the  remaining  of  the  variables  suggested  above,  namely: 


Page  2-48 


Name 

Contents 

Type 

a 

-0.25 

real 

A12 

3xl05 

real 

Q 

V(m+r)' 

algebraic 

R 

[3,2,1] 

vector 

zl 

3+5i 

complex 

pl 

«  ->  r  '7i*rA2'  » 

program 

•    Algebraic  mode 

Use  the  following  keystrokes  to  store  the  value  of  -0.25  into  variable 

a:  CBCUCBCSC^  C^3  (^CrDffl  .  AT  this  point,  the  screen 
will  look  as  follows:  

This  expression  means  that  the  value  -0.25  is  being  stored  into  a  (the 
symbol  ►  suggests  the  operation).  Press  [enter)  to  create  the  variable. 
The  variable  is  now  shown  in  the  soft  menu  key  labels: 

The  following  are  the  keystrokes  required  to  enter  the  remaining 
variables: 

A12:  CEC^CSC^SSCTJCE^ 

Q:  CDeosSOlL-  

5^QD(SCSS^QI)(3CE)CD  S^t^Dg] (enter} 

R:  QD{Z  CTDCB  'CDCB  'CDCD  (^{^(S  (enter) 

zl:  CBCBCSCKDCS;       (^{^(^(S(jJ(ente^  (if  needed, 

accept  change  to  Complex  mode) 

pl:  CB^CS^SEDSCZDED^S 

The  screen,  at  this  point,  will  look  as  follows: 


m+r 

:[3  2  1»R 

C3  2  1] 

:3+5-ifrzl 

3+5- i 

:  #  ■+  r  1 

TT*r^2'  3>Pl 

Page  2-49 


You  will  see  six  of  the  seven  variables  listed  at  the  bottom  of  the 
screen:  p7,  zl,  R,  Q,  A12,  a. 


•  RPNmode 

Use  the  following  keystrokes  to  store  the  value  of  -0.25  into  variable 

a:  QDCZ^CBCSCDt^  (^CBSQwra).  At  this  point,  the 
screen  will  look  as  follows: 


£: 

-.25 

l: 

1  o:  1 

This  expression  means  that  the  value  -0.25  is  ready  to  be  stored  into 
a.  Press  jsroy J  to  create  the  variable.  The  variable  is  now  shown  in 
the  soft  menu  key  labels:  


To  enter  the  value  3x1 05  into  Al  2,  we  can  use  a  shorter  version  of 
the  procedure:  CBC^CSCZJt^fflCDCII]^) 
Here  is  a  way  to  enter  the  contents  of  Q: 

Q:  CDeOSSSlL-  

To  enter  the  value  of  R,  we  can  use  an  even  shorter  version  of  the 
procedure: 

Notice  that  to  separate  the  elements  of  a  vector  in  RPN  mode  we  can 

use  the  space  key  (jspc J ),  rather  than  the  comma  (LiU  »    )  used 

above  in  Algebraic  mode. 

zl:CDQD(SCXD(SCSD^—  CD^JEDSCD  SI  (if 
needed,  accept  change  to  Complex  mode) 

pi:  CS^CS^SCTSCZDED^CxD 
The  screen,  at  this  point,  will  look  as  follows: 


You  will  see  six  of  the  seven  variables  listed  at  the  bottom  of  the 
screen:  p7,  z7,  R,  Q,  A12,  a. 


Page  2-50 


Checking  variables  contents 

As  an  exercise  on  peeking  into  the  contents  of  variables  we  will  use  the  seven 
variables  entered  in  the  exercise  above.  We  showed  how  to  use  the  FILES 
menu  to  view  the  contents  of  a  variable  in  an  earlier  exercise  when  we 
created  the  variable  A.  In  this  section  we  will  show  a  simple  way  to  look  into 
the  contents  of  a  variable. 

Pressing  the  soft  menu  key  label  for  the  variable 

This  procedure  will  show  the  contents  of  a  variable  as  long  as  the  variable 
contains  a  numerical  value  or  an  algebraic  value,  or  an  array.  For  example, 
for  the  variables  listed  above,  press  the  following  keys  to  see  the  contents  of 
the  variables: 

Algebraic  mode 

Type  these  keystrokes:  [^rJMM(enter)  jjjjjj m  [enter) (enter)  .  At  this  point,  the 
screen  looks  as  follows: 


=  Z1 

3+5- i 

:R 

C3  2  1] 

:Q 

r 

m+r 

Next,  type  these  keystrokes:  ElE3I(^)  ^M(enter)  [nxtJ  mm  (enter)  .  At  this  point, 
the  screen  looks  as  follows: 


A. 

m+r 

:fll2 

300000. 

-.25 

12.5 

Pressing  the  soft  menu  key  corresponding  to  pi  will  provide  an  error  message 
(try  (nxt]  lllllll  [enter)  ) ; 


:fl 

300000. 

A  ■*  Error: 
Too  Few 
Arguments 

.25 
2.5 

=  pl 

"Too  Few  Argument s" 

Page  2-51 


Note:  By  pressing         (enter)  we  are  trying  to  activate  (run)  the  pi  program. 
However,  this  program  expects  a  numerical  input.  Try  the  following  exercise: 

®B!E][i  ^JJ(enter)  .  The  result  is: 


:plC5) 

TT-25 

The  program  has  the  following  structure:  «      r  '7i*rA2'  » 
The  ■£  &  symbols  indicate  a  program  in  User  RPL  language  (the  original 
programming  language  of  the  HP  28/48  calculators,  and  available  in  the  HP 
49G  series).  The  characters  ->  r  indicate  that  an  input,  to  be  referred  to  as  r, 
is  to  be  provided  to  the  program.  The  action  from  the  program  is  to  take  that 
value  of  r  and  evaluate  the  algebraic  '7i*rA2'.  In  the  example  shown  above, 
r  took  the  value  of  5,  and  thus  the  value  of  7ir2  =  7i-25  is  returned.  This 
program,  therefore,  calculates  the  area  of  a  circle  given  its  radius  r. 

RPNmode 

In  RPN  mode,  you  only  need  to  press  the  corresponding  soft  menu  key  label  to 
get  the  contents  of  a  numerical  or  algebraic  variable.  For  the  case  under 
consideration,  we  can  try  peeking  into  the  variables  z7,  /?,  Q,  A  7  2,  a,  and  A, 
created  above,  as  follows:    L  m  J  BUI  BUS 311 11815- §111  IKIES  11111 
At  this  point,  the  screen 


ooks  like  this: 

b  = 

5: 

3+5- i 

4: 

C3  2  1] 

3: 

r 

m+r 

£: 

300000. 

l: 

-.25 

To  see  the  contents  of  A,  use:  jNxrj  j 
To  run  program  pi  with  r  =  5,  use:  Ij^JLiJ  P^ri 


b  = 

Id  ^  1J 

5: 

r 

m+r 

4: 

300000. 

3: 

-.25 

2: 

12.5 

l: 

TT-25 

Notice  that  to  run  the  program  in  RPN  mode,  you  only  need  to  enter  the  input 
(5)  and  press  the  corresponding  soft  menu  key.  (In  algebraic  mode,  you  need 
to  place  parentheses  to  enter  the  argument). 


Page  2-52 


Using  the  right-shift  key  C3  followed  by  soft  menu  key  labels 

This  approach  for  viewing  the  contents  of  a  variable  works  the  same  in  both 
Algebraic  and  RPN  modes.  Try  the  following  examples  in  either  mode: 

[j^J  [j-M  ill!!  CxD  !!!!!::!!.!!!!!!  Lr^J  llill  (  rMlllffl  (xLJ  llIEJEli 
This  produces  the  following  screen  (Algebraic  mode  in  the  left,  RPN  in  the 
right) 


b  = 

*  ■*  r  ,TT*r^2'  s- 

5: 

*  +  r  ,TT*r^2'  » 

3+5- i 

4: 

3+5- i 

C3  2  1] 

3: 

C3  2  1] 

r 

2- 

r 

m+r 

m+r 

 30080S. 

l: 

300000. 

Notice  that  this  time  the  contents  of  program  pi  are  listed  in  the  screen.  To 
see  the  remaining  variables  in  this  directory,  use: 

IIS  (W)  C3  111 
Listing  the  contents  of  all  variables  in  the  screen 

Use  the  keystroke  combination  L  r>  J^or?  to  list  the  contents  of  all  variables  in 
the  screen.  For  example: 


*  +  r  'TT*r^2. 
zl:  <3.i5-> 
R:  [3.?2.?1.] 

R12:  300000. 
ot ■  -.25 


Press  jQN J  to  return  to  normal  calculator  display. 


Replacing  the  contents  of  variables 

Replacing  the  contents  of  a  variable  can  be  thought  of  as  storing  a  different 
value  in  the  same  variable  name.  Thus,  the  examples  for  creating  variables 
shown  above  can  be  used  to  illustrate  the  replacement  of  a  variable's  content. 

Using  the  STO^  command 

Using  as  illustration  the  six  variables,  pi,  zl,  R,  Q,  Al 2,  a,  and  A,  created 
earlier,  we  will  proceed  to  change  the  contents  of  variable  A12  (currently  a 
numerical  variable)  with  the  algebraic  expression  'p/2',  using  the  STO^ 
command.  First,  using  the  Algebraic  operating  mode: 

CDt^CS^GDCDCE)  d™D  mEm [enter] 


Page  2-53 


Check  the  new  contents  of  variable  A  72  by  using  L r>  J II 
Using  the  RPN  operating  mode: 

CD^^CS(SC±JCID(^}  CD^Sm (enter)  (sW) 

or,  in  a  simplified  way, 

Using  the  left-shift  C3D  key  followed  by  the  variable's  soft  menu  key  (RPN) 

This  is  a  very  simple  way  to  change  the  contents  of  a  variable,  but  it  only 
works  in  the  RPN  mode.  The  procedure  consists  in  typing  the  new  contents  of 
the  variable  and  entering  them  into  the  stack,  and  then  pressing  the  left-shift 
key  followed  by  the  variable's  soft  menu  key.  For  example,  in  RPN,  if  we 
want  to  change  the  contents  of  variable  z7  to  'a+b-\ ',  use: 

This  will  place  the  algebraic  expression  'a+b-'i '  in  level  7:  in  the  stack.  To 
enter  this  result  into  variable  z7,  use:  L m  J L 5  J IBll 
To  check  the  new  contents  of  z  7,  use:  L  r>  J IIIIIIIB 

An  equivalent  way  to  do  this  in  Algebraic  mode  is  the  following: 

[jT]  ®  C+J  GD  CXD  C5D ;  (^D  CZ^t)  (^D 

To  check  the  new  contents  of  z7,  use:  L  r>  J llllllllllllllllll 

Using  the  ANS(l)  variable  (Algebraic  mode) 

In  Algebraic  mode  one  can  use  the  ANS(1 )  variable  to  replace  the  contents  of 
a  variable.  For  example,  the  procedure  for  changing  the  contents  of  z7  to 
'a+b\'  is  the  following:      {^Jans_  fjroT)  111  [enter]  .  To  check  the  new 
contents  of  z7,  use:  Ll?J IE™ 

Copying  variables 

The  following  exercises  show  different  ways  of  copying  variables  from  one 
sub-directory  to  another. 

Using  the  FILES  menu 

To  copy  a  variable  from  one  directory  to  another  you  can  use  the  FILES  menu. 
For  example,  within  the  sub-directory  {HOME  MANS  INTRO},  we  have 


Page  2-54 


variables  p7,  zl ,  R,  Q,  AT  2,  a,  and  A.  Suppose  that  we  want  to  copy 
variable  A  and  place  a  copy  in  sub-directory  {HOME  MANS}.  Also,  we  will 
copy  variable  R  and  place  a  copy  in  the  HOME  directory.  Here  is  how  to  do 
it:  Press  CZD5^_  JEM  to  produce  the  following  list  of  variables: 

rffgHorn:    31533  I  Select:  fl  I 


Use  the  down-arrow  key  ^3?  to  select  variable  A  (the  last  in  the  list),  then 
press  The  calculator  will  respond  with  a  screen  labeled  PICK 

DESTINATION: 


MRhH 
L :  ERflH 
2: FLASH 

\nnws 

L!iltii!] 
kflSDIR 


PICK  DESTinATIOn 

2  5  5  HE: 
31GKG 


Use  the  up  arrow  key        to  select  the  sub-directory  MANS  and  press  BIS.  If 
you  now  press  l<n  jf™  ,  the  screen  will  show  the  contents  of  sub-directory 
MANS  (notice  that  variable  A  is  shown  in  this  list,  as  expected): 


Press  C^D  Hlllffl  (^(Algebraic  mode),  or  Q^D  IliMilli  (RPN  mode)  to  return 
to  the  INTRO  directory.  Press  CZD^L_  BIB  to  produce  the  list  of  variables  in 
{HOME  MANS  INTRO}.  Use  the  down  arrow  key  (^7 )  to  select  variable  R, 
then  press  Eillillilll.  Use  the  up  arrow  key  (<^aN> )  to  select  the  HOME  directory, 
and  press  BIB.  If  you  now  press  LjnJ^™-  ,  twice,  the  screen  will  show  the 
contents  of  the  HOME  directory,  including  a  copy  of  variable  R: 


□CflSDIR 


DIR 


3H5 


EDIT   COPY   HOVE    RCL    E'.'HL  TREE 


Page  2-55 


Using  the  history  in  Algebraic  mode 

Here  is  a  way  to  use  the  history  (stack)  to  copy  a  variable  from  one  directory 
to  another  with  the  calculator  set  to  the  Algebraic  mode.  Suppose  that  we  are 
within  the  sub-directory  {HOME  MANS  INTRO},  and  want  to  copy  the 
contents  of  variable  zl  to  sub-directory  {HOME  MANS}.  Use  the  following 
procedure:  LrLJ BKiro^J B3It^?J  This  simply  stores  the  contents  of  z?  into 
itself  (no  change  effected  on  zl).  Next,  use  {j^Jupdr_  [enter)  to  move  to  the 


a+i-b 

:flNS(l)*zl 

a+i-b 

:UPDIR 

HOVflL 

Next,  use  the  delete  key  three  times,  to  remove  the  last  three  lines  in  the 
display:  (3D  L*J  L*J  •  At  this  point,  the  stack  is  ready  to  execute  the 
command  ANS(1  )^zl .    Press  (enter)  to  execute  this  command.  Then,  use 
L  r>  j El,  to  verify  the  contents  of  the  variable. 


Using  the  stack  in  RPN  mode 

To  demonstrate  the  use  of  the  stack  in  RPN  mode  to  copy  a  variable  from  one 
sub-directory  to  another,  we  assume  you  are  within  sub-directory  {HOME 
MANS  INTRO},  and  that  we  will  copy  the  contents  of  variable  zl  into  the 
HOME  directory.  Use  the  following  procedure:CfQi3 
This  procedure  lists  the  contents  and  the  name  of  the  variable  in  the  stack. 
The  calculator  screen  will  look  like  this: 


2: 

a+i-b 

l: 

'zl' 

Now,  use  (jT]l™_  (jT]l™_  to  move  to  the  HOME  directory,  and  press  C^T) 
to  complete  the  operation.    Use  LrLJ E!I,  to  verify  the  contents  of  the  variable. 


Copying  two  or  more  variables  using  the  stack  in  Algebraic  mode 

The  following  is  an  exercise  to  demonstrate  how  to  copy  two  or  more 
variables  using  the  stack  when  the  calculator  is  in  Algebraic  mode.  Suppose, 
once  more,  that  we  are  within  sub-directory  {HOME  MANS  INTRO}  and  that 
we  want  to  copy  the  variables  R  and  Q  into  sub-directory  {HOME  MANS}. 
The  keystrokes  necessary  to  complete  this  operation  are  shown  following: 


Page  2-56 


(    s  \  see;  ===:======  rr^r\  =======1======  t  rk  ,Trn^ 

Lr^  J  iiii       If^J  KS  [ENTER) 

QD™  (flVTER) 

3D  SH3D(^ 
3D  (3D  (3D  (3D 
To  verify  the  contents  of  the  variables,  use  U^JIIIII  and  LxDlB. 
This  procedure  can  be  generalized  to  the  copying  of  three  or  more  variables. 

Copying  two  or  more  variables  using  the  stack  in  RPN  mode 

The  following  is  an  exercise  to  demonstrate  how  to  copy  two  or  more 
variables  using  the  stack  when  the  calculator  is  in  RPN  mode.  We  assume, 
again,  that  we  are  within  sub-directory  {HOME  MANS  INTRO}  and  that  we 
want  to  copy  the  variables  R  and  Q  into  sub-directory  {HOME  MANS}.  The 
keystrokes  necessary  to  complete  this  operation  are  shown  following: 

C3 1  HIS  CLD  ■IS 

f      v    1  ::::  iiviiiii    I  \  :::E:EvE:E:E  (  rklTrn\ 

Lj^J  iiii  iiviiiii  L_LJ  iiiiii=I=iiiii  l^^J 

To  verify  the  contents  of  the  variables,  use  L  r»  JIEI  and  LiUlE. 

This  procedure  can  be  generalized  to  the  copying  of  three  or  more  variables. 

Reordering  variables  in  a  directory 

In  this  section  we  illustrate  the  use  of  the  ORDER  command  to  reorder  the 
variables  in  a  directory.  We  assume  we  start  within  the  sub-directory  {HOME 
MANS}  containing  the  variables,  A 7 2,  R,  Q,  z7,  A,  and  the  sub-directory 
INTRO,  as  shown  below   (Copy  A1  2  from  INTRO  into  MANS). 


Algebraic  mode 

In  this  case,  we  have  the  calculator  set  to  Algebraic  mode.  Suppose  that  we 
want  to  change  the  order  of  the  variables  to  INTRO,  A,  z7,  Q,  R,  A12. 
Proceed  as  follows  to  activate  the  ORDER  function: 

UiJ prg_  <^r?IEI]jIli  Select  MEMORY  from  the  programming  menu 

^P^P^P^P  ill!!     Select  DIRECTORY  from  the  MEMORY  menu 
MM  Select  ORDER  from  the  DIRECTORY  menu 

The  screen  will  show  the  following  input  line: 


Page  2-57 


Next,  well  list  the  new  order  of  the  variables  by  using  their  names  typed 
between  quotes: 

C5D  {j      CD  HUES  CD  CB  ?  CD 

CDC3  !  CZJEB(T)  CB  ?  CZJIillllCE) 

GD  !  CZCiSiSiillSI  CD  GD  j  CZDEa  S 

The  screen  now  shows  the  new  ordering  of  the  variables: 


ORDERS  INTRO'  'FT  'zl'  'Q'  ► 
HOVRL 


RPN  mode 

In  RPN  mode,  the  list  of  re-ordered  variables  is  listed  in  the  stack  before 
applying  the  command  ORDER.  Suppose  that  we  start  from  the  same  situation 
as  above,  but  in  RPN  mode,  i.e., 


The  reordered  list  is  created  by  using: 

CfDU 


........... . 

liLliliiii  IIILL1II  i 


Then,  enter  the  command  ORDER,  as  done  before,  i.e., 

Ijnj^—  <s3?E3i  Select  MEMORY  from  the  programming  menu 

^P^P^P^P  mM     Select  DIRECTORY  from  the  MEMORY  menu 

MM  Select  ORDER  from  the  DIRECTORY  menu 

The  result  is  the  following  screen: 


Moving  variables  using  the  FILES  menu 

To  move  a  variable  from  one  directory  to  another  you  can  use  the  FILES  menu. 
For  example,  within  the  sub-directory  {HOME  MANS  INTRO},  we  have 
variables  p7,  zl ,  /?,  Q,  A 12,  a,  and  A.  Suppose  that  we  want  to  move 
variable  A  72  to  sub-directory  {HOME  MANS}.  Here  is  how  to  do  it:  Press 
Ij^Jfiles_        to  show  a  variable  list.  Use  the  down-arrow  key  ^3?  to  select 
variable  A  72,  then  press  EEZ9.  The  calculator  will  respond  with  a  PICK 
DESTINATION  screen.    Use  the  up  arrow  key  <^>  to  select  the  sub-directory 
MANS  and  press  lliilllli.    The  screen  will  now  show  the  contents  of  sub- 
directory {HOME  MANS  INTRO}: 


Page  2-58 


[[]]R 
IR  A 


hLG 
HflTRK 
hLG 
REAL 


1? 
23 
23 
ID 


Notice  that  variable  Al  2  is  no  longer  there.    If  you  now  press  L<i  J{™_  ,  the 


HLG 

1H 

[[]]R 

HflTRK 

12 

HLG 

23 

Eff  2i 

HLG 

21 

IR  fl 

REAL 

10 

Note:  You  can  use  the  stack  to  move  a  variable  by  combining  copying  with 
deleting  a  variable.  Procedures  for  deleting  variables  are  demonstrated  in  the 
next  section. 


Deleting  variables 

Variables  can  be  deleted  using  function  PURGE.  This  function  can  be 
accessed  directly  by  using  the  TOOLS  menu  (\joolJ),  or  by  using  the  FILES 
menu  {^Jfiles_  MM. 


Using  the  FILES  command 

The  FILES  command  can  be  used  to  purge  one  variable  at  a  time.  To  delete  a 
variable  from  a  given  directory  you  can  use  the  FILES  menu.  For  example, 
within  the  sub-directory  {HOME  MANS  INTRO},  we  have  variables  pi,  zl,  /?, 
Q,  a,  and  A  left.  Suppose  that  we  delete  variable  A.  Here  is  how  to  do  it: 
Press  ( jnj files         to  produce  the  variable  list.  Use  the  down-arrow  key  <\j? 
to  select  variable  A  (the  last  in  the  list),  then  press  C^lHIEEai  1EEEIII.  The 


screen  will  now  show  the  contents  of  sub-directory  IN 

^Hor^:    31735  I  Select:  Q 


[[]]R 
IR  « 


HflTRK 

flLG 

REAL 


RO  without  variable  A. 


EanmaMEmMfflMHEiMiEa 


Page  2-59 


Using  function  PURGE  in  the  stack  in  Algebraic  mode 

We  start  again  at  subdirectory  {HOME  MANS  INTRO}  containing  now  only 
variables  p7,  z7 ,  Q,  /?,  and  a.  We  will  use  command  PURGE  to  delete 
variable  pi .    Press  {tool)  illllilElI  (j^IRIII  [enter]  .    The  screen  will  now  show 
variable  pi  removed: 


:PURGE('pn 

HOVRL 

You  can  use  the  PURGE  command  to  erase  more  than  one  variable  by  placing 
their  names  in  a  list  in  the  argument  of  PURGE.  For  example,  if  now  we 
wanted  to  purge  variables  R  and  Q,  simultaneously,  we  can  try  the  following 
exercise.  Press  : 

[tool)  wmm  GTJi^CD  Chilli  CD  CB  :  CD  C^iilili 

At  this  point,  the  screen  will  show  the  following  command  ready  to  be 
executed: 


HOVflL 


iPURGECpl') 

To  finish  deleting  the  variables,  press  (enter}  .  The  screen  will  now  show  the 
remaining  variables: 


:PURGE('pn 

HOVflL 

:PURGEC{'R'  'Q'H 

HOVflL 

Using  function  PURGE  in  the  stack  in  RPN  mode 

We  start  again  at  subdirectory  {HOME  MANS  INTRO}  containing  variables 
p7,  z7 ,  Q,  /?,  and  a.  We  will  use  command  PURGE  to  delete  variable  pi . 
Press  Lj_JIH  [enter)  [tool)  iil!31i:ili.   The  screen  will  now  show  variable  pi 
removed: 


EDIT  |  '.' I E H  |:"ThlK|  F;lL  |PURGE|CLEAR 


To  delete  two  variables  simultaneously,  say  variables  R  and  Q,  first  create  a 
list  (in  RPN  mode,  the  elements  of  the  list  need  not  be  separated  by  commas 

as  in  Algebraic  mode):    (j^T)  CSDU  CZD  1111  CD  CZD  i^i  (enter).  Then, 

press  [roajllinili  use  to  purge  the  variables. 


Page  2-60 


UNDO  and  CMD  functions 

Functions  UNDO  and  CMD  are  useful  for  recovering  recent  commands,  or  to 
revert  an  operation  if  a  mistake  was  made.  These  functions  are  associated 
with  the  HIST  key:  UNDO  results  from  the  keystroke  sequence  (j^J       f  while 
CMD  results  from  the  keystroke  sequence  l<n ~]om  . 


To  illustrate  the  use  of  UNDO,  try  the  following  exercise  in  algebraic  (ALG) 
mode:  CZDCZDCEGDCBt^  •  The  UNDO  command  (CS mo)  will 
simply  erase  the  result.  The  same  exercise  in  RPN  mode,  will  follow  these 
keystrokes:  CS^^CO^^CKDCZIl^^CS  •  Using  C3_wo  at  this  point 
will  undo  the  most  recent  operation  (20/3),  leaving  the  original  terms  back  in 
the  stack: 


2: 

20 

l: 

3 

AECUY|CHinR|CYCLO|  DIY3 

EGCD  IFftCTO 

To  illustrate  the  use  of  CMD,  let's  enter  the  following  entries  in  ALG  mode. 
Press  [enter)  after  each  entry. 


Next,  use  the  CMD  function  (Q ~]om  )  to  show  the  four  most  recent 
commands  entered  by  the  user,  i.e., 


:TF 

:SI 

:3J27 

You  can  use  the  up  and  down  arrow  keys  (^><\3?)  to  navigate  through  these 
commands  and  highlight  any  of  them  that  you  want  to  entry  anew.  Once  you 
have  selected  the  command  to  enter,  press  III31IBL 

The  CMD  function  operates  in  the  same  fashion  when  the  calculator  is  in  RPN 
mode,  except  that  the  list  of  commands  only  shows  numbers  or  algebraics.  It 
does  not  show  functions  entered.  For  example,  try  the  following  exercise  in 
RPN  mode: 


Page  2-61 


Pressing      ~]om    produces  the  following  selection  box: 


4: 
3: 
£: 

l: 


SIHC5-21 


As  you  can  see,  the  numbers  3,  2,  and  5,  used  in  the  first  calculation  above, 
are  listed  in  the  selection  box,  as  well  as  the  algebraic  /SIN(5x2)/,  but  not  the 
SIN  function  entered  previous  to  the  algebraic. 


Flags 

A  flag  is  a  Boolean  value,  that  can  be  set  or  cleared  (true  or  false),  that 
specifies  a  given  setting  of  the  calculator  or  an  option  in  a  program.  Flags  in 
the  calculator  are  identified  by  numbers.  There  are  256  flags,  numbered  from 
-128  to  128.  Positive  flags  are  called  user  flags  and  are  available  for 
programming  purposes  by  the  user.  Flags  represented  by  negative  numbers 
are  called  system  flags  and  affect  the  way  the  calculator  operates. 
To  see  the  current  system  flag  setting  press  the  LM0P£J  button,  and  then  the 
DEQ1  soft  menu  key  (i.e.,  Fl).  You  will  get  a  screen  labeled  SYSTEM  FLAGS 
listing  flag  numbers  and  the  corresponding  setting. 

MMMSSYSTEH  FLAGS 

02  Constant  ■+  f yHb 

03  Function  ■+  jyHb 
iH  Payticnt  at  end 
15        ■+  vector 

20  Uridyl- h  i. oh  -i-  o 

(Note:  In  this  screen,  as  only  system  flags  are  present,  only  the  absolute  value 
of  the  flag  number  Is  displayed).  A  flag  is  said  to  be  set  if  you  see  a  check 
mark  (^)  in  front  of  the  flag  number.  Otherwise,  the  flag  is  not  set  or  cleared. 
To  change  the  status  of  a  system  flag  press  the  Tjjj !  soft  menu  key  while  the 
flag  you  want  to  change  is  highlighted,  or  use  the  L^J  key.  You  can  use  the 
up  and  down  arrow  keys  (^><\3?)  to  move  about  the  list  of  system  flags. 
Although  there  are  1  28  system  flags,  not  all  of  them  are  used,  and  some  of 
them  are  used  for  internal  system  control.  System  flags  that  are  not  accessible 
to  the  user  are  not  visible  in  this  screen.  A  complete  list  of  flags  is  presented 
in  Chapter  24. 


Page  2-62 


Example  of  flag  setting:  general  solutions  vs.  principal  value 

For  example,  the  default  value  for  system  flag  01  is  General  solutions.  What 
this  means  is  that,  if  an  equation  has  multiple  solutions,  all  the  solutions  will  be 
returned  by  the  calculator,  most  likely  in  a  list.  By  pressing  the  soft 
menu  key  you  can  change  system  flag  01  to  Principal  value.  This  setting  will 
force  the  calculator  to  provide  a  single  value  known  as  the  principal  value  of 
the  solution. 


To  see  this  at  work,  first  set  system  flag  01  (i.e.,  select  Principal  Value).  Press 
EH!  twice  to  return  to  normal  calculator  display.  We  will  try  solving  a 
quadratic  equation  solution,  say,  t2+5t+6  =  0,  with  command  QUAD. 


Algebraic  mode 

Use  the  following  keystroke  sequence:  L  r>  J  cat  [alpha}  (q}  (use  the  up  and  down 
arrow  keys,  S+^^r?  ,  to  select  command  QUAD)  ,  press  E9I . 

To  enter  the  equation  as  the  first  argument  of  function  QUAD,  use  the 
following  keystrokes: 

CgJ  l  {alpha}  CB(3(^D 

The  result  is: 


QUflD(t2+5-t+6=0,t) 


t=-3 


Now,  change  the  setting  of  flag  1  to  General  solutions:  [mode) MSSSm 
I^SX]  E9  EH .  And  try  the  solution  again:  ^^(^Qwra) .  The  solution 
now  includes  two  values: 


:QUflD(t£+5-t+6=0,t) 
QUflD(t£+5-t+6=0,t) 


t=-3 
it  =-2  t=-3> 


RPNmode 

First  set  system  flag  01  (i.e.,  Principal  Value).  Press  EH  twice  to  return  to 
normal  calculator  display.    Then,  type  the  quadratic  equation  as  follows: 


Page  2-63 


(enter}  (keeping  a  second  copy  in  the  RPN  stack) 


3: 
l: 


t  +5-t+6=0 

t2+5-t+6=0 
't 


Use  the  following  keystroke  sequence  to  enter  the  QUAD  command: 
L  r>  J  cat  [alpha}@  (use  the  up  and  down  arrow  keys,  </aN><\t>?  f  to  select 
command  QUAD)  ,  press  ll!:!!!!! .  The  screen  shows  the  principal  solution: 


3: 
2: 


t  +5-t+6=0 
t=-3 


Now,  change  the  setting  of  flag  01   to  General  solutions:  [mode) E3!33i 
ISHaMSMMBM.  And  try  the  solution  again:       CSCZD    S  QDS^ 
(j^J_ar  [alpha} @     (use  the  up  and  down  arrow  keys,    ^^^3?  ,  to  select 
command  QUAD)  ,  press  lilll!!! .  The  screen  now  shows  the  two  solutions: 


Other  flags  of  interest 

Bring  up  once  more  the  current  flag  setting  by  pressing  the  (mode)  button,  and 
then  the  HUE! soft  menu  key.    Make  sure  to  clear  system  flag  01 ,  which  was 
left  set  from  the  previous  exercise.  Use  the  up  and  down  arrow  keys  (^^^P) 
to  move  about  the  system  flag  list. 

Some  flags  of  interest  and  their  preferred  value  for  the  purpose  of  the 
exercises  that  follow  in  this  manual  are: 

02  Constant  ->  symb:     Constant  values  (e.g.,  n)  are  kept  as  symbols 

03  Function  —>  symb:     Functions  are  not  automatically  evaluated,  instead 

they  are  loaded  as  symbolic  expressions. 
27  'X+Y*i'  —>  (X,Y):       Complex  numbers  are  represented  as  ordered  pairs 
60  [a] [a]  locks:  The  sequence  [ai^{alpha}  locks  the  alphabetic  keyboard 

Press  EH  twice  to  return  to  normal  calculator  display. 


Page  2-64 


CHOOSE  boxes  vs.  Soft  MENU 

In  some  of  the  exercises  presented  in  this  chapter  we  have  seen  menu  lists  of 
commands  displayed  in  the  screen.  This  menu  lists  are  referred  to  as 
CHOOSE  boxes.  For  example,  to  use  the  ORDER  command  to  reorder 
variables  in  a  directory,  we  used: 

GT3^_  ^P  Show  PROG  menu  list  and  select  MEMORY 


&: 
5: 

PROG  HEHU 

i. STACK..  | 

4: 
3: 
£: 
l: 

3 .  BRftnCH..  1 
H .  TEST..  1 

5.  TYPE..  I 

6.  LIST.. 

EH  ^P^P^P^P    Show  the  MEMORY  menu  list  and  select  DIRECTORY 


&: 
5: 
4: 
3: 

MEMORY  HEnU 

1 . PURGE  I 
a.  MEM 

3 . BYTES  A 
H.nEHOE  1 

£: 
l: 

6 . ARITHMETIC. 

Show  the  DIRECTORY  menu 

7'. 

&: 
5: 
4: 
3: 

DIRECTORY  MEHU 

S . CRDIR  | 
C.PGDIR 

7.YARS  y 

£: 

l: 

±0 . HEHORY  $ 

ist  and  select  ORDER 


activate  the  ORDER  command 
There  is  an  alternative  way  to  access  these  menus  as  soft  MENU  keys,  by 
setting  flag  7  77.  To  set  this  flag  try  the  following: 


::  ;•:  III  \ :  :  :::!:: 


The  screen  shows  flag  1  1  7  not  set  (CHOOSE  boxes),  as  shown  here: 


115  Rigorous  on 

±20  Silent  Hod*  oFF 

133  riUou  SHitch  Hod* 

135  Recur.  Sisn-SturH 

13G  rreF  us  (.ast  col. 

/  12S  Varj  are  reals 


Press  the  soft  menu  key  to  set  flag  1  1  7  to  soft  MENU.  The  screen  will 

reflect  that  change: 


Page  2-65 


113  Rigorous  on 

120  Silent  HOd«  off 

133  hI.i.oh  S Hitch  Hod* 

135  flccur.  Sijn-SturH 

136  rr*F  H--"  (.aft  col. 

Press  twice  to  return  to  normal  calculator  display. 

Now,  well  try  to  find  the  ORDER  command  using  similar  keystrokes  to  those 
used  above,  i.e.,  we  start  with  l<n  J^g  . 

Notice  that  instead  of  a  menu  list,  we  get  soft  menu  labels  with  the  different 
options  in  the  PROG  menu,  i.e., 


Press  CZD  to  select  the  MEMORY  soft  menu  (IICIEIII).  The  display  now  shows: 


Press  CZD  to  select  the  DIRECTORY  soft  menu  (63) 


The  ORDER  command  is  not  shown  in  this  screen.  To  find  it  we  use  the  LnxtJ 
key  to  find  it: 

To  activate  the  ORDER  command  we  press  the  QD  (1333333)  soft  menu  key. 
Although  not  applied  to  a  specific  example,  this  exercise  shows  the  two 
options  for  menus  in  the  calculator  (CHOOSE  boxes  and  soft  MENUs). 

Selected  CHOOSE  boxes 

Some  menus  will  only  produce  CHOOSE  boxes,  e.g., 

•     The  APPS  (APPlicationS  menu),  activated  with  the  [apps)  key,  first  key  in 


3. 1-^0  Functions.. 

3. Constants  Vib.. 

H.nuHcric  fol.u«r..  j 

S.TiHC  &  date.. 

6. Equation  writer 

F.Fil.*  Hanger  | 

The  CAT  (CATalog  menu),  activated  with  the  L  r>  j  cat  key,  second 
key  in  the  fourth  row  of  keys  from  the  top  of  the  keyboard: 


Page  2-66 


CATALOG :  F63  COHHflnDS 

V. 

The  HELP  menu,  activated  with  {tool}  {mj  jjljlll 


CflS  h«l.p  on  = 

fi&CUY 

flDDTHOD 

HDDTOREHL 

HEL 

HL'jE: 

•     The  CMDS  (CoMmanDS)  menu,  activated  within  the  Equation  Writer, 

i.e.,  cs-^r  (nxl)  mm  


fi&CUY 

flDDTHOD 

HDDTnREHL 

HL'jE: 

Page  2-67 


Chapter  3 

Calculation  with  real  numbers 

This  chapter  demonstrates  the  use  of  the  calculator  for  operations  and 
functions  related  to  real  numbers.    Operations  along  these  lines  are  useful  for 
most  common  calculations  in  the  physical  sciences  and  engineering.  The  user 
should  be  acquainted  with  the  keyboard  to  identify  certain  functions  available 
in  the  keyboard  (e.g.,  SIN,  COS,  TAN,  etc.).  Also,  it  is  assumed  that  the 
reader  knows  how  to  adjust  the  calculator's  operation,  i.e.,  select  operating 
mode  (see  Chapter  1),  use  menus  and  choose  boxes  (see  Chapter  1),  and 
operate  with  variables  (see  Chapter  2). 


Checking  calculators  settings 

To  check  the  current  calculator  and  CAS  settings  you  need  to  just  look  at  the 
top  line  in  the  calculator  display  in  normal  operation.  For  example,  you  may 
see  the  following  setting:  RAD  XYZ  DEC  R  =  'X' 

This  stands  for  RADians  for  angular  measurements,  XYZ  for  Rectangular 
(Cartesian)  coordinates,  DECimal  number  base,  Real  numbers  preferred,  = 
means  "exact"  results,  and  'X'  is  the  value  of  the  default  independent  variable. 

Another  possible  listing  of  options  could  be         DEG  RZZ  HEX  C  ~  Y 

This  stands  for  DEGrees  as  angular  measurements,  RZZ  for  Polar  coordinates, 
HEXagesimal  number  base,  Complex  numbers  allowed,  ~  stands  for 
"approximate"  results,  and  Y  as  the  default  independent  variable. 

In  general,  this  part  of  the  display  contains  seven  elements.  Each  element  is 
identified  next  under  numbers  1  through  7.    The  possible  values  for  each 
element  are  shown  between  parentheses  following  the  element  description. 
The  explanation  of  each  of  those  values  is  also  shown: 
1 .   Angle  measure  specification  (DEG,  RAD,  GRD) 

DEG:  degrees,  360  degrees  in  a  complete  circle 

RAD:  radians,  2n  radians  in  a  complete  circle 

GRD:  grades,  400  grades  in  a  complete  circle 


Page  3-1 


2.  Coordinate  system  specification  (XYZ,  RZZ,  RZZ).    The  symbol  Z 
stands  for  an  angular  coordinate. 

XYZ:    Cartesian  or  rectangular  (x,y,z) 
RZZ:  cylindrical  Polar  coordinates  (r,0,z) 
RZZ:  Spherical  coordinates  (p,0,(|)) 

3.  Number  base  specification  (HEX,  DEC,  OCT,  BIN) 
HEX:    hexadecimal  numbers  (base  16) 

DEC:  decimal  numbers  (base  10) 
OCT:  octal  numbers  (base  8) 
BIN:    binary  numbers  (base  2) 

4.  Real  or  complex  mode  specification  (R,  C) 
R:      real  numbers 

C:      complex  numbers 

5.  Exact  or  approximate  mode  specification  (=,  ~) 
=        exact  (symbolic)  mode 

approximate  (numerical)  mode 

6.  Default  CAS  independent  variable  (e.g.,  'X',  Y,  etc.) 

Checking  calculator  mode 

When  in  RPN  mode  the  different  levels  of  the  stack  are  listed  in  the  left-hand 
side  of  the  screen.  When  the  ALGEBRAIC  mode  is  selected  there  are  no 
numbered  stack  levels,  and  the  word  ALG  is  listed  in  the  top  line  of  the  display 
towards  the  right-hand  side.  The  difference  between  these  operating  modes 
was  described  in  detail  in  Chapter  1 . 

Real  number  calculations 

To  perform  real  number  calculations  it  is  preferred  to  have  the  CAS  set  to  Real 
(as  opposite  to  Complex)  mode.  In  some  cases,  a  complex  result  may  show 
up,  and  a  request  to  change  the  mode  to  Complex  will  be  made  by  the 
calculator.  Exact  mode  is  the  default  mode  for  most  operations.  Therefore, 
you  may  want  to  start  your  calculations  in  this  mode.  Any  change  to  Approx 
mode  required  to  complete  an  operation  will  be  requested  by  the  calculator. 
There  is  no  preferred  selection  for  the  angle  measure  or  for  the  number  base 
specification.    Real  number  calculations  will  be  demonstrated  in  both  the 
Algebraic  (ALG)  and  Reverse  Polish  Notation  (RPN)  modes. 


Page  3-2 


Changing  sign  of  a  number,  variable,  or  expression 

Use  the  L^J  key.  In  ALG  mode,  you  can  press  L^J  before  entering  the 
number,  e.g.,  CD CD CD CD (^) .  Result  =  -2.5.  In  RPN  mode,  you  need 
to  enter  at  least  part  of  the  number  first,  and  then  use  the  L+/-  J  key,  e.g., 
L2JCDCDCD  •  Result  =  -2.5.  If  you  use  the  L^J  function  while  there  is  no 
command  line,  the  calculator  will  apply  the  NEG  function  (inverse  of  sign)  to 
the  object  on  the  first  level  of  the  stack. 

The  inverse  function 

Use  the  L  ?*  J  key.  In  ALG  mode,  press  L  ?*  J  first,  followed  by  a  number  or 
algebraic  expression,  e.g.,  UDLU  •  Result  =  Vi  or  0.5.  In  RPN  mode,  enter 
the  number  first,  then  use  the  key,  e.g.,  14  J[enter)1  h  J .  Result  =  %  or  0.25. 

Addition,  subtraction,  multiplication,  division 

Use  the  proper  operation  key,  namely,  L±J  CD  L><J  L±J  •  In  ALG  mode, 

press  an  operand,  then  an  operator,  then  an  operand,  followed  by  an  {enter}  to 

obtain  a  result.  Examples: 

CXJCD3CD}  C±D  CD  CD  CD  ^ 
(XJCD3CTD  (CD  CSCDGD  ^ 
CDJCDJCE  CX3  CD  CD  CD  (^) 
CD3CD3CD)  GD  CD  CD  CD  (*?D 


: 3. 7+5. 2 

8.9 

■  & ■ 3— S ■ 5 

-2-2 

:4.2-2.5 

10.5 

In  RPN  mode,  enter  the  operands  one  after  the  other,  separated  by  an  [svraj , 

then  press  the  operator  key.  Examples: 

□DHSe  CDCZDCX)  33 
3J333)(™™)  333)33  33 
CZDCZDCX)^  3D  (3  3D  (X) 
333333(^)  3D3D3D  (3 

Alternatively,  in  RPN  mode,  you  can  separate  the  operands  with  a  space 

(Lfrcj)  before  pressing  the  operator  key.  Examples: 

(3  3D 3) 3) 33 3D  3D  (33 
3D  3D  3D  33  (DID  3D  3D  33 


Page  3-3 


Using  parentheses 

Parentheses  can  be  used  to  group  operations,  as  well  as  to  enclose  arguments 
of  functions.  The  parentheses  are  available  through  the  keystroke 

combination  Cjjp^  •  Parentheses  are  always  entered  in  pairs.  For 

example,  to  calculate  (5+3.2)/(7-2.2): 

In  ALG  mode: 

In  RPN  mode,  you  do  not  need  the  parenthesis,  calculation  is  done  directly  on 
the  stack: 

In  RPN  mode,  typing  the  expression  between  quotes  will  allow  you  to  enter 
the  expression  like  in  algebraic  mode: 

For  both,  ALG  and  RPN  modes,  using  the  Equation  Writer: 

The  expression  can  be  evaluated  within  the  Equation  writer,  by  using: 

&  &  &  &  heu  or,  C3  &  iiini 

Absolute  value  function 

The  absolute  value  function,  ABS,  is  available  through  the  keystroke 
combination:  [j^Jabs_  .   When  calculating  in  the  stack  in  ALG  mode,  enter 
the  function  before  the  argument,  e.g.,  C5D^—        CX3 CZD CD CD (^3) 

In  RPN  mode,  enter  the  number  first,  then  the  function,  e.g., 

(XJraCXJCXJCDED^ 

Squares  and  square  roots 

The  square  function,  SQ,  is  available  through  the  keystroke  combination: 

L*i )XJ-  .   When  calculating  in  the  stack  in  ALG  mode,  enter  the  function 

before  the  argument,  e.g.,  GT3£  CDCDCDCD(^D 


Page  3-4 


In  RPN  mode,  enter  the  number  first,  then  the  function,  e.g., 

rorarocDE}^ 

The  square  root  function,  V,  is  available  through  the  R  key.  When  calculating 
in  the  stack  in  ALG  mode,  enter  the  function  before  the  argument,  e.g., 

In  RPN  mode,  enter  the  number  first,  then  the  function,  e.g., 

Powers  and  roots 

The  power  function,  A,  is  available  through  the  L  yx  J  key.   When  calculating 
in  the  stack  in  ALG  mode,  enter  the  base  (y)  followed  by  the  CD  key,  and 
then  the  exponent  (x),  e.g.,  CD  CD  CD  CD  CD  CD  CD  CD 
In  RPN  mode,  enter  the  number  first,  then  the  function,  e.g., 

cscD^^roracxJcxj^ra 

The  root  function,  XROOT(y,x),  is  available  through  the  keystroke  combination 
CD^S  •  When  calculating  in  the  stack  in  ALG  mode,  enter  the  function 
XROOT  followed  by  the  arguments  (y,x),  separated  by  commas,  e.g., 

In  RPN  mode,  enter  the  argument  y,  first,  then,  x,  and  finally  the  function  call, 
e.g.,  CDCDC^CD^GD^? 

Base- 10  logarithms  and  powers  of  10 

Logarithms  of  base  10  are  calculated  by  the  keystroke  combination  CD— ^5 

(function  LOG)  while  its  inverse  function  (ALOG,  or  antilogarithm)  is  calculated 

by  using      )[QX    .  In  ALG  mode,  the  function  is  entered  before  the  argument: 

GD-^CDCDCDCDC§D 
ED^CDCXJraCX}^ 

In  RPN  mode,  the  argument  is  entered  before  the  function 

CDCDCDCDC^  CD-^S 
CD  CD  CD  CD  (^™D  C5D£— 

Using  powers  of  1 0  in  entering  data 

Powers  of  ten,  i.e.,  numbers  of  the  form  -4.5x1  0"2,  etc.,  are  entered  by  using 
the  CD  key.    For  example,  in  ALG  mode: 

CDCXJSSD^JCDCXJ^ 


Page  3-5 


Or,  in  RPN  mode: 

(^(dj^cdcd^jcdcd 

Natural  logarithms  and  exponential  function 

Natural  logarithms  (i.e.,  logarithms  of  base  e  =  2.71 8281 8282)  are 

calculated  by  the  keystroke  combination  L  r>  J  m  (function  LN)  while  its 

inverse  function,  the  exponential  function  (function  EXP)  is  calculated  by  using 
L*i   .  In  ALG  mode,  the  function  is  entered  before  the  argument: 

C3 -JH  CZD  CD  GD  CD  (£5) 

ED  £L_  GE)  CD  CD  CD  CD 

In  RPN  mode,  the  argument  is  entered  before  the  function 

CDCDGDCD^™)  CB— ^ 
{JJCDtJJCDCD  GD^— 

Trigonometric  functions 

Three  trigonometric  functions  are  readily  available  in  the  keyboard:  sine 
(CD),  cosine  (CD),  and  tangent  (L^J).  The  arguments  of  these  functions  are 
angles,  therefore,  they  can  be  entered  in  any  system  of  angular  measure 
(degrees,  radians,  grades).  For  example,  with  the  DEG  option  selected,  we 
can  calculate  the  following  trigonometric  functions: 
In  ALG  mode: 

CDCDUDCD 
CDCDCDCD 
CDCDCDCDCD 
In  RPN  mode:  

CX3CD(^™DGD 
CX3CDC^GD 
CDCDCDCDCD 

Inverse  trigonometric  functions 

The  inverse  trigonometric  functions  available  in  the  keyboard  are  the  arcsine 
(ASIN),  arccosine  (ACOS),  and  arctangent  (ATAN),  available  through  the 
keystroke  combinations  C5J^sw_  f  [J^Jacos_  f  and  [j^J^an_  t  respectively. 
Since  the  inverse  trigonometric  functions  represent  angles,  the  answer  from 
these  functions  will  be  given  in  the  selected  angular  measure  (DEG,  RAD, 
GRD).  Some  examples  are  shown  next: 
In  ALG  mode: 

CD^CDCDCDCDC(™D 
{^jAcgs_  UDCDCDCDCD 


Page  3-6 


{^r)AjAN_  cmcmcmcm^ 

In  RPN  mode: 

□D  Cm  CTD  CTD  (^™D  C3D 
□D  Cm  dD  CD  Qmzk)  cm  £os_ 

CD  CZD  CD  CD  (^TER)  *[AN_ 

All  the  functions  described  above,  namely,  ABS,  SQ,  V,  A,  XROOT,  LOG, 
ALOG,  LN,  EXP,  SIN,  COS,  TAN,  ASIN,  ACOS,  ATAN,  can  be  combined 
with  the  fundamental  operations  (C±J CD C><J C±J )  to  form  more  complex 
expressions.  The  Equation  Writer,  whose  operations  is  described  in  Chapter 
2,  is  ideal  for  building  such  expressions,  regardless  of  the  calculator 
operation  mode. 

Differences  between  functions  and  operators 

Functions  like  ABS,  SQ,  V,  LOG,  ALOG,  LN,  EXP,  SIN,  COS,  TAN,  ASIN, 
ACOS,  ATAN  require  a  single  argument.   Thus,  their  application  is  ALG 
mode  is  straightforward,  e.g.,  ABS(x).     Some  functions  like  XROOT  require 
two  arguments,  e.g.,  XROOT(x,y).   This  function  has  the  equivalent  keystroke 
sequence  (13—5  • 

Operators,  on  the  other  hand,  are  placed  after  a  single  argument  or  between 
two  arguments.  The  factorial  operator  (!),  for  example,  is  placed  after  a 
number,  e.g.,  CmC™)CmCm^^-  Since  this  operator  requires  a  single 
argument  it  is  referred  to  as  a  unary  operator.  Operators  that  require  two 
arguments,  such  as  L±J  Cm  CD  C±J  L  yX  J,  are  binary  operators,  e.g., 

cm  cm  cm,  orcmcmcm. 

Real  number  functions  in  the  MTH  menu 

The  MTH  (MaTHematics)  menu  include  a  number  of  mathematical  functions 
mostly  applicable  to  real  numbers.  To  access  the  MTH  menu,  use  the 
keystroke  combination  (jT}miw_  .   With  the  default  setting  of  CHOOSE  boxes 
for  system  flag  1  1  7  (see  Chapter  2),  the  MTH  menu  is  shown  as  the  following 
menu  list: 


Page  3-7 


HATH  METlU 

3.HATRIK.. 
3.  LIST.. 

H. HYPERBOLIC.  f 

5 .  REAL.. 

6.  BASE.. 

HATH  HEHU 


6.  BASE.. 

? .  PROBABILITY.. 
S.FFT.. 
3.C0HPLEK.. 
iD.COnSTAnTS.. 


ii .  special  FuncTions.. 


As  they  are  a  great  number  of  mathematic  functions  available  in  the  calculator, 
the  MTH  menu  is  sorted  by  the  type  of  object  the  functions  apply  on.  For 
example,  options  1 .  VECTOR..,  2.  MATRIX.,  and  3.  LIST.,  apply  to  those  data 
types  (i.e.,  vectors,  matrices,  and  lists)  and  will  discussed  in  more  detail  in 
subsequent  chapters.  Options  4.  HYPERBOLIC,  and  5.  REAL.,  apply  to  real 
numbers  and  will  be  discussed  in  detailed  herein.  Option  6.  BASE.,  is  used 
for  conversion  of  numbers  in  different  bases,  and  is  also  to  be  discussed  in  a 
separate  chapter.  Option  7.  PROBABILITY.,  is  used  for  probability 
applications  and  will  be  discussed  in  an  upcoming  chapter.    Option  8.  FFT.. 
(Fast  Fourier  Transform)  is  an  application  of  signal  processing  and  will  be 
discussed  in  a  different  chapter.    Option  9.  COMPLEX.,  contains  functions 
appropriate  for  complex  numbers,  which  will  be  discussed  in  the  next  chapter. 
Option  70.  CONSTANTS  provides  access  to  the  constants  in  the  calculator. 
This  option  will  be  presented  later  in  this  section.  Finally,  option  7  7.  SPECIAL 
FUNCTIONS.,  includes  functions  of  advanced  mathematics  that  will  be 
discussed  in  this  section  also. 


In  general,  to  apply  any  of  these  functions  you  need  to  be  aware  of  the 
number  and  order  of  the  arguments  required,  and  keep  in  mind  that,  in  ALG 
mode  you  should  select  first  the  function  and  then  enter  the  argument,  while  in 
RPN  mode,  you  should  enter  the  argument  in  the  stack  first,  and  then  select 
the  function. 


Using  calculator  menus: 

1 .  Since  the  operation  of  MTH  functions  (and  of  many  other  calculator 
menus)  is  very  similar,  we  will  describe  in  detail  the  use  of  the  4. 
HYPERBOLIC,  menu  in  this  section,  with  the  intention  of  describing  the 
general  operation  of  calculator  menus.  Pay  close  attention  to  the  process 
for  selecting  different  options. 

2.  To  quickly  select  one  of  the  numbered  options  in  a  menu  list  (or  CHOOSE 
box),  simply  press  the  number  for  the  option  in  the  keyboard.  For 


Page  3-8 


example,  to  select  option  4.  HYPERBOLIC  in  the  MTH  menu,  simply 
press  LU . 

Hyperbolic  functions  and  their  inverses 

Selecting  Option  4.  HYPERBOLIC.  ,  in  the  MTH  menu,  and  pressing  EH, 
produces  the  hyperbolic  function  menu: 


HYPERBOLIC  HEHU 


3. COSH 
H.flCOSH 
F.TrtriH 
6 .  HTHriH 


HYPERBOLIC  HEHU 


H.RCOSH 
F.TrtriH 
C .  hThTiH 
7.EHFH 
g.LnPi 


The  hyperbolic  functions  are: 

Hyperbolic  sine,  SINH,  and  its  inverse,  ASINH  or  sinh"1 
Hyperbolic  cosine,  COSH,  and  its  inverse,  ACOSH  or  cosh"1 
Hyperbolic  tangent,  TANH,  and  its  inverse,  ATANH  or  tanh"1 
This  menu  contains  also  the  functions: 

EXPM(x)  =  exp(x)  -  1, 
LNPl(x)  =  ln(x+l). 
Finally,  option  9.  MATH,  returns  the  user  to  the  MTH  menu. 

For  example,  in  ALG  mode,  the  keystroke  sequence  to  calculate,  say, 

tanh(2.5),  is  the  following: 

^Jmjh_  Select  MTH  menu 

CTD  EH  Select  the  4.  HYPERBOLIC  menu 

CD  IliOI  Select  the  5.  TANH  function 

CZDCZDCX]^  Evaluate  tanh(2.5) 

The  screen  shows  the  following  output:  


:TFINH(2.5) 

.986614298151 

In  the  RPN  mode,  the  keystrokes  to  perform  this  calculation  are  the  following: 
CX3CZI)CTD(^D  Enter  the  argument  in  the  stack 

GT]m™_  Select  MTH  menu 

GD  MM  Select  the  4.  HYPERBOLIC  menu 

CD  EH  Select  the  5.  TANH  function 


Page  3-9 


Th 


e  result  is: 


,986614298151 


The  operations  shown  above  assume  that  you  are  using  the  default  setting  for 
system  flag  1  1  7  (CHOOSE  boxes).  If  you  have  changed  the  setting  of  this  flag 
(see  Chapter  2)  to  SOFT  menu,  the  MTH  menu  will  show  as  labels  of  the  soft 
menu  keys,  as  follows  (left-hand  side  in  ALG  mode,  right  -hand  side  in  RPN 
mode): 


VECTR  HhTRX  LIST    HYP    REAL  ERSE 


Pressing  jNxrj  shows  the  remaining  options: 


l:  

nflaaftiaffimum  1 1  im  1 1 hflaaftiaffimum  1 1  im  1 1 


Note:  Pressing  LjnJ^_  will  return  to  the  first  set  of  MTH  options.  Also,  using 
the  combination  C3<\3?  will  list  all  menu  functions  in  the  screen,  e.g. 


FREE 
rFT 

CMPLX 
CONST 

SPECIAL  FUNCTIONS 


Thus,  to  select,  for  example,  the  hyperbolic  functions  menu,  with  this  menu 
format  press  EMM  ,  to  produce: 


l:  

^j.iiirm.iiiw.^iirw.Miiin.iiirWii.i!!  ^j.iiinm.iiiw.Hiirw.Hiiin.iiirWii.i!! 


Finally,  in  order  to  select,  for  example,  the  hyperbolic  tangent  (tanh)  function, 
simply  press 


Note:  To  see  additional  options  in  these  soft  menus,  press  the  jNxrj  key  or  the 
y,REV  keystroke  sequence. 


Page  3-10 


For  example,  to  calculate  tanh(2.5),  in  the  ALG  mode,  when  using  SOFT 
menus  over  CHOOSE  boxes,  follow  this  procedure: 


jjjjj.jj  : 


Select  MTH  menu 
Select  the  HYPERBOLIC  menu 
Select  the  TANH  function 
Evaluate  tanh(2.5) 


In  RPN  mode,  the  same  value  is  calculated  using: 

CBCZDCS(^D  Enter  argument  in  the  stack 

^Jmjh_  Select  MTH  menu 

Select  the  HYPERBOLIC,  menu 
EEDCI  Select  the  TANH  function 


As  an  exercise  of  applications  of  hyperbolic  functions,  verify  the  following 
values: 


SINH  (2.5)  =  6.05020.. 
COSH  (2.5)  =  6.1  3228.. 
TANH(2.5)  =  0.98661.. 
EXPM(2.0)  =  6.38905.... 


ASINH(2.0)=  1.4436... 
ACOSH(2.0)=  1.3169. 
ATANH(0.2)  =  0.2027.. 
LNPl(l.O)  =  0.69314.... 


Once  again,  the  general  procedure  shown  in  this  section  can  be  applied  for 
selecting  options  in  any  calculator  menu. 

Real  number  functions 

Selecting  option  5.  REAL.,  in  the  MTH  menu,  with  system  flag  1  1  7  set  to 


REAL  HEHU 


3. V.J 
H.Min 

5.  HHK 

6.  HOD 


'I 


REAL  HEnU 

7.  RES  n 
s.sian  I 

ii.lP  | 

Page  3-1 1 


REAL  HERU 

13.RnD  f 
IH.TRnC 

15.  FLOOR 

16.  CEIL 

1F.D-*  j 

REAL  HERU 

IH.TRnC 

15.  FLOOR 

16.  CEIL 

17.  D-* 

1S.R-*  f 

Option  7  9.  MATH.,  returns  the  user  to  the  MTH  menu.  The  remaining 
functions  are  grouped  into  six  different  groups  described  below. 

If  system  flag  1  1  7  is  set  to  SOFT  menus,  the  REAL  functions  menu  will  look  like 
this  (ALG  mode  used,  the  same  soft  menu  keys  will  be  available  in  RPN 
mode): 


ElMliaiaHMaEBMMffiMlffil 

The  very  last  option,  Iljilll,  returns  the  user  to  the  MTH  menu. 
Percentage  functions 

These  functions  are  used  to  calculate  percentages  and  related  values  as 
follows: 

%  (y,x)     :  calculates  the  x  percentage  of  y 

%CH(y,x)  :  calculates  100(y-x)/x,  i.e.,  the  percentage  change,  the 
difference  between  two  numbers. 

%T(y,x)    :  calculates  100  x/y,  i.e.,  the  percentage  total,  the  portion 

that  one  number  (x)  is  of  another  (y). 
These  functions  require  two  arguments,  we  illustrate  the  calculation  of 
%T(15,45),  i.e.,  calculation  15%  of  45,  next.  We  assume  that  the  calculator 
is  set  to  ALG  mode,  and  that  system  flag  1  1  7  is  set  to  CHOOSE  boxes.  The 
procedure  is  as  follows: 


GT]m™_  Select  MTH  menu 

CD  EB  Select  the  5.  REAL,  menu 

CD  EB  Select  the  5.  %T  function 

LUL5J  Enter  first  argument 

L  r>  J  \  Enter  a  comma  to  separate  arguments 

14  JL5  J  Enter  second  argument 


Page  3-12 


[enter]  Calculate  function 

The  result  is  shown  next: 


:*T(15,45) 

300 

In  RPN  mode,  recall  that  argument  y  is  located  in  the  second  level  of  the  stack, 
while  argument  x  is  located  in  the  first  level  of  the  stack.  This  means,  you 
should  enter  x  first,  and  then,  y,  just  as  in  ALG  mode.  Thus,  the  calculation  of 
%T(15,45),  in  RPN  mode,  and  with  system  flag  1  17  set  to  CHOOSE  boxes, 
we  proceed  as  follows: 

CDCD(^D  Enter  first  argument 

14  JL5  J[enter)  Enter  second  argument 

CT}mth_  Select  A/177-/  menu 

CD  DB  Select  the  5.  REAL,  menu 

CD  Select  the  5.  %T  function 


Note:  The  exercises  in  this  section  illustrate  the  general  use  of  calculator 
functions  having  2  arguments.  The  operation  of  functions  having  3  or  more 
arguments  can  be  generalized  from  these  examples. 


As  an  exercise  for  percentage-related  functions,  verify  the  following  values: 
70(5,20)  =  I  %CH(22,25)  =  13.6363..,  %T(500,20)  =  4 

Minimum  and  maximum 

Use  these  functions  to  determine  the  minimum  or  maximum  value  of  two 
arguments. 

MIN(x,y)  :  minimum  value  of  x  and  y 

MAX(x,y)  :  maximum  value  of  x  and  y 
As  an  exercise,  verify  that  MIN(-2,2)  =  -2,  MAX(-2,2)  =  2 

Modulo 

MOD:  y  mod  x  =  residual  of  y/x,  i.e.,  if  x  and  y  are  integer  numbers,  y/x 
d  +  r/x,  where  d  =  quotient,  r  =  residual.  In  this  case,  r  =  y  mod  x. 


Page  3-13 


Please  notice  that  MOD  is  not  a  function,  but  rather  an  operator,  i.e.,  in  ALG 
mode,  MOD  should  be  used  as  y  x,  and  not  as  MODCy  x  )  .  Thus, 

the  operation  of  MOD  is  similar  to  that  of  L±J ,       /  L><J ,  L±J  • 

As  an  exercise,  verify  that  15  MOD  4=15  mod  4  =  residual  of  15/4  =  3 

Absolute  value,  sign,  mantissa,  exponent,  integer  and  fractional  parts 

ABS(x)  :  calculates  the  absolute  value,  |x| 

SIGN(x) :  determines  the  sign  of  x,  i.e.,  -1 ,  0,  or  1 . 

MANT(x):  determines  the  mantissa  of  a  number  based  on  log10. 

XPON(x):  determines  the  power  of  10  in  the  number 

IP(x)      :  determines  the  integer  part  of  a  real  number 

FP(x)     :  determines  the  fractional  part  of  a  real  number 

As  an  exercise,  verify  that  ABS(-3)  =  \-3\  =  3,  SIGN(-5)  =  -1,  MANT(2540)  = 

2.540,  XPON(2540)  =  3,  IP(2.35)  =  2,  FP(2.35)  =  0.35. 

Rounding,  truncating,  floor,  and  ceiling  functions 

RND(x,y)     :  rounds  up  y  to  x  decimal  places 

TRNC(x,y)  :  truncate  y  to  x  decimal  places 

FLOOR(x)    :  closest  integer  that  is  less  than  or  equal  to  x 

CEIL(x)       :  closest  integer  that  is  greater  than  or  equal  to  x 

As  an  exercise,  verify  that  RND(1 .4567,2)  =  1.46,  TRNQ1 .4567,2)  =  1.45, 

FLOOR(2.3)  =  2,  CEIL(2,3)  =  3 

Radians-to-degrees  and  degrees-to-radians  functions 

D->R  (x)  :  converts  degrees  to  radians 

R^D  (x)  :  converts  radians  to  degrees. 

As  an  exercise,  verify  that  D^R(45)  =  0.78539  (i.e.,  45°  =  0.78539rad), 
R^D(1.5)  =  85.943669..  (i.e.,  1 .5rad  =  85.943669..°). 

Special  functions 

Option  1  1 .  Special  functions...  in  the  MTH  menu  includes  the  following 
functions: 


Page  3-14 


SPECIAL  FUnCTIOnS  METlU 

a.  psi 

3.  Psi 
H .  HATH.. 

GAMMA:         The  Gamma  function  r(a) 

PSI:  N-th  derivative  of  the  digamma  function 

Psi:  Digamma  function,  derivative  of  the  In(Gamma) 

The  Gamma  function  is  defined  by  T(a)  =      xa~le~xdx .  This  function  has 

applications  in  applied  mathematics  for  science  and  engineering,  as  well  as 
in  probability  and  statistics. 


Factorial  of  a  number 

The  factorial  of  a  positive  integer  number  n  is  defined  as  n!=n-(n-l)-(n- 
2)  ... 3-2-1 ,  with  0!  =  7.  The  factorial  function  is  available  in  the  calculator  by 
using  ld™Jlr?JL2J .  In  both  ALG  and  RPN  modes,  enter  the  number  first, 
followed  by  the  sequence  (^)CBCX3.  Example:  CZD {alpha} (TQ £jJ (enter)  . 
The  Gamma  function,  defined  above,  has  the  property  that 

r(a)  =  (a-1)  r(a-l),  for  a  >  1 . 
Therefore,  it  can  be  related  to  the  factorial  of  a  number,  i.e.,  r(a)  =  (a-l)\, 
when  a  is  a  positive  integer.  We  can  also  use  the  factorial  function  to 
calculate  the  Gamma  function,  and  vice  versa.  For  example,  r(5)  =  4!  or, 
QZ3S^CBCTD(^].   The  factorial  function  is  available  in  the  MTH  menu, 
through  the  7.  PROBABILITY.,  menu.  


The  PSI  function,  ^(x.y),  represents  the  y-th  derivative  of  the  digamma  function, 

dn 

i.e.,  ^{n.x)  =  y(x)f  where  y/(x)  is  known  as  the  digamma  function,  or 

dx11 

Psi  function.  For  this  function,  y  must  be  a  positive  integer. 

The  Psi  function,  y/(x),  or  digamma  function,  is  defined  as  y/{x)  =  ln[T(x)] . 


Page  3-15 


Examples  of  these  special  functions  are  shown  here  using  both  the  ALG  and 
RPN  modes.   As  an  exercise,  verify  that  GAMMA(2.3)  =  1.16671 1..., 
PSI(1.5,3)  =  1.40909..,  and  Psi(l. 5)=  3.64899739. .E-2. 


These  calculations  are  shown  in  the  following  screen  shot: 


GRMMRC2-  33 

1. 1667119052 

PSICl.5,3) 

1-409091034 

:Psi(1.5) 

3.64899739786E-2 


Calculator  constants 

The  following  are  the  mathematical  constants  used  by  your  calculator: 

•  e:         the  base  of  natural  logarithms. 

•  /:  the  imaginary  unit,  P 2  =  -1 . 

•  TV.         the  ratio  of  the  length  of  the  circle  to  its  diameter. 

•  MINR:  the  minimum  real  number  available  to  the  calculator. 

•  MAXR:  the  maximum  real  number  available  to  the  calculator. 

To  have  access  to  these  constants,  select  option  1  1 .  CONSTANTS.,  in  the 
MTH  menu,   


HATH  HEnU 

6.  BASE.. 

F. PROBABILITY.. 

S.FFT..  . 

3.C0HPLEK..  t 

ii.  special  Funaions..  | 

The  constants  are  listed  as  follows: 


consTAnTS  hehu 


a.a.FiSaSiSaSHG 
H .  ui .  .i  i . !' 

e.3.iHi55aes355 


consTAnTS  mehu 

e.3.iHi55aes355  f 

F.HinR 
3.1.E-H55 

3  HA MR  I 

Selecting  any  of  these  entries  will  place  the  value  selected,  whether  a  symbol 
(e.g.,  e,  /,  n,  MINR,  or  MAXR)  or  a  value  (2.77..,  (0,1),  3. 14..,  1E-499, 
9.99. .E499)  in  the  stack. 


Page  3-16 


Please  notice  that  e  is  available  from  the  keyboard  as  exp(l),  i.e., 
C5D^CD(^),  in  ALG  mode,  or  CD(§nter)  C5D^ ,  in  RPN  mode.  Also, 

71  is  available  directly  from  the  keyboard  as   •  Finally,  /  is  available 

by  using  LjnJ  ;  • 

Operations  with  units 

Numbers  in  the  calculator  can  have  units  associated  with  them.  Thus,  it  is 
possible  to  calculate  results  involving  a  consistent  system  of  units  and  produce 
a  result  with  the  appropriate  combination  of  units. 


The  UNITS  menu 

The  units  menu  is  launched  by  the  keystroke  combination  (j^JJMJ  (associated 
with  the  LU  key).  With  system  flag  1  1  7  set  to  CHOOSE  boxes,  the  result  is 
the  following  menu:  


UniTS  HETlU 


2.  Length.. 

3.  Area. . 

S.Tim.. 

e.speed.. 


UniTS  HEMJ 

7.H055..  jj 
E.  Force..  | 
5.  Energy..  | 
lD.Pouer..  | 

ii. Pressure..  jj 

UniTl"  METlU 


12.  TeHperature.. 

13.  E^ctric  Current.. 
iH.An^e.. 

15.  Lisht.. 

16.  Radiation.. 


Option  7.  Tools.,  contains  functions  used  to  operate  on  units  (discussed  later). 
Options  3.  Length.,  through  7 7. Viscosity.,  contain  menus  with  a  number  of 
units  for  each  of  the  quantities  described.  For  example,  selecting  option  8. 
Force.,  shows  the  following  units  menu: 


FORCE  HEnU 

IjKII 

H.hip 
5.  LbF 

FORCE  HEHU 


2.'jyn 

3-3F 

H.hip 

SAbf 

C.pdl 


The  user  will  recognize  most  of  these  units  (some,  e.g.,  dyne,  are  not  used 
very  often  nowadays)  from  his  or  her  physics  classes:  N  =  newtons,  dyn  = 
dynes,  gf  =  grams  -  force  (to  distinguish  from  gram-mass,  or  plainly  gram,  a 


Page  3-17 


unit  of  mass),  kip  =  kilo-poundal  (1000  pounds),  Ibf  =  pound-force  (to 
distinguish  from  pound-mass),  pdl  =  poundal. 


To  attach  a  unit  object  to  a  number,  the  number  must  be  followed  by  an 
underscore.  Thus,  a  force  of  5  N  will  be  entered  as  5_N. 
For  extensive  operations  with  units  SOFT  menus  provide  a  more  convenient 
way  of  attaching  units.    Change  system  flag  1  1  7  to  SOFT  menus  (see 
Chapter  1 ),  and  use  the  keystroke  combination  (j^JJMJ  to  get  the  following 
menus.  Press  [nxtJ  to  move  to  the  next  menu  page. 


nffr;i?ifti¥fifHiiJ 

Pressing  on  the  appropriate  soft  menu  key  will  open  the  sub-menu  of  units  for 
that  particular  selection.  For  example,  for  the  iEilllllii  sub-menu,  the  following 
units  are  available: 


Pressing  the  soft  menu  key  will  take  you  back  to  the  UNITS  menu. 


Recall  that  you  can  always  list  the  full  menu  labels  in  the  screen  by  using 
LeD<s3?  /  e.g.,  for  the  llllllllll  set  of  units  the  following  labels  will  be  listed: 


erg 
Kcal 
cal 
Btu 

ft*lbf 


EgMEMBEMflHE 


therm 
MeV 
eV 


UNITS 


Note:  Use  the  jNxrj  key  or  the  l<n  J?REV  keystroke  sequence  to  navigate 
through  the  menus. 


Available  units 

The  following  is  a  list  of  the  units  available  in  the  UNITS  menu.  The  unit 
symbol  is  shown  first  followed  by  the  unit  name  in  parentheses: 


Page  3-18 


LENGTH 

m  (meter),  cm  (centimeter),  mm  (millimeter),  yd  (yard),  ft  (feet),  in  (inch),  Mpc 
(Mega  parsec),  pc  (parsec),  lyr  (light-year),  au  (astronomical  unit),  km 
(kilometer),  mi  (international  mile),  nmi  (nautical  mile),  miUS  (US  statute  mile), 
chain  (chain),  rd  (rod),  fath  (fathom),  ftUS  (survey  foot),  Mil  (Mil),  u.  (micron), 
A  (Angstrom),  fermi  (fermi) 

AREA 

mA2  (square  meter),  cmA2  (square  centimeter),  b  (barn),  ydA2  (square  yard), 
ftA2  (square  feet),  inA2  (square  inch),  kmA2  (square  kilometer),  ha  (hectare), 
a  (are),  miA2  (square  mile),  miUSA2  (square  statute  mile),  acre  (acre) 

VOLUME 

mA3  (cubic  meter),  st  (stere),  cmA3  (cubic  centimeter),  ydA3  (cubic  yard),  ftA3 
(cubic  feet),  inA3  (cubic  inch),  I  (liter),  galUK  (UK  gallon),  galC  (Canadian 
gallon),  gal  (US  gallon),  qt  (quart),  pt  (pint),  ml  (mililiter),  cu  (US  cup),  ozfl  (US 
fluid  ounce),  ozUK  (UK  fluid  ounce),  tbsp  (tablespoon),  tsp  (teaspoon),  bbl 
(barrel),  bu  (bushel),  pk  (peck),  fbm  (board  foot) 

TIME 

yr  (year),  d  (day),  h  (hour),  min  (minute),  s  (second),  Hz  (hertz) 
SPEED 

m/s  (meter  per  second),  cm/s  (centimeter  per  second),  ft/s  (feet  per  second), 
kph  (kilometer  per  hour),  mph  (mile  per  hour),  knot  (nautical  miles  per  hour),  c 
(speed  of  light),  ga  (acceleration  of  gravity  ) 

MASS 

kg  (kilogram),  g  (gram),  Lb  (avoirdupois  pound),  oz  (ounce),  slug  (slug),  Ibt 
(Troy  pound),  ton  (short  ton),  tonUK  (long  ton),  t  (metric  ton),  ozt  (Troy  ounce), 
ct  (carat),  grain  (grain),  u  (unified  atomic  mass),  mol  (mole) 

FORCE 

N  (newton),  dyn  (dyne),  gf  (gram-force),  kip  (kilopound-force),  Ibf  (pound- 
force),  pdl  (poundal) 


Page  3-19 


ENERGY 

J  (joule),  erg  (erg),  Kcal  (kilocalorie),  Cal  (calorie),  Btu  (International  table  btu), 
ftxlbf  (foot-pound),  therm  (EEC  therm),  MeV  (mega  electron-volt),  eV  (electron- 
volt) 

POWER 

W  (watt),  hp  (horse  power), 
PRESSURE 

Pa  (pascal),  atm  (atmosphere),  bar  (bar),  psi  (pounds  per  square  inch),  torr 
(torr),  mmHg  (millimeters  of  mercury),  inHg  (inches  of  mercury),  inH20  (inches 
of  water), 

TEMPERATURE 

°C  (degree  Celsius),  °F  (degree  Fahrenheit),  K  (Kelvin),  °R  (degree  Rankine), 
ELECTRIC  CURRENT  (Electric  measurements) 

V  (volt),  A  (ampere),  C  (coulomb),  Q  (ohm),  F  (farad),  W  (watt),  Fdy  (faraday), 
H  (henry),  mho  (mho),  S  (siemens),  T  (tesla),  Wb  (weber ) 

ANGLE  (planar  and  solid  angle  measurements) 

°  (sexagesimal  degree),  r  (radian),  grad  (grade),  arcmin  (minute  of  arc),  arcs 
(second  of  arc),  sr  (steradian) 

LIGHT  (Illumination  measurements) 

fc  (footcandle),  flam  (footlambert),  Ix  (lux),  ph  (phot),  sb  (stilb),  Im  (lumem),  cd 
(candela),  lam  (lambert) 

RADIATION 

Gy  (gray),  rad  (rad),  rem  (rem),  Sv  (sievert),  Bq  (becquerel),  Ci  (curie),  R 
(roentgen) 

VISCOSITY 

P  (poise),  St  (stokes) 


Page  3-20 


Units  not  listed 

Units  not  listed  in  the  Units  menu,  but  available  in  the  calculator  include:  gmol 
(gram-mole),  Ibmol  (pound-mole),  rpm  (revolutions  per  minute),  dB  (decibels). 
These  units  are  accessible  through  menu  1  1  7.02,  triggered  by  using 
MENU(1 1  7.02)  in  ALG  mode,  or  1  1 7.02  [enter}  MENU  in  RPN  mode.  The 
menu  will  show  in  the  screen  as  follows  (use  LrU^3?to  show  labels  in 
display): 


TNu 
rnol 
bmol 

dB 

EQLIB 


rpH  |   41  |Eul_IE: 


These  units  are  also  accessible  through  the  catalog,  for  example: 

gmol:  Ir^J— ^  ^™j(jnj(3 

Ibmol:  CS_cat  (S^CfDS 

rpm:  Q^J _w  [alph^I^^^ 

dB:  CS_c^(^(jT3© 

Converting  to  base  units 

To  convert  any  of  these  units  to  the  default  units  in  the  SI  system,  use  the 
function  UBASE.  For  example,  to  find  out  what  is  the  value  of  7  poise  (unit  of 
viscosity)  in  the  SI  units,  use  the  following: 

In  ALG  mode,  system  flag  1  1  7  set  to  CHOOSE  boxes: 
[Jfjmrs  Select  the  UNITS  menu 

1SIB  Select  the  TOOLS  menu 

<\3?  liilllll  Select  the  UBASE  function 

LU  LcD  -     Enter  1  and  underline 

[Jfjmrs  Select  the  UNITS  menu 

^  MM         Select  the  VISCOSITY  option 
Select  the  UNITS  menu 
Convert  the  units 


Page  3-21 


This  results  in  the  following  screen  (i.e.,  7  poise  =  0. 7  kg/(m-s)) 


UBflSE(M_P) 

1  ^ 


In  RPN  mode,  system  flag  1  1  7  set  to  CHOOSE  boxes: 
LU  Enter  1  (no  underline) 

[Jfjmrs  Select  the  UNITS  menu 

^  D9  Select  the  VISCOSITY  option 
111!  Select  the  unit  P  (poise) 

{J^jjnijs  Select  the  UNITS  menu 

Select  the  TOOLS  menu 
^r?  EB  Select  the  UBASE  function 

In  ALG  mode,  system  flag  1  1  7  set  to  SOFT  menus: 
{J^jjnijs  Select  the  UNITS  menu 

Select  the  TOOLS  menu 
Select  the  UBASE  function 

L  /  J  L  r>  J  -     Enter  1  and  underline 

[Jfjmrs  Select  the  UNITS  menu 

tK)PREv_  mm    Select  the  VISCOSITY  option 
Select  the  unit  P  (poise) 
Convert  the  units 


In  RPN  mode,  system  flag  1  1  7  set  to  SOFT  menus: 

LU  Enter  1  (no  underline) 

{J^jjnijs  Select  the  UNITS  menu 

^prev_  mm  Select  the  VISCOSITY  option 

1IE1I!  Select  the  unit  P  (poise) 

{Jf^mi  Select  the  UNITS  menu 

D39B  Select  the  TOOLS  menu 

IfllH:!!  Select  the  UBASE  function 


Attaching  units  to  numbers 

To  attach  a  unit  object  to  a  number,  the  number  must  be  followed  by  an 
underscore  (LrU  - ,  key(8,5)).  Thus,  a  force  of  5  N  will  be  entered  as  5_N. 


Page  3-22 


Here  is  the  sequence  of  steps  to  enter  this  number  in  ALG  mode,  system  flag 
1 1  7  set  to  CHOOSE  boxes: 


UJlxLJ  -  Enter  number  and  underscore 

[^Jmrs  Access  the  UNITS  menu 

Select  units  of  force  (8.  Force..) 
MM  Select  Newtons  (N) 

(enter}  Enter  quantity  with  units  in  the  stack 


The  screen  will  look  like  the 

Allowing: 

:5-l_H 

Note:  If  you  forget  the  underscore,  the  result  is  the  expression  5*N,  where  N 
here  represents  a  possible  variable  name  and  not  Newtons. 


To  enter  this  same  quantity,  with  the  calculator  in  RPN  mode,  use  the  following 
keystrokes: 

L  5  J  Enter  number  (do  not  enter  underscore) 

[^Jmrs  Access  the  UNITS  menu 

CSIIIISil  Select  units  of  force  (8.  Force..) 

Select  Newtons  (N) 

Notice  that  the  underscore  is  entered  automatically  when  the  RPN  mode  is 
active.  The  result  is  the  following  screen: 


As  indicated  earlier,  if  system  flag  1  1  7  is  set  to  SOFT  menus,  then  the  UNITS 
menu  will  show  up  as  labels  for  the  soft  menu  keys.  This  set  up  is  very 
convenient  for  extensive  operations  with  units. 

The  keystroke  sequences  to  enter  units  when  the  SOFT  menu  option  is  selected, 
in  both  ALG  and  RPN  modes,  are  illustrated  next.  For  example,  in  ALG  mode, 
to  enter  the  quantity  5_N  use: 

L  5  JL  r>  J  -  Enter  number  and  underscore 


Page  3-23 


CSJM?  Access  the  UNITS  menu 

(nxt)  WS333i  Select  units  of  force 

Select  Newtons  (N) 
Enter  quantity  with  units  in  the  stack 


The  same  quantity,  entered  in  RPN  mode  uses  the  following  keystrokes: 
L  5  J  Enter  number  (no  underscore) 

□3  JM*  Access  the  UNITS  menu 

®  tSSESi  Select  units  of  force 

iltil!  Select  Newtons  (N) 

Note:  You  can  enter  a  quantity  with  units  by  typing  the  underline  and  units 
with  the  {alpha}  keyboard,  e.g.,  CZDCB  ~(alpha}(n}  will  produce  the  entry:  5.....N 

Unit  prefixes 

You  can  enter  prefixes  for  units  according  to  the  following  table  of  prefixes 
from  the  SI  system. 

The  prefix  abbreviation  is  shown  first,  followed  by  its  name,  and  by  the 
exponent  x  in  the  factor  1  0X  corresponding  to  each  prefix: 


Prefix 

Name 

X 

Prefix 

Name 

X 

Y 

yotta 

+24 

d 

deci 

-1 

Z 

zetta 

+21 

c 

centi 

-2 

E 

exa 

+  18 

m 

milli 

-3 

P 

peta 

+  15 

micro 

-6 

T 

tera 

+  12 

n 

nano 

-9 

G 

g'ga 

+9 

P 

pico 

-12 

M 

mega 

+6 

f 

femto 

-15 

k,K 

kilo 

+3 

a 

atto 

-18 

h,H 

hecto 

+2 

z 

zepto 

-21 

D(*) 

deka 

+  1 

y 

yocto 

-24 

(*)  In  the  SI  system,  this  prefix  is  da  rather  than  D.  Use  D  for  deka  in  the 
calculator,  however. 


Page  3-24 


To  enter  these  prefixes,  simply  type  the  prefix  using  the  {alpha}  keyboard.  For 
example,  to  enter  123  pm  (1  picometer),  use: 

Using  UBASE  to  convert  to  the  default  unit  (1  m)  results  in: 


123-l_pm 

123_ppi 

UBflSE(flNS(lH 

.  000000000 123_m 


COnYE  UEASE  UYrtL  UFhCT  -HJnIT  UnIT: 


Operations  with  units 

Once  a  quantity  accompanied  with  units  is  entered  into  the  stack,  it  can  be 
used  in  operations  similar  to  plain  numbers,  except  that  quantities  with  units 
cannot  be  used  as  arguments  of  functions  (say,  SQ  or  SIN).  Thus,  attempting 
to  calculate  LN(10_m)  will  produce  an  error  message:  Error:  Bad  Argument 
Type. 


Here  are  some  calculation  examples  using  the  ALG  operating  mode.  Be 
warned  that,  when  multiplying  or  dividing  quantities  with  units,  you  must 
enclosed  each  quantity  with  its  units  between  parentheses.  Thus,  to  enter,  for 
example,  the  product  1  2.5m  x  5.2_yd,  type  it  to  read  (1  2.5_m)*(5.2_yd) 


:  12.5_m-5.2_yd 

a]flAJJIIi™j|ILAn«IIIJ^JEIIrt*J(TTft*a 


which  shows  as  65_(m-yd).  To  convert  to  units  of  the  SI  system,  use  function 
UBASE: 


:  12.5_m-5.2_yd 

UBflSEtflHSdH 


2 


59.436_m 


COnVE  UEflSE  UYHL  UFflCT  -KJniT  UniT: 


Note:  Recall  that  the  ANS(1 )  variable  is  available  through  the  keystroke 
combination  CjjD^_  (associated  with  the  (enter}  key). 


Page  3-25 


To  calculate  a  division,  say,  3250  mi  /  50  h,  enter  it  as 
(3250_mi)/(50_h)  [enter}: 


3250-1-pu 
50-l_h 


which  transformed  to  SI  units,  with  function  UBASE,  produces: 

 b~_- 


:UBflSE(flHS(lH 


29.0576_^ 


COnVE  UEflSE  UYrtL  UFhCT  -HJnIT  UnIT: 


Addition  and  subtraction  can  be  performed,  in  ALG  mode,  without  using 
parentheses,  e.g.,  5  m  +  3200  mm,  can  be  entered  simply  as  5_m  + 
3200_mm  (enter}:   


UBflSE(flNS(lH 


5-l_m+3200-l_mm 

S200._mrn 


More  complicated  expression  require  the  use  of  parentheses,  e.g., 
(12_mm)*(l_cmA2)/(2_s)(^  


12'1-mm'l'l-cm 
2-l_s 


Stack  calculations  in  the  RPN  mode,  do  not  require  you  to  enclose  the 
different  terms  in  parentheses,  e.g., 

1  2_m  [enter}  1 .5_yd  [enter}  QT] 
3250_mi  [enter}  50_h  [enter}  C±J 


These  operations  produce  the  following  output: 


Page  3-26 


l: 


lS_Cm-yd) 


Also,  try  the  following  operations: 

5_m  [enter)  3200_mm  [enter}  [TJ 
]  2_rr\rr\  [enter)  l_cmA2  [enter)[^J  2_s  [enter) 

These  last  two  operations  produce  the  following  output: 


3: 


lo-invyaj 
2 

^  roro'cro 


Note:  Units  are  not  allowed  in  expressions  entered  in  the  equation  writer. 

Units  manipulation  tools 

The  UNITS  menu  contains  a  TOOLS  sub-menu,  which  provides  the  following 
functions: 

CONVERT(x,y):  convert  unit  object  x  to  units  of  object  y 
UBASE(x):         convert  unit  object  x  to  SI  units 
UVAL(x):  extract  the  value  from  unit  object  x 

UFACT(x,y):       factors  a  unit  y  from  unit  object  x 
->UNIT(x,y):      combines  value  of  x  with  units  of  y 

The  UBASE  function  was  discussed  in  detail  in  an  earlier  section  in  this 
chapter.  To  access  any  of  these  functions  follow  the  examples  provided 
earlier  for  UBASE.  Notice  that,  while  function  UVAL  requires  only  one 
argument,  functions  CONVERT,  UFACT,  and  ->UNIT  require  two  arguments. 

Try  the  following  exercises,  in  your  favorite  calculator  settings.  The  output 
shown  below  was  developed  in  ALG  mode  with  system  flat  1  1  7  set  to  SOFT 
menu: 


Examples  of  CONVERT 


Page  3-27 


These  examples  produce  the  same  result,  i.e.,  to  convert  33  watts  to  btu's 

CONVERT(33_W,l_hp)  [enter} 
CONVERT(33_W,l  l_hp)  (enter) 
These  operations  are  shown  in  the  screen  as:  


Examples  of  UVAL: 


CON  VERTC33- 1  _N,  1  ■  1  _hj=0 

4. 42537289566E-2_hp 
C0NVERT£33-1-1_N,1 

4.42537289566E-2_hp 


UVAL(25_ft/s)  (ENTER} 
UVAL(0.021_cmA3)(^ 


UVflLlSS-l-^r1 


UVfll_(.021_cm3] 


25. 
.021 


Examples  of  UFACT 


UFACT(l_ha,18_kmA2)(^) 
UFACT(1_mm,15.1_cm)  (enter} 


UFRCTl 


(l-l_ha, 


lS-l-km' 


2) 


.01_km 
:UFfiCT(l-l_mm,15-l_cm) 

.  l_cm 


Examples  of  ^UNIT 


^UNIT(25,l_m)(^) 
^UNIT(11.3,1_mph)(^ 


-HJNIT(25,M_mi 


25_m 


-^UHITC11.3,12-l-l_mph3 

1 1 . 3_mph 


COnVE  UEflSE  UVHL  UFhCT  -KJnIT  UnlT: 


Page  3-28 


Physical  constants  in  the  calculator 

Following  along  the  treatment  of  units,  we  discuss  the  use  of  physical 
constants  that  are  available  in  the  calculator's  memory.  These  physical 
constants  are  contained  in  a  constants  library  activated  with  the  command 
CONLIB.   To  launch  this  command  you  could  simply  type  it  in  the  stack: 

[alpha]  [alpha)  (Q  (g\  (n)(l)(T)  (b)  [alpha)  (enter) 

or,  you  can  select  the  command  CONLIB  from  the  command  catalog,  as 
follows:  First,  launch  the  catalog  by  using:  L  r>  J  cat  [alpha)®  .  Next,  use  the  up 
and  down  arrow  keys  (j^^z?  to  select  CONLIB.  Finally,  press  the  (JL)  (iii£i=Ll:iiiM) 
soft  menu  key.  Press  [enter) t  if  needed. 


The  constants  library  screen  will  look  like  the  following  (use  the  down  arrow 
key  to  navigate  through  the  library): 


olTzmann 
Vm:  molar  volume 
R:  universal  gas 
StdT:  std  temperature 
StdP:  std  pressure  1 


SI  -|  EH GL  | U Ti I T - 1 '.■  H L U E |  -+5TK  |  GUIT 


h:  Planck's  f 
hbar:  Dirac's 
q:  electronic  charge 
me:  electron  mass 


consTflms  library 
(j:  Stef  an-Boltzmann 
c:  speed  of  light 
e0:  permittivity 
ul0 :  permeability 
q_5  accel  of  gravity 


:  gravitation 


:i  -|  ETmjL  |UniT"|VHLUE|  -+5TK  |  CUIT 


nonsTftriTS  library  ^^^^^ 
mpme=  mp^me  ratio 
fine  structure 
0~  mag  flux  quantum 
F:  Faraday 
R«s  RudberQ 


consTflms  library 
ijlB  =  Bohr  magneton  f 
ijlH  =  nuclear  magneton 
^0:  photon  wavelength 
f0:  photon  frequency 
Compton  uavelen 


1  radian 


VALUE  -+:"TH  GUIT 


consTRms  library  § 
twoir:  2tt  radians 
angl:  ^  in  trig  mode 
c3:  Mien's 
kq:  k^c 


qe0 
ESI 


consTAms  library 

k^q 
e0^"q 
q*£0 

dielectric  const 


Page  3-29 


The  soft  menu  keys  corresponding  to  this  CONSTANTS  LIBRARY  screen 

include  the  following  functions: 

SI         when  selected,  constants  values  are  shown  in  SI  units 

ENGL    when  selected,  constants  values  are  shown  in  English  units  (*) 

UNIT     when  selected,  constants  are  shown  with  units  attached  (*) 

VALUE  when  selected,  constants  are  shown  without  units 

->STK   copies  value  (with  or  without  units)  to  the  stack 

QUIT     exit  constants  library 

(*)  Active  only  if  the  function  VALUE  is  active. 


This  is  the  way  the  top  of  the  CONSTANTS  LIBRARY  screen  looks  when  the 
option  VALUE  is  selected  (units  in  the  SI  system): 


Ub58E-23_J.-1 
Vm:  22.4141_l/gmol 
R:  S.31451_J^C9mol*K> 
StdT:  £73.15_K 
StdP:   101.3£5_kPa  4 


SI  ■  EriuL  uniT-  UhLU-  -+5TK  quit 


To  see  the  values  of  the  constants  in  the  English  (or  Imperial)  system,  press  the 


ESm  option: 


0063E-27_Btu/„ 
Vm:  359.0394_ftA3^1b„ 
R:  18.73164_psi*ftA3„ 
StdT:  491.67_nR 
StdP:  14.6959_psi 


SI   I E Tl ij L ■  I U Tl I T ■  I '.' H L U ■  I  -+STK  QUIT 


If  we  de-select  the  UNITS  option  (press  [QMiS  )  only  the  values  are  shown 
(English  units  selected  in  this  case): 


S063E-27 
Vm:  359.0394 
R:  10-73164 
StdT:  491.67 
StdP:   14.6959  i 


Page  3-30 


To  copy  the  value  of  Vm  to  the  stack,  select  the  variable  name,  and 

press  then,  press  EIIIEEil.  For  the  calculator  set  to  the  ALG,  the  screen 

will  look  like  this: 


COHLIB 

Vm:  359.  0394^ 


The  display  shows  what  is  called  a  togged  value,  Vm«  8394.  In  here, 
Vm,  is  the  tag  of  this  result.  Any  arithmetic  operation  with  this  number  will 

ignore  the  tag.  Try,  for  example:  CgJ  ^  CTD QD QD i™—       ,  which 

produces: 


: COHLIB 

Vm: 359. 0394 

:|_N(2-flNSm) 

57657931233 

The  same  operation  in  RPN  mode  will  require  the  following  keystrokes  (after 
the  value  of  Vm  was  extracted  from  the  constants  library):  ClI33^CZDC3 

LN 


Special  physical  functions 

Menu  1  1  7,  triggered  by  using  MENU(1  1  7)  in  ALG  mode,  or  1  1  7  [enter}  MENU 

in  RPN  mode,  produces  the  following  menu  (labels  listed  in  the  display  by 

using  C3^3? ):   

ZFhuTuR  I 

FANNING 

DflRCY 

F0^. 

SI DENS 
TDELTfl 


Z  F  H  H  T  |  F  H  Ti  fi  1 1 D  H  F;  H  'i'  |  F  Li ;-.  I L"  I  [i  E  Ti  I T  [i  E  L  T 


The  functions  include: 

ZFACTOR:  gas  compressibility  Z  factor  function 
FANNING:  Fanning  friction  factor  for  fluid  flow 
DARCY:  Darcy-Weisbach  friction  factor  for  fluid  flow 
FOX:  Black  body  emissive  power  function 
SIDENS:  Silicon  intrinsic  density 
TDELTA:  Temperature  delta  function 


Page  3-31 


In  the  second  page  of  this  menu  (press  Lnxt J )  we  find  the  following  items: 


I^.UB:l!r.M[rTTCTITTT»M.|:Jl7.J^:j 

In  this  menu  page,  there  is  one  function  (TINC)  and  a  number  of  units 
described  in  an  earlier  section  on  units  (see  above).  The  function  of  interest  is: 

TINC:    temperature  increment  command 

Out  of  all  the  functions  available  in  this  MENU  (UTILITY  menu),  namely, 
Z FACTOR,  FANNING,  DARCY,  FOA,  SIDENS,  TDELTA,  and  TINC,  functions 
FANNING  and  DARCY  are  described  in  Chapter  6  in  the  context  of  solving 
equations  for  pipeline  flow.  The  remaining  functions  are  described  following. 

Function  ZFACTOR 

Function  ZFACTOR  calculates  the  gas  compressibility  correction  factor  for 
nonideal  behavior  of  hydrocarbon  gas.  The  function  is  called  by  using 
ZFACTOR(xT,  yP),  where  xT  is  the  reduced  temperature,  i.e.,  the  ratio  of  actual 
temperature  to  pseudo-critical  temperature,  and  yP  is  the  reduced  pressure,  i.e., 
the  ratio  of  the  actual  pressure  to  the  pseudo-critical  pressure.   The  value  of  xT 
must  be  between  1 .05  and  3.0,  while  the  value  of  yP  must  be  between  0  and 
30.    Example,  in  ALG  mode: 

:ZFflCT0RE2.5,12.5) 

1 . 25980762398 


Z F M H T | F M Ti Ti 1 1 Ci H F; H 'i' |  FlU-.  |£IDEn|TDELT 


Function  fOX 

Function  FOA,  (T,  X)  calculates  the  fraction  (dimensionless)  of  total  black-body 
emissive  power  at  temperature  T  between  wavelengths  0  and  X.  If  no  units 
are  attached  to  T  and  X,  it  is  implied  that  T  is  in  K  and  X  in  m.  Example,  in 
ALG  mode: 

:F0^(452.,.  000013 

. 567343728392 


Z F M H T | F M Ti Ti 1 1 Ci H F; H 'i' |  FlU-.  |£IDEn|TDELT 


Page  3-32 


Function  SIDENS 

Function  SIDENS(T)  calculates  the  intrinsic  density  of  silicon  (in  units  of  1/cm3) 
as  a  function  of  temperature  T  (T  in  K),  for  T  between  0  and  1  685  K.  For 
example, 

:SIDENS(450.) 

6.07995618238E13 


Function  TDELTA 

Function  TDELTA(T0,Tf)  yields  the  temperature  increment  Tf  -  T0.  The  result  is 
returned  with  the  same  units  as  T0/  if  any.  Otherwise,  it  returns  simply  the 
difference  in  numbers.  For  example, 


:TDELTFl(25_nF,52_nC) 

-100.6_nF 


Z  F  H  H  T  |  F  H  Ti  fi  1 1 D  H  F;  H  'i'  |  F  Li ;-.  I L"  I  [i  E  Ti  I T  [i  E  L  T 


The  purpose  of  this  function  is  to  facilitate  the  calculation  of  temperature 
differences  given  temperatures  in  different  units.  Otherwise,  it's  simply 
calculates  a  subtraction,  e.g.,  

:TDELTffl250.,520.) 

-270. 

^JMilfflflfHIffl^lWilFtifliilliJiU 


Function  TINC 

Function  TINC(T0,AT)  calculates  T0+DT.  The  operation  of  this  function  is  similar 
to  that  of  function  TDELTA  in  the  sense  that  it  returns  a  result  in  the  units  of  T0. 
Otherwise,  it  returns  a  simple  addition  of  values,  e.g., 


TINC(125_nF,-25_fO 

S0_nF 

TIHCC256..25J 

2S1 . 

iHiHtlitHMttfflWTlWmSaiJ 


Page  3-33 


Defining  and  using  functions 

Users  can  define  their  own  functions  by  using  the  DEF  command  available 
thought  the  keystroke  sequence  l<n  JPEF     (associated  with  the  L  2  j  key).  The 
function  must  be  entered  in  the  following  format: 

Function_name(arguments)  =  expression_contoining_orguments 

For  example,  we  could  define  a  simple  functionH(x]  =  ln(x+l)  +  exp(-x). 
Suppose  that  you  have  a  need  to  evaluate  this  function  for  a  number  of 
discrete  values  and,  therefore,  you  want  to  be  able  to  press  a  single  button 
and  get  the  result  you  want  without  having  to  type  the  expression  in  the  right- 
hand  side  for  each  separate  value.  In  the  following  example,  we  assume  you 
have  set  your  calculator  to  ALG  mode.  Enter  the  following  sequence  of 
keystrokes: 

The  screen  will  look  like  this: 


: DEFIHEt HCx)=LHCx+l)+e  ) 
HOVflL 

Press  the  Cjjg  j  key,  and  you  will  notice  that  there  is  a  new  variable  in  your  soft 
menu  key  (1111).  To  see  the  contents  of  this  variable  press  Lr^  JO.  The  screen 
will  show  now: 

:  DEFIHE('HCx3=LHCx+13+eX') 
HOVflL 

*  ■*  x  'LhKx+l>+EXP<x>' 

■MHfflJIflilflgTMiaiia 

Thus,  the  variable  H  contains  a  program  defined  by: 
«^x'LN(x+l)  +  EXP(x)'  » 

This  is  a  simple  program  in  the  default  programming  language  of  the  HP  48 
G  series,  and  also  incorporated  in  the  HP  49  G  series.   This  programming 
language  is  called  UserRPL.   The  program  shown  above  is  relatively  simple 
and  consists  of  two  parts,  contained  between  the  program  containers  <<  >>  : 


Page  3-34 


•  Input:  ->  x      ->  x 

•  Process:  'LN(x+l)  +  EXP(x)  ' 

This  is  to  be  interpreted  as  saying:  enter  a  value  that  is  temporarily  assigned 
to  the  name  x  (referred  to  as  a  local  variable),  evaluate  the  expression 
between  quotes  that  contain  that  local  variable,  and  show  the  evaluated 
expression. 

To  activate  the  function  in  ALG  mode,  type  the  name  of  the  function  followed 

by  the  argument  between  parentheses,  e.g.,  IIEBQD^  L2jt™L)  •  Some 

examples  are  shown  below: 


HC2) 


HC1.2) 


LN(3)+e 
4. 1085742831 


FFhF;  EGHn 


LH[|+l|+e 


MMBannsniiEMMMCina 

In  the  RPN  mode,  to  activate  the  function  enter  the  argument  first,  then  press 
the  soft  menu  key  corresponding  to  the  variable  name  IIIII .    For  example, 
you  could  try:  L2J  [enter) ESI .  The  other  examples  shown  above  can  be 

entered  by  usi ng :  CD  CD  CD       MM ,  CD  CD  CD  G^ra)  ill . 


Functions  can  have  more  than  2  arguments.  For  example,  the  screen  below 
shows  the  definition  of  the  function  K(a,p)  =  a+fi,  and  its  evaluation  with  the 
arguments  K(^2fn),  and  K(] .2,2.3):  


iDEFINECKtocF^ffl+F1) 
:  K(J2  ,tt) 
:K(1.2,2.3) 


HOVRL 

J2+TT 

3.5 


The  contents  of  the  variable  K  are:  «  ->  a  p  7a+p'  ». 

Functions  defined  by  more  than  one  expression 

In  this  section  we  discuss  the  treatment  of  functions  that  are  defined  by  two  or 
more  expressions.  An  example  of  such  functions  would  be 

f  2  •  x  -  1,     x  <  0] 
jc    —  1 ,     x  >  0  I 


Page  3-35 


The  calculator  provides  the  function  IFTE  (IF-Then-Else)  to  describe  such 
functions. 

The  IFTE  function 

The  IFTE  function  is  written  as  IFTE(cond/f/on,  operationjfjrue,  operation_if_false) 
If  condition  is  true  then  operotion_if_true  is  performed,  else  operotion_if_folse 
is  performed.  For  example,  we  can  write  'f(x)  =  IFTE(x>0,  xA2-l,  2*x-l)/,  to 
describe  the  function  listed  above.  Function  IFTE  is  accessible  from  the 
function  catalog  (L J  cat  ).  The  symbol  '>'  (greater  than)  is  available  as 
(associated  with  the  L  ?*  J  key).  To  define  this  function  in  ALG  mode  use  the 
command:  DEF(f(x)  =  IFTE(x>0,  xA2-l ,  2*x-l )) 

then,  press  [enter]  .  In  RPN  mode,  type  the  function  definition  between 
apostrophes: 

'f(x)  =  IFTE(x>0,  xA2-l,  2*x-l)' 

then  press  C3D?^-  . 

Press  jvAR j  to  recover  your  variable  menu.    The  function  llllllli  should  be 
available  in  your  soft  key  menu.  Press  LiU ■III!  to  see  the  resulting  program: 
«  ^  x  /IFTE(x>0,  xA2-l ,  2*x-l )'  » 

To  evaluate  the  function  in  ALG  mode,  type  the  function  name,  f,  followed  by 
the  number  at  which  you  want  to  evaluate  the  function,  e.g.,  f(2),  then  press 
[enter]  .  In  RPN  mode,  enter  a  number  and  press  llllllli.  Check,  for  example, 
that  f(2)  =  3,  while  f(-2)  =  -5. 

Combined  IFTE  functions 

To  program  a  more  complicated  function  such  as 

-  x,    x  <  -2 
x  +  1,  -2<x<0 
x-1,  0<x<2 
x2,  x>2 

you  can  combine  several  levels  of  the  IFTE  function,  i.e., 

'g(x)  =  IFTE(x<-2,  -x,  IFTE(x<0,  x+1,  IFTE(x<2,  x-1,  xA2))),/ 


g(x)  =  < 


Page  3-36 


Define  this  function  by  any  of  the  means  presented  above,  and  check  that 
g(-3)  =  3,g(-l)  =  0,g(l)  =  0,g(3)  =  9. 


Page  3-37 


Chapter  4 

Calculations  with  complex  numbers 

This  chapter  shows  examples  of  calculations  and  application  of  functions  to 
complex  numbers. 

Definitions 

A  complex  number  z  is  a  number  written  as  z  =  x  +  iy,  where  x  and  y  are 
real  numbers,  and  /  is  the  imaginary  unit  defined  by  i2  =  -1 .    The  complex 
number  x+iy  has  a  real  part,  x  =  Re(z),  and  an  imaginary  par\,  y  =  lm(z). 
We  can  think  of  a  complex  number  as  a  point  P(x,y)  in  the  x-y  plane,  with  the 
x-axis  referred  to  as  the  real  axis,  and  the  y-axis  referred  to  as  the  imaginary 
axis.  Thus,  a  complex  number  represented  in  the  form  x+iy  is  said  to  be  in  its 
Cartesian  representation.  An  alternative  Cartesian  representation  is  the 
ordered  pair  z  =  (x,y).  A  complex  number  can  also  be  represented  in  polar 
coordinates  (polar  representation)  as  z  =  re  10  =  r-cos6  +  i  r-sinO,  where  r  = 

jz I  =  tJx2  +  y2  is  the  magnitude  of  the  complex  number  z,  and  6=  Arg(z)  = 

arctan(y/x)  is  the  argument  of  the  complex  number  z.  The  relationship 
between  the  Cartesian  and  polar  representation  of  complex  numbers  is  given 
by  the  Euler  formula:  e'e=  cos  6+  i  sin  0.    The  complex  conjugate  of  a 
complex  number  z  =  x  +  iy  =  re  10 ,  is  z  =  x  -  iy  =  re  'l0 .  The  complex 
conjugate  of  /  can  be  thought  of  as  the  reflection  of  z  about  the  real  (x)  axis. 
Similarly,  the  negative  of  z,  -z  =  -x-iy  =  -  re  10 ,  can  be  thought  of  as  the 
reflection  of  z  about  the  origin. 

Setting  the  calculator  to  COMPLEX  mode 

When  working  with  complex  numbers  it  is  a  good  idea  to  set  the  calculator  to 
complex  mode,  use  the  following  keystrokes:  (w*|)fiSH  CZD^^^^CDl^HIO 
The  COMPLEX  mode  will  be  selected  if  the  CAS  MODES  screen  shows  the 
option  _Complex  checked  off,  i.e., 


CAS  MODEL" 

Iri'j-zp  uar:K 
HoduU:  13 

_nuH«ric  _flpprox  QtOHpLcx 
^  Verbose  _St«p'St«p  _Incr  Poh 
^Risorous _SiHp  non-Rational. 
ftUoH  coHpUx  nuHbcrs?  


Page  4-1 


Press  IliEIII,  twice,  to  return  to  the  stack. 


Entering  complex  numbers 

Complex  numbers  in  the  calculator  can  be  entered  in  either  of  the  two 
Cartesian  representations,  namely,  x+iy,  or  (x,y).    The  results  in  the  calculator 
will  be  shown  in  the  ordered-pair  format,  i.e.,  (x,y).  For  example,  with  the 
calculator  in  ALG  mode,  the  complex  number  (3.5,-1 .2),  is  entered  as: 


A  complex  number  can  also  be  entered  in  the  form  x+iy.  For  example,  in 
ALG  mode,  3.5-7.2/  is  entered  as: 


The  following  screen  results  after  entering  these  complex  numbers: 


(3-5,-1.23 
3.5-1.2-i 


(3.5,-1.2) 
(3.5,-1.2) 


EDIT   YIEH    F;HL    £Tqp-  PURGE  CLEAR 


In  RPN  mode,  these  numbers  will  be  entered  using  the  following  keystrokes: 

(Notice  that  the  change-sign  keystroke  is  entered  after  the  number  1 .2  has 
been  entered,  in  the  opposite  order  as  the  ALG  mode  exercise),  and 

(Notice  the  need  to  enter  an  apostrophe  before  typing  the  number  3.5-1 .2i  in 
RPN  mode).  The  resulting  RPN  screen  will  be:  


-:: 

2: 

(3.5,-1.2) 

l: 

3.5-1.2-i 

EDIT  |  VIEH  | 

RCL  |  £TDt  |PURGE|CLERR 

Notice  that  the  last  entry  shows  a  complex  number  in  the  form  x+iy.  This  is  so 
because  the  number  was  entered  between  apostrophes,  which  represents  an 
algebraic  expression.  To  evaluate  this  number  use  the  EVAL  key(  [eval)). 


::: 

£: 

(3.5,-1.2) 

l: 

(3.5,-1.2) 

EDIT  |  VIEH  | 

RCL  |  STOHPURGEICLEflR 

Once  the  algebraic  expression  is  evaluated,  you  recover  the  complex  number 
(3.5,1.2). 


Page  4-2 


Polar  representation  of  a  complex  number 

The  result  shown  above  represents  a  Cartesian  (rectangular)  representation  of 
the  complex  number  3.5-1 .2i.   A  polar  representation  is  possible  if  we 
change  the  coordinate  system  to  cylindrical  or  polar,  by  using  function  CYLIN. 
You  can  find  this  function  in  the  catalog  (L    J  cat  ).    Changing  to  polar 
shows  the  result: 

|l:  t3.7^.33S297354329l| 


EDIT  |  '.' I E H  |:"ThlK|  F;lL  |PURGE|CLEflR 


For  this  result  the  angular  measure  is  set  to  radians  (you  can  always  change  to 
radians  by  using  function  RAD).  The  result  shown  above  represents  a 
magnitude,  3.7,  and  an  angle  0.33029....  The  angle  symbol  (Z)  is  shown  in 
front  of  the  angle  measure. 

Return  to  Cartesian  or  rectangular  coordinates  by  using  function  RECT 
(available  in  the  catalog,  Lr>J  cat  ).   A  complex  number  in  polar 
representation  is  written  as  z  =  r-e'e.  You  can  enter  this  complex  number  into 
the  calculator  by  using  an  ordered  pair  of  the  form  (r,  Z0).  The  angle  symbol 
(Z)  can  be  entered  as  ^™)Lr^JLU  •     For  example,  the  complex  number  z  = 
5.2e]  5l,  can  be  entered  as  follows  (the  figures  show  the  stack,  before  and 
after  entering  the  number): 


l:  (3.5,1.2) 
<5.2Pil.5> 

2:  (3.5,1.2) 
l:  C. 367333443672, 5. 18> 

HatilUH!lHiraMIHHlllftHUI*H 

Because  the  coordinate  system  is  set  to  rectangular  (or  Cartesian),  the 
calculator  automatically  converts  the  number  entered  to  Cartesian  coordinates, 
i.e.,  x  =  r  cos  0,  y  =  r  sin  0,  resulting,  for  this  case,  in  (0.3678...,  5.18...). 

On  the  other  hand,  if  the  coordinate  system  is  set  to  cylindrical  coordinates 
(use  CYLIN),  entering  a  complex  number  (x,y),  where  x  and  y  are  real 
numbers,  will  produce  a  polar  representation.  For  example,  in  cylindrical 
coordinates,  enter  the  number  (3. ,2.).  The  figure  below  shows  the  RPN  stack, 
before  and  after  entering  this  number: 


Page  4-3 


Simple  operations  with  complex  numbers 

Complex  numbers  can  be  combined  using  the  four  fundamental  operations 
(L  +  JL  —  JL x  JL-*- J).  The  results  follow  the  rules  of  algebra  with  the  caveat  that 
i2=  -7.    Operations  with  complex  numbers  are  similar  to  those  with  real 
numbers.  For  example,  with  the  calculator  in  ALG  mode  and  the  CAS  set  to 
Complex,  well  attempt  the  following  sum:  (3+5i)  +  (6-3i): 

:3.+5.-i+6.-3.-i 

(9.  ,2J 


EDIT  |  '.'IEH  |  RCL  |  £T0t  |FUF;ijE|CLEHF; 


Notice  that  the  real  parts  (3+6)  and  imaginary  parts  (5-3)  are  combined 
together  and  the  result  given  as  an  ordered  pair  with  real  part  9  and 
imaginary  part  2.    Try  the  following  operations  on  your  own: 

(5-2i)  -  (3+4i)  =  (2,-6) 
(3-i)-(2-4i)  =  (2,-14) 
(5-2i)/(3+4i)  =  (0.28,-1 .04) 
l/(3+4i)  =  (0.12,  -0.16) 

Notes: 

The  product  of  two  numbers  is  represented  by:  (x1+iy1)(x2+iy2)  =  (x^  -  yiy2)  + 
i  (x!y2  +  x2yi). 

The  division  of  two  complex  numbers  is  accomplished  by  multiplying  both 
numerator  and  denominator  by  the  complex  conjugate  of  the  denominator, 
i.e., 

xx  +  iyx  =  xx  +  iyx    x2  -iy2  =  xxx2  +  yxy2  |     x2yx  -xxy2 
x2  +  iy2     x2  +  iy2   x2  -  iy2       x\  +  y\  x2  +  y\ 

Thus,  the  inverse  function  INV  (activated  with  the  L  '/*  J  key)  is  defined  as 
1  1      x-iy         x  y 

  —  r-  —  1-  /  

x  +  iy     x  +  iy  x-iy     x2  +  y2       x2  +  y2 
Changing  sign  of  a  complex  number 

Changing  the  sign  of  a  complex  number  can  be  accomplished  by  using  the 
C^D  key,  e.g.,  -(5-3i)  =  -5  +  3i 


Page  4-4 


i-(5.-3.-i) 


(-5.  ,3.) 


EDIT  I '.'IEH  I  RCL    STOfr  PURGE  CLEAR 


Entering  the  unit  imaginary  number 

To  enter  the  unit  imaginary  number  type  :  ), 


(0. ,  1.) 

 i 


Notice  that  the  number  /  is  entered  as  the  ordered  pair  [0, 1)  if  the  CAS  is  set 
to  APPROX  mode.  In  EXACT  mode,  the  unit  imaginary  number  is  entered  as  /. 

Other  operations 

Operations  such  as  magnitude,  argument,  real  and  imaginary  parts,  and 
complex  conjugate  are  available  through  the  CMPLX  menus  detailed  later. 

The  CMPLX  menus 

There  are  two  CMPLX  (CoMPLeX  numbers)  menus  available  in  the  calculator. 
One  is  available  through  the  MTH  menu  (introduced  in  Chapter  3)  and  one 
directly  into  the  keyboard  (Lr*  Jcmplx  ),  The  two  CMPLX  menus  are  presented 
next. 


CMPLX  menu  through  the  MTH  menu 

Assuming  that  system  flag  1  1  7  is  set  to  CHOOSE  boxes  (see  Chapter  2), 
the  CMPLX  sub-menu  within  the  MTH  menu  is  accessed  by  using: 
(jT]/ww_  L_9j         .  The  following  sequence  of  screen  shots  illustrates  these 
steps: 


HATH  HE TlLI 

H. HYPERBOLIC.  I 

5 .  REAL.. 

CEASE.. 

F.  PROBABILITY.. 

S .  FFT..  | 

i  ■  1 1 1  ii  i  ^^^^m\ 

COHPLEK  HEAU 

iTi'h™     "  "  "~[ 

3.C-* 
H.R-K 
5.  AGS 

C.ARG  1 

The  first  menu  (options  1  through  6)  shows  the  following  functions: 


Page  4-5 


RE(z)     :  Real  part  of  a  complex  number 
IM(z)     :  Imaginary  part  of  a  complex  number 

C^R(z)  :  Takes  a  complex  number  (x,y)  and  separates  it  into  its  real  and 
imaginary  parts 

R^C(x,y):  Forms  the  complex  number  (x,y)  out  of  real  numbers  x  and  y 
ABS(z)  :  Calculates  the  magnitude  of  a  complex  number  or  the  absolute  value 
of  a  real  number. 

ARG(z):  Calculates  the  argument  of  a  complex  number. 


>ptions  7  through  10)  are 

the 

LUMFLEM  HEMJ 

6.ARG 

S.nEG 

5. con j  J 

ollowmg: 


SIGN(z)  :  Calculates  a  complex  number  of  unit  magnitude  as  z/ 1  z  | . 

NEG     :  Changes  the  sign  of  z 

CONJ(z):  Produces  the  complex  conjugate  of  z 

Examples  of  applications  of  these  functions  are  shown  next.  Recall  that,  for 
ALG  mode,  the  function  must  precede  the  argument,  while  in  RPN  mode,  you 
enter  the  argument  first,  and  then  select  the  function.  Also,  recall  that  you  can 
get  these  functions  as  soft  menus  by  changing  the  setting  of  system  flag  1  1  7 
(See  Chapter  3). 


This  first  screen  shows  functions  RE,  IM,  and  C->R.    Notice  that  the  last 
function  returns  a  list  {3.  5.}  representing  the  real  and  imaginary  components 
of  the  complex  number: 


:RE(3.-2.-i) 
:  IIK3.-2.-i) 
:OR(3.+5.-i) 


3. 
-2. 


<3.  5.) 


The  following  screen  shows  functions  R->C,  ABS,  and  ARG.  Notice  that  the 
ABS  function  gets  translated  to  |  3.+5.-i  | ,  the  notation  of  the  absolute  value. 


Page  4-6 


Also,  the  result  of  function  ARG,  which  represents  an  angle,  will  be  given  in 
the  units  of  angle  measure  currently  selected.  In  this  example,  ARG(3.+5.-i) 
1 .0303...  is  given  in  radians. 


<3.  5 
(5.  ?2.) 


R+C(5.,2.:i 
l3.+5.-il 


5.83095189483 
FlRG(3.+5.-i) 

1 . 03037632652 


In  the  next  screen  we  present  examples  of  functions  SIGN,  NEG  (which  shows 
up  as  the  negative  sign  - ),  and  CONJ. 


1 . 03037682652) 
:SIGN(-2.+3.-i) 
-.554700196225, . S320=> 
:-C-2.+3.-i) 

C2.  ,-3.) 

:C0HJC-2.+3.-i) 

C-2. 3-3.) 


CMPLX  menu  in  keyboard 

A  second  CMPLX  menu  is  accessible  by  using  the  right-shift  option  associated 
with  the  CD  key,  i.e.,  ^Jcmx  .    With  system  flag  117  set  to  CHOOSE 


COHPLEK  METlU 

a.  flES 
3. con J 

H.i 
5.IH 

e.nEG 

COHPLEK  METlU 

3. con j 

H.i 

S.IH 

C.nEG 

F.P.E 

The  resulting  menu  include  some  of  the  functions  already  introduced  in  the 
previous  section,  namely,  ARG,  ABS,  CONJ,  IM,  NEG,  RE,  and  SIGN.  It  also 
includes  function   /  which  serves  the  same   purpose  as  the  keystroke 

combination  l<n ~)j  ,  i.e.,  to  enter  the  unit  imaginary  number  /  in  an 

expression. 

The  keyboard-base  CMPLX  menu  is  an  alternative  to  the  MTH-based  CMPLX 
menu  containing  the  basic  complex  number  functions.  Try  the  examples 
shown  earlier  using  the  keyboard-based  CMPLX  menu  for  practice. 


Page  4-7 


Functions  applied  to  complex  numbers 

Many  of  the  keyboard-based  functions  defined  in  Chapter  3  for  real  numbers, 
e.g.,  SQ,  ,LN,  ex,  LOG,  10x,  SIN,  COS,  TAN,  ASIN,  ACOS,  ATAN,  can  be 
applied  to  complex  numbers.  The  result  is  another  complex  number,  as 
illustrated  in  the  following  examples.  To  apply  this  functions  use  the  same 


:SQ(3.+4.-i) 

-7. ,24.) 

:J3.+4.-i 

(2. ,  1.) 

:RL0G(2.-i) 

[-66.32S151B19, ■ 

-74.39E* 

SIN(4.-3.-i) 
[-7. 61923172032, 6. 543 1> 
:C0SC-5.+7.-i) 
[155.  536303519,  -525.  7^ 
:TRN(3.+3.-i) 
[-1.4340S15S162E-3, 1 .  £► 


L0G(5.+3.-i) 
[.  765739453521 ,  .  23470 1* 
.  5.-4.-i 

[-e97.  0093146996,  1 12.  31> 

LN(5.-6.-i) 
!2 .  05543693209,  -■  3760^ 


„  R3IN(7.+3.-i) 
.71663915401, 3. 057141* 
,  RC0S(8.+3.-i) 
[.  361040042712,  -2.  3357^ 
:RTRN(-l.+2.-i) 
[-1.33397252229,-4023 


Note:  When  using  trigonometric  functions  and  their  inverses  with  complex 
numbers  the  arguments  are  no  longer  angles.  Therefore,  the  angular  measure 
selected  for  the  calculator  has  no  bearing  in  the  calculation  of  these  functions 
with  complex  arguments.  To  understand  the  way  that  trigonometric  functions, 
and  other  functions,  are  defined  for  complex  numbers  consult  a  book  on 
complex  variables. 


Functions  from  the  MTH  menu 

The  hyperbolic  functions  and  their  inverses,  as  well  as  the  Gamma,  PSI,  and 
Psi  functions  (special  functions)  were  introduced  and  applied  to  real  numbers 
in  Chapter  3.  These  functions  can  also  be  applied  to  complex  numbers  by 
following  the  procedures  presented  in  Chapter  3.  Some  examples  are  shown 
below: 


:SIHHC4.-6.-i3 

[26.  2029676173,  7.  6303^ 

:C0SHCl.-i3 

[.  333730025131 ,-.  9333^ 
:TflNH(-l.+i) 

[- 1 .  03392332734,  .27 1 7=> 


RSINH(7.-9.-i) 
|[3.  12644592412,  -.  9073E* 
RC0SHC3.-i) 

1.31344645923, 1.5707^ 
:RTRNH(l.-6.-i) 
C2.  634012S9145E-2, -1 .  4* 
b*<n:ir:^<n:iM.H;ir:Tro;iBfnn;ir:ifnn; 


Page  4-8 


The  following  screen  shows  that  functions  EXPM  and  LNP1  do  not  apply  to 
complex  numbers.  However,  functions  GAMMA,  PSI,  and  Psi  accept  complex 
numbers: 


:GAMMA(4.+5.-i) 


:  LKKru^.  -D.  -1J 

"Bad  Argument  Type" 
:LNPl(-9.-i) 

"Bad  _Argument  Type" 


[.  149655327961,  ■314603* 
:PSI(l.-i,3.:i 


[-1 .  52237444395,  .  3172E* 
:Psi(5.+9.-i) 


[2.30854964207,  !■  10681* 


Function  DROITE:  equation  of  a  straight  line 

Function  DROITE  takes  as  argument  two  complex  numbers,  say,  x^iy!  and 
x2+iy2/  and  returns  the  equation  of  the  straight  line,  say,  y  =  a+bx,  that 
contains  the  points  (x^yj  and  (x2,y2).  For  example,  the  line  between  points 
A(5,-3)  and  B(6,2)  can  be  found  as  follows  (example  in  Algebraic  mode): 


:DR0ITEC5-3-i,6+2-i) 

Y=5-(X-5)+-3 


Function  DROITE  is  found  in  the  command  catalog  (Lr>J  cat  ). 


Using  EVAL(ANS(1))  simplifies  the  result  to: 


:DR0ITE(5-3-i,6+2-i) 


Y=5-(X-5)+-3 
:EVfiL(fiNS(D) 

Y=5-X-28 


Page  4-9 


Chapter  5 

Algebraic  and  arithmetic  operations 

An  algebraic  object,  or  simply,  algebraic,  is  any  number,  variable  name  or 
algebraic  expression  that  can  be  operated  upon,  manipulated,  and  combined 
according  to  the  rules  of  algebra.  Examples  of  algebraic  objects  are  the 
following: 

•  A  number:  1  2.3,  1 5.2_m,  V,  V,  "\ 

•  A  variable  name:  'a',  'ux',  'width',  etc. 

•  An  expression:     V°A2/4Vf  WD)*(VA2/(2*g))' 

•  An  equation:  /Q=(Cu/n)*A(y)*R(y)A(2/3)*SoA0.5, 


Entering  algebraic  objects 

Algebraic  objects  can  be  created  by  typing  the  object  between  single  quotes 
directly  into  stack  level  1  or  by  using  the  equation  writer  L  r>  j  eqw  .  For 
example,  to  enter  the  algebraic  object  /7i*DA2/4/  directly  into  stack  level  1 

use:CZI3C5D5  CKD5^@CZDCII]C±I)C£l)^^)  •   The   resulting  screen  is 

shown  next  for  both  the  ALG  mode  (left-hand  side)  and  the  RPN  mode  (right- 
hand  side):   


l: 


TT-D 


EaaBBnaMBBianMHBna 

An  algebraic  object  can  also  be  built  in  the  Equation  Writer  and  then  sent  to 
the  stack.  The  operation  of  the  Equation  Writer  was  described  in  Chapter  2. 
As  an  exercise,  build  the  following  algebraic  object  in  the  Equation  Writer: 


After  building  the  object,  press  to  show  it  in  the  stack  (ALG  and  RPN  modes 
shown  below): 


a    A     l_  1 

D  2-9 

2 

f-V  -L 

3- 

l: 

2 

f-V  -L 

2-9-D 

2-9-D 

Page  5-1 


Simple  operations  with  algebraic  objects 

Algebraic  objects  can  be  added,  subtracted,  multiplied,  divided  (except  by 
zero),  raised  to  a  power,  used  as  arguments  for  a  variety  of  standard 
functions  (exponential,  logarithmic,  trigonometry,  hyperbolic,  etc.),  as  you 
would  any  real  or  complex  number.    To  demonstrate  basic  operations  with 
algebraic  objects,  let's  create  a  couple  of  objects,  say  /7i*RA2/  and  'g*tA2/4', 
and  store  them  in  variables  Al  and  A2  (See  Chapter  2  to  learn  how  to  create 
variables  and  store  values  in  them).  Here  are  the  keystrokes  for  storing 

variables  Al  in  ALG  mode:CDC3D5  CZD(^f3CZDCDCD  Hz^t) 

^™J(3L/J  (^ter},  resulting  in: 


The  keystrokes  corresponding  to  RPN  mode  are:  CZDGDS  [j^[alpha}(r} 

After  storing  the  variable  A2  and  pressing  the  key,  the  screen  will  show  the 
variables  as  follows: 


2' 


>Fl2 


4 


In  ALG  mode,  the  following  keystrokes  will  show  a  number  of  operations  with 
the  algebraics  contained  in  variables  EI3III  and  IE5II  (press  (j^J  to  recover 
variable  menu): 


iiii-s-if-fMiii  L+J  iUi:::lllll  [ENTER} 


:fll+fl2 


2  2 

4'R   'TT+t  '9 


IKIQIl  [3D  iEEII  [enter) 


2  2 

4'R   ■TT+t  '9 

:ni-n2 

4 

2  2 

4-R   -TT-t  -9 

4 

Page  5-2 


Ill  QD 

iiii.:.!:::!i|n  [ENTER] 

2  2 

4-R   -TT-t  -9 

:fll-fl2 

4 

2  2 

t    -9-R  -TT 

4 

:::::;:::*:::::: 

!;!!.:.!:.:!!!!! 

1 

:  LHCRli 

2 

4-R  -TT 

Lh(r£-tt. 

Sill  GD 


A. 

.  m 

4 

"H2 

4-R  -TT 

t£.9 

L  ^  J  f  jjjj. 


FI2 


LnIr^-tt 
,  2 


The  same  results  are  obtained  in  RPN  mode  if  using  the  following  keystrokes: 

liThiij  [enter)    iljiliilil [_  -|-  j  lITFlil  {Jnter] liTPIl  I  —  ] 

ilifiii  f  fa/7-0?  i  lliTilll  f  v  i  IElIIS  [fly7»jj  ISEOiil!  GlD 


lEIDljj  [enter]  q=>j  


Functions  in  the  ALG  menu 

The  ALG  (Algebraic)  menu  is  available  by  using  the  keystroke  sequence 
LrU^S  (associated  with  the  LiU  key).   With  system  flag  1  1  7  set  to 


ALG  HEnU 


2 .  EMFHTlD 

3.  FACTOR 
H.LnCOLLECT 
5.  Lin 

G.PHRTFRHC 


RLG  HEnU 

H.LnCOLLECT  0 
5.  Lin 

G.PARTFRRC  I 
? . SOLVE  1 
3.SUEST  1 

™  ^^^^"1 

Rather  than  listing  the  description  of  each  function  in  this  manual,  the  user  is 
invited  to  look  up  the  description  using  the  calculator's  help  facility:  [tool]  [nxt] 
SIE1III  [enter)  .  To  locate  a  particular  function,  type  the  first  letter  of  the  function. 
For  example,  for  function  COLLECT,  we  type  [alpha)®  ,  then  use  the  up  and 
down  arrow  keys,  <^>^3? ,  to  locate  COLLECT  within  the  help  window. 

To  complete  the  operation  press  139.  Here  is  the  help  screen  for  function 
COLLECT: 


Page  5-3 


COLLECT: 
Recursive      f actoriza- 
tion  of  a  polynomial 
oyer  integers 
COLLECT  t:X*2-4> 

See :  EXFAHD  FACTOR  


We  notice  that,  at  the  bottom  of  the  screen,  the  line  See:  EXPAND  FACTOR 
suggests  links  to  other  help  facility  entries,  the  functions  EXPAND  and 
FACTOR.  To  move  directly  to  those  entries,  press  the  soft  menu  key  for 
EXPAND,  and  BIS  for  FACTOR.    Pressing  EMM,  for  example,  shows  the 
following  information  for  EXPAND: 

EXPAND:  I 
Expands  and  simplifies 
an  algebraic  expr- 
EXPflND<<X+2>*<X-2>> 

X--2-4 

See :  COLLECT  SIMPLIFY 

The  help  facility  provides  not  only  information  on  each  command,  but  also 
provides  an  example  of  its  application.  This  example  can  be  copied  onto 
your  stack  by  pressing  the  B333  soft  menu  key.  For  example,  for  the  EXPAND 
entry  shown  above,  press  the         soft  menu  key  to  get  the  following  example 
copied  to  the  stack  (press  [enter]  to  execute  the  command): 


:HELP 

:  EXPANDECX+2MX-2H 


Thus,  we  leave  for  the  user  to  explore  the  applications  of  the  functions  in  the 
ALG  (or  ALGB)  menu.  This  is  a  list  o 


HP]  HEHU 


2 .  e::fhnd 

3.  FACTOR 
H.mCOLLECT 
5.  Lin 

G.PHRTFRhC 


f  the 

commands: 

hLG  HEnu 

H . LRCOLLECT  p 

5.  Lin 

6 . PftRTFRflC  I 

7. SOLVE  1 

Page  5-4 


The  help  facility  will  show  the  following  information  on  the  commands: 


COLLECT: 


COLLECT: 
Recurs  i  u e      f  act  or  i  za- 
tion  of  a  polynomial 
oyer  integers 
COLLECT <X*2-4> 

<X+2>*<X-2> 
See:  EXPAND  FACTOR 


EXPAND: 


EXPAND: 
expands  and  simplifies 
an  algebraic  expr- 
:XPAHD((X+2>*t:X-2>> 

X--2-4] 

See:  COLLECT  SIMPLIFY 


:ACTOR: 


FACTOR: 

Factorizes  an  integer 
□r  a  polynomial 
FACTOR <X*2-2> 

<X+J"2XX-J"2> 

See:  EXPAND  COLLECT 

i^i^imiMH^im^mtiLiiUji 


NCOLLECT: 


LNCOLLECT: 
Col lects  logarithms 
LNCOLLECT < LN t X > +LN < Y > > 
LN(X*Y> 


See:  TEXPAND 


UN: 


Linearization  of 
exponent  ials 
LIN<EXP<X>A2> 

EXP(2*X> 
See:  TEXPAND  TLIH 


PARTFRAC: 


WTFRAC: 
Performs  partial  frac- 
■  ion  decomposition  on 

fraction 
sflRTFRflC<2XA2AXA2-l  >  > 
2+l^<X-l>-l^<X+l> 
iee:  PROPFRAC 


SOLVE: 


SOLVE: 

Solves  a  (.or  a  set  of) 
o 1 y nom  i  a 1  equat  i  on 
0LVE<XA4-l=3iX> 

£X=T2  X=--J"2 } 

See:  LINSOLVE  SOLVEVX 


SUBST: 


SUBST: 

Substitutes  a  ualue 
for  a  variable  in  an 
expression 
SUBST CA^2+1?A=2> 

2^2+1 

See:  


TEXPAND: 


TEXPAND: 

Expands  transcendental 
f unct  ions 
TEXPAND < EXP (X+Y>> 

EXP<X>*EXP(Y> 

See:  LIN  TLIN 


Page  5-5 


Note:  Recall  that,  to  use  these,  or  any  other  functions  in  the  RPN  mode, 
you  must  enter  the  argument  first,  and  then  the  function.  For  example,  the 
example  for  TEXPAND,  in  RPN  mode  will  be  set  up  as: 

CD  C5D      GD  S™)  (3  GD  S™)  (9  G^r) 

At  this  point,  select  function  TEXPAND  from  menu  ALG  (or  directly  from 
the  catalog  L J  cat  ),  to  complete  the  operation. 


Other  forms  of  substitution  in  algebraic  expressions 

Functions  SUBST,  shown  above,  is  used  to  substitute  a  variable  in  an 
expression.  A  second  form  of  substitution  can  be  accomplished  by  using  the 

L  r> J  ^   (associated  with  the  I  key).     For  example,  in  ALG  mode,  the 

following  entry  will  substitute  the  value  x  =  2  in  the  expression  x+x2.  The 
figure  to  the  left  shows  the  way  to  enter  the  expression  (the  substituted  value, 
x=2,  must  be  enclosed  in  parentheses)  before  pressing  [enter)  .  After  the  (enter) 
key  is  pressed,  the  result  is  shown  in  the  right-hand  figure: 


:x+x^ 

x=2 

2 

2+2 

In  RPN  mode,  this  can  be  accomplished  by  entering  first  the  expression  where 
the  substitution  will  be  performed  (x+x2),  followed  by  a  list  (see  Chapter  8) 
containing  the  substitution  variable,  an  space,  and  the  value  to  be  substituted, 
i.e.,  {x  2}.  The  final  step  is  to  press  the  keystroke  combination:  LxU  I  . 


T  ■ 

3: 
2: 

X+X 

l:  

<x  2> 

CftSCMl HELP  |         |         I  1 

T  ■ 

3: 
2: 

l: 

2 

2+2 

The  required  keystrokes  are  the  following: 


□J  (ALPHA)  [JT]  (X)  [TJ  [ALPHA)  [jT]  (xj  £YJ  (^TER) 

C5D{j  (^[^(^(j^(^(enter)   ^(enter) 


Page  5-6 


In  ALG  mode,  substitution  of  more  than  one  variable  is  possible  as  illustrated 


in  the  fo 


lowing  example  (shown  before  and  after  pressing  [enter}) 


WEICfl=2,B=3) 


fl+JBI <fl=2,B=3> 


2+J3 


In  RPN  mode,  it  is  also  possible  to  substitute  more  than  one  variable  at  a  time, 
as  illustrated  in  the  example  below.  Recall  that  RPN  mode  uses  a  list  of 
variable  names  and  values  for  substitution. 


4: 

3: 

2: 

n+JE 

l: 

{R  2  E  3) 

CflSCHl HELP  III! 

4: 

3: 

2: 

l: 

2+J3 

A  different  approach  to  substitution  consists  in  defining  the  substitution 
expressions  in  calculator  variables  and  placing  the  name  of  the  variables  in 
the  original  expression.  For  example,  in  ALG  mode,  store  the  following 
variables: 


a+b 

:'tt+1'K 

TT+1 

Then,  enter  the  expression  A+B: 


a+b 

:'tt+1>E 

TT+1 

:fl+B 

Tr+a+b+1 

The  last  expression  entered  is  automatically  evaluated  after  pressing  the  [enter} 
key,  producing  the  result  shown  above. 

Operations  with  transcendental  functions 

The  calculator  offers  a  number  of  functions  that  can  be  used  to  replace 
expressions  containing  logarithmic,  exponential,  trigonometric,  and 


Page  5-7 


hyperbolic  functions  in  terms  of  trigonometric  identities  or  in  terms  of 
exponential  functions.  The  menus  containing  functions  to  replace 
trigonometric  functions  can  be  obtained  directly  from  the  keyboard  by 
pressing  the  right-shift  key  followed  by  the  8  key,  i.e.,  LHJ_™^  .  The 
combination  of  this  key  with  the  left-shift  key,  i.e.,  L  r>  J  exp&ln  r  produces  a 
menu  that  lets  you  replace  expressions  in  terms  of  exponential  or  natural 
logarithm  functions.  In  the  next  sections  we  cover  those  menus  in  more  detail. 


Expansion  and  factoring  using  log-exp  functions 

The  Cji  Jexp&ln  produces  the  following  menu: 


EKFEin  HERU 


2 .  E!!FM 

3.  Lin 

H.LRCOLLECT 

s.lrpi 
g.tekprrd 


EKFSJ.R  HERU 


2 .  E!!FM 

3.  Lin 

H.LRCOLLECT 

s.lrfi 
g.tekfrrd 


Information  and  examples  on  these  commands  are  available  in  the  help 

facility  of  the  calculator.  Some  of  the  command  listed  in  the  EXP&LN  menu, 

i.e.,  LIN,  LNCOLLECT,  and  TEXPAND  are  also  contained  in  the  ALG  menu 

presented  earlier.  Functions  LNP1  and  EXPM  were  introduced  in  menu 

HYPERBOLIC,  under  the  MTH  menu  (See  Chapter  2).  The  only  remaining 

function  is  EXPLN.  Its  description  is  shown  in  the  left-hand  side,  the  example 

from  the  help  facility  is  shown  to  the 
EXPLN: 

Rewrites  transcendent 
functions  in  terns  of 
EXP  and  LH 
EXPLH(COS(X>> 
(EXPCi*X>+l^EXP(i*X>> 
See:  SI NCOS  EXP2HYP 


:HELP 

:  EXPLHCC0SCX3) 

ei,X+  1 

e  i-X 

e 

2 

Expansion  and  factoring  using  trigonometric  functions 


TRIG  HEMJ 

H.flsinaT 

S  .  H  T  H  Tl  2  L" 

e.HftLFTfln  L 

TRIG  HERU  | 

7.SIRC0S 

S.TRRaSC 

s.THRasca 

10.TC0LLECT 

ii.TEKPRRD 

|i2.TLin  |l| 

Page  5-8 


TRIG  HERU 


12 

TLin  f 

13 

TRIG 

1H 

TRIGCOS 

15 

TRiGsin  y 

1G 

TRIGTfln  1 

17.TSIHF  11 

These  functions  allow  to  simplify  expressions  by  replacing  some  category  of 
trigonometric  functions  for  another  one.  For  example,  the  function  ACOS2S 
allows  to  replace  the  function  arccosine  (acos(x))  with  its  expression  in  terms 
of  orcsine  (asin(x)). 


Description  of  these  commands  and  examples  of  their  applications  are 
available  in  the  calculator's  help  facility  ((j^(j^CHD).  The  user  is  invited  to 
explore  this  facility  to  find  information  on  the  commands  in  the  TRIG  menu. 


Notice  that  the  first  command  in  the  TRIG  menu  is  the  HYPERBOLIC  menu, 
whose  functions  were  introduced  in  Chapter  2. 


Functions  in  the  ARITHMETIC  menu 

The  ARITHMETIC  menu  contains  a  number  of  sub-menus  for  specific 
applications  in  number  theory  (integers,  polynomials,  etc.),  as  well  as  a 
number  of  functions  that  apply  to  general  arithmetic  operations.  The 
ARITHMETIC  menu  is  triggered  through  the  keystroke  combination  l<n  J^rith 
(associated  with  the  CD  key).    With  system  flag  1  1  7  set  to  CHOOSE 


HRITH  HERU 

1 .  IRTEGER..                    1  i 

2.F0LYR0HIRL.  | 

3.H0DUL0.. 

H.FERHUTRTIOR..  1 

:FF 

5.DIYIS 

G. FACTORS  |j 

!       8       i       1  ■: h n ■: l |  ok 

RRITH  HERU 

H.FERHUTATIOR..  | 

5.DIYIS 

G. FACTORS  I 

7.LGCD 

:FF 

S.FROFFRAC  j 

Out  of  this  menu  list,  options  5  through  9  (D/V7S,  FACTORS,  LGCD, 
PROPFRAC,  SIMP2)  correspond  to  common  functions  that  apply  to  integer 
numbers  or  to  polynomials.   The  remaining  options  (7.  INTEGER,  2. 
POLYNOMIAL,  3.  MODULO,  and  4.  PERMUTATION)  are  actually  sub-menus 


Page  5-9 


of  functions  that  apply  to  specific  mathematical  objects.    This  distinction 
between  sub-menus  (options  1  through  4)  and  plain  functions  (options  5 
through  9)  is  made  clear  when  system  flag  1  1  7  is  set  to  SOFT  menus. 
Activating  the  ARITHMETIC  menu  ((Jnj arith_  );  under  these  circumstances, 
produces: 


Following,  we  present  the  help  facility  entries  for  the  functions  of  options  5 
through  9  in  the  ARITHMETIC  menu: 


DIVIS: 


DIVIS:  ~ 

List  of  divisors  of  a 

polynomial  or  integer 

:>IVlS<6> 

{6  3  2 

See:  FACTOR  


FACTORS: 


FACTORS: 

Returns  irreduct ible 
factors  of  an  integer 
or  a  polynomial 
FACTORS  ^2-1  > 

£  X+l  1.  X-l  1.  -J\ 
See:  FACTOR  


GCD  (Greatest  Common  Denominator): 

GCD  of  a  list  of 
objects 

LGCD<£125,75,35» 


See:  CCD  


PROPFRAC  (proper  fraction) 


:'RuPFRAC 
Splits  a  fraction  into 
an  integer  part  and  a 
■raction  part 
^ROPFRFlCt:  43^12} 

3+7^13 

iee:  PARTFRAC 


SIMP2: 


EIMP2: 

Simplifies  2  objects 
by  dividing  them  by 
their  GCD 

SIMP2<XA3-liXA2-l> 

£X*-2+X+l?X+l} 
See:  


The  functions  associated  with  the  ARITHMETIC  submenus:  INTEGER, 
POLYNOMIAL,  MODULO,  and  PERMUTATION,  are  the  following: 

INTEGER  menu 

EULER         Number  of  integers  <  n,  co  -prime  with  n 


Page  5-10 


IABCUV      Solves  au  +  bv  =  c,  with  a,b,c  =  integers 

IBERNOULLI  n-th  Bernoulli  number 

ICHINREM  Chinese  reminder  for  integers 

IDIV2  Euclidean  division  of  two  integers 

IEGCD        Returns  u,v,  such  that  au  +  bv  =  gcd(a,b) 

IQUOT       Euclidean  quotient  of  two  integers 

IREMAINDER  Euclidean  remainder  of  two  integers 

ISPRIME?     Test  if  an  integer  number  is  prime 

NEXTPRIME  Next  prime  for  a  given  integer  number 

PA2B2        Prime  number  as  square  norm  of  a  complex  number 

PREVPRIME  Previous  prime  for  a  given  integer  number 


POLYNOMIAL  menu 

ABCUV       Bezout  polynomial  equation  (au+bv=c) 

CHINREM   Chinese  remainder  for  polynomials 

CYCLOTOMIC  n-th  cyclotomic  polynomial 

DIV2  Euclidean  division  of  two  polynomials 

EGDC         Returns  u,v,  from  au+bv=gcd(a,b) 

FACTOR      Factorizes  an  integer  number  or  a  polynomial 

FCOEF        Generates  fraction  given  roots  and  multiplicity 

FROOTS      Returns  roots  and  multiplicity  given  a  fraction 

GCD  Greatest  common  divisor  of  2  numbers  or  polynomials 

HERMITE     n-th  degree  Hermite  polynomial 

HORNER     Horner  evaluation  of  a  polynomial 

LAGRANGE  Lagrange  polynomial  interpolation 

LCM  Lowest  common  multiple  of  2  numbers  or  polynomials 

LEGENDRE  n-th  degree  Legendre  polynomial 

PARTFRAC  Partial-fraction  decomposition  of  a  given  fraction 

PCOEF        (help-facility  entry  missing) 

PTAYL         Returns  Q(x-a)  in  Q(x-a)  =  P(x),  Taylor  polynomial 

QUOT        Euclidean  quotient  of  two  polynomials 

RESULTANT  Determinant  of  the  Sylvester  matrix  of  2  polynomials 

REMAINDER  Euclidean  reminder  of  two  polynomials 

STURM       Sturm  sequences  for  a  polynomial 

STURMAB  Sign  at  low  bound  and  number  of  zeros  between  bounds 


Page  5-1 1 


MODULO  menu 

ADDTMOD    Add  two  expressions  modulo  current  modulus 
DIVMOD       Divides  2  polynomials  modulo  current  modulus 
DIV2MOD     Euclidean  division  of  2  polynomials  with  modular 
coefficients 

EXPANDMOD  Expands/simplify  polynomial  modulo  current  modulus 
FACTORMOD  Factorize  a  polynomial  modulo  current  modulus 
GCDMOD     GCD  of  2  polynomials  modulo  current  modulus 
INVMOD      inverse  of  integer  modulo  current  modulus 
MOD  (not  entry  available  in  the  help  facility) 

MODSTO      Changes  modulo  setting  to  specified  value 
MULTMOD    Multiplication  of  two  polynomials  modulo  current  modulus 
POWMOD    Raises  polynomial  to  a  power  modulo  current  modulus 
SUBTMOD     Subtraction  of  2  polynomials  modulo  current  modulus 


Applications  of  the  ARITHMETIC  menu 

This  section  is  intended  to  present  some  of  the  background  necessary  for 
application  of  the  ARITHMETIC  menu  functions.  Definitions  are  presented  next 
regarding  the  subjects  of  polynomials,  polynomial  fractions  and  modular 
arithmetic.    The  examples  presented  below  are  presented  independently  of 
the  calculator  setting  (ALG  or  RPN) 

Modular  arithmetic 

Consider  a  counting  system  of  integer  numbers  that  periodically  cycles  back 
on  itself  and  starts  again,  such  as  the  hours  in  a  clock.  Such  counting  system 
is  called  a  ring.  Because  the  number  of  integers  used  in  a  ring  is  finite,  the 
arithmetic  in  this  ring  is  called  finite  arithmetic.  Let  our  system  of  finite  integer 
numbers  consists  of  the  numbers  0,  7 ,  2,  3,  n-1 ,  n.  We  can  also  refer  to 
the  arithmetic  of  this  counting  system  as  modular  arithmetic  of  modulus  n.  In 
the  case  of  the  hours  of  a  clock,  the  modulus  is  12.  (If  working  with  modular 
arithmetic  using  the  hours  in  a  clock,  however,  we  would  have  to  use  the 
integer  numbers  0,  1,2,  3,       10,  11,  rather  than  1,  2,  3,...,1  1,  12). 


Page  5-12 


Operations  in  modular  arithmetic 

Addition  in  modular  arithmetic  of  modulus  n,  which  is  a  positive  integer, 
follow  the  rules  that  if  /  and  k  are  any  two  nonnegative  integer  numbers,  both 
smaller  than  n,  if  j+k>  n,  then  j+k  is  defined  as  j+k-n.  For  example,  in  the 
case  of  the  clock,  i.e.,  for  n  =  1  2,  6+9  "="  3.  To  distinguish  this  'equality' 
from  infinite  arithmetic  equalities,  the  symbol  =  is  used  in  place  of  the  equal 
sign,  and  the  relationship  between  the  numbers  is  referred  to  as  a  congruence 
rather  than  an  equality.  Thus,  for  the  previous  example  we  would  write  6+9  = 
3  (mod  12),  and  read  this  expression  as  "six  plus  nine  is  congruent  to  three, 
modulus  twelve."  If  the  numbers  represent  the  hours  since  midnight,  for 
example,  the  congruence  6+9  =  3  (mod  1  2),  can  be  interpreted  as  saying 
that  "six  hours  past  the  ninth  hour  after  midnight  will  be  three  hours  past 
noon."  Other  sums  that  can  be  defined  in  modulus  1  2  arithmetic  are:  2+5  = 
7  (mod  1  2);  2+10  =  0  (mod  1  2);  7+5  =  0  (mod  1  2);  etcetera. 

The  rule  for  subtraction  will  be  such  that  if  /  -  k  <  0,  then  j-k  is  defined  as  j-k+n. 
Therefore,  8-10  =  2  (mod  12),  is  read  "eight  minus  ten  is  congruent  to  two, 
modulus  twelve."  Other  examples  of  subtraction  in  modulus  12  arithmetic 
would  be  10-5  =  5  (mod  1  2);  6-9  =  9  (mod  1 2);  5  -  8  =  9  (mod  1  2);  5-10 
=  7  (mod  1  2);  etcetera. 

Multiplication  follows  the  rule  that  if  j-k  >  n,  so  that  j-k  =  m-n  +  r,  where  m  and 
rare  nonnegative  integers,  both  less  than  n,  then  j-k  =  r  (mod  n).  The  result  of 
multiplying  /  times  k  in  modulus  n  arithmetic  is,  in  essence,  the  integer 
remainder  of  j-k/ n  in  infinite  arithmetic,  if  j-k>n.  For  example,  in  modulus  12 
arithmetic  we  have  7-3  =  21  =  12  +  9,  (or,  7-3/12  =  21/12  =  1  +  9/12, 
i.e.,  the  integer  reminder  of  21/12  is  9).  We  can  now  write  7-3  =  9  (mod 
1  2),  and  read  the  latter  result  as  "seven  times  three  is  congruent  to  nine, 
modulus  twelve." 

The  operation  of  division  can  be  defined  in  terms  of  multiplication  as  follows, 
r/k  =  j  (mod  n),  if,  j-k  =  r  (mod  n).  This  means  that  r  must  be  the  remainder 
of  j-k/n.  For  example,  9/7  =  3  (mod  1  2),  because  7-3  =  9  (mod  1  2).  Some 
divisions  are  not  permitted  in  modular  arithmetic.  For  example,  in  modulus  12 
arithmetic  you  cannot  define  5/6  (mod  1  2)  because  the  multiplication  table  of 


Page  5-13 


6  does  not  show  the  result  5  in  modulus  1  2  arithmetic.  This  multiplication 
table  is  shown  below: 


6*0  (mod  12) 

0 

6*6  (mod  12) 

0 

6*1  (mod  12) 

6 

6*7  (mod  12) 

6 

6*2  (mod  12) 

0 

6*8  (mod  12) 

0 

6*3  (mod  12) 

6 

6*9  (mod  12) 

6 

6*4  (mod  12) 

0 

6*10  (mod  12) 

0 

6*5  (mod  12) 

6 

6*11  (mod  12) 

6 

Formal  definition  of  a  finite  arithmetic  ring 

The  expression  a  =  b  (mod  n)  is  interpreted  as  "a  is  congruent  to  b,  modulo 
n,"  and  holds  if  (b-a)  is  a  multiple  of  n.    With  this  definition  the  rules  of 
arithmetic  simplify  to  the  following: 

If  a  =  b  (mod  n)  and  c  =  d  (mod  n), 

then 

o+c  =  b+d  (mod  nj, 
a-c  =  b  -  d  (mod  n), 
axe  =  bxd  (mod  n). 

For  division,  follow  the  rules  presented  earlier.  For  example,  17  =  5  (mod  6), 
and  21=3  (mod  6).  Using  these  rules,  we  can  write: 

17  +  21  =5  +  3  (mod  6)  =>   38  =  8  (mod  6)    =>  38  =  2  (mod  6) 
17-21=5-3  (mod  6)  =>     -4=2  (mod  6) 

17x21  =  5x3(mod6)=>   357  =  15  (mod  6)  =>  357  =  3  (mod  6) 

Notice  that,  whenever  a  result  in  the  right-hand  side  of  the  "congruence" 
symbol  produces  a  result  that  is  larger  than  the  modulo  (in  this  case,  n  =  6), 
you  can  always  subtract  a  multiple  of  the  modulo  from  that  result  and  simplify 
it  to  a  number  smaller  than  the  modulo.  Thus,  the  results  in  the  first  case  8 
(mod  6)  simplifies  to  2  (mod  6),  and  the  result  of  the  third  case,  75  (mod  6) 
simplifies  to  3  (mod  6).  Confusing?  Well,  not  if  you  let  the  calculator  handle 
those  operations.  Thus,  read  the  following  section  to  understand  how  finite 
arithmetic  rings  are  operated  upon  in  your  calculator. 


Page  5-14 


Finite  arithmetic  rings  in  the  calculator 

All  along  we  have  defined  our  finite  arithmetic  operation  so  that  the  results 
are  always  positive.  The  modular  arithmetic  system  in  the  calculator  is  set  so 
that  the  ring  of  modulus  n  includes  the  numbers  -n/2+1,  ...,-1,  0,  1  ,...,n/2-l , 
n/2,  if  n  is  even,  and  -(n-l)/2,  -(n-3)/2,...,-l ,0,1  ,...,(n-3)/2,  (n-l)/2,  if  n  is 
odd.  For  example,  for  n  =  8  (even),  the  finite  arithmetic  ring  in  the  calculator 
includes  the  numbers:  (-3,-2,-1,0,1,3,4),  while  for  n  =  7  (odd),  the 
corresponding  calculator's  finite  arithmetic  ring  is  given  by  (-3,-2,-1,0,1,2,3). 

Modular  arithmetic  in  the  calculator 

To  launch  the  modular  arithmetic  menu  in  the  calculator  select  the  MODULO 
sub-menu  within  the  ARITHMETIC  menu  (Ij^Jarith_  ).  The  available  menu 
includes  functions:  ADDTMOD,  DIVMOD,  DIV2MOD,  EXPANDMOD, 
FACTORMOD,  GCDMOD,  INVMOD,  MOD,  MODSTO,  MULTMOD, 
POWMOD,  and  SUBTMOD.  Brief  descriptions  of  these  functions  were 
provided  in  an  earlier  section.  Next  we  present  some  applications  of  these 
functions. 

Setting  the  modulus  (or  MODULO) 

The  calculator  contains  a  variable  called  MODULO  that  is  placed  in  the 
{HOME  CASDIR}  directory  and  will  store  the  magnitude  of  the  modulus  to  be 
used  in  modular  arithmetic. 

The  default  value  of  MODULO  is  1  3.    To  change  the  value  of  MODULO,  you 
can  either  store  the  new  value  directly  in  the  variable  MODULO  in  the  sub- 
directory {HOME  CASDIR}  Alternatively,  you  can  store  a  new  MODULO 
value  by  using  function  MODSTO. 

Modular  arithmetic  operations  with  numbers 

To  add,  subtract,  multiply,  divide,  and  raise  to  a  power  using  modular 
arithmetic  you  will  use  the  functions  ADDTMOD,  SUBTMOD,  MULTMOD, 
DIV2MOD  and  DIVMOD  (for  division),  and  POWMOD.  In  RPN  mode  you 
need  to  enter  the  two  numbers  to  operate  upon,  separated  by  an  [ENTER]  or 
an  [SPC]  entry,  and  then  press  the  corresponding  modular  arithmetic  function. 
For  example,  using  a  modulus  of  1  2,  try  the  following  operations: 


Page  5-15 


ADDTMOD  examples 

6+5  =  -1  (mod  12) 
11+5  =  4  (mod  1  2) 

SUBTMOD  examples 

5  -  7  =  -2  (mod  1  2) 
11-8  =  3  (mod  1  2) 

MULTMOD  examples 

68  =  0  (mod  1  2) 
5-6  =  6  (mod  12) 

DIVMOD  examples 

12/3  =  4  (mod  1  2) 
25/5  =  5  (mod  1  2) 
66/6  =  -1  (mod  12) 

DIV2MOD  examples 

2/3  (mod  1  2)  does  not  exist 

26/1  2  (mod  1  2)  does  not  exist 

1  25/1  7  (mod  1  2)  =  1  with  remainder  =  0 

68/7  =  -4  (mod  1  2)  with  remainder  =  0 

7/5  =  -1  (mod  1  2)  with  remainder  =  0 


Note:  DIVMOD  provides  the  quotient  of  the  modular  division  j/k  (mod  n), 
while  DIMV2MOD  provides  no  only  the  quotient  but  also  the  remainder  of  the 
modular  division  j/k  (mod  n). 


POWMOD  examples 

23=  -4  (mod  1  2)  35=  3  (mod  1  2)  510=  1  (mod  1  2) 

1  18=  1  (mod  12)  62=0  (mod  12)  99=-3  (mod  12) 

In  the  examples  of  modular  arithmetic  operations  shown  above,  we  have  used 
numbers  that  not  necessarily  belong  to  the  ring,  i.e.,  numbers  such  as  66, 
125,  17,  etc.    The  calculator  will  convert  those  numbers  to  ring  numbers 


6+6  =  0  (mod  1  2)  6+7  =  1  (mod  1  2) 
8+10  =  -6  (mod  12) 


8-4=4  (mod  12)  5  -1 0  =  -5  (mod  1  2) 
8  -  1  2  =  -4  (mod  1  2) 


9-8  =  0  (mod  12)  3-2  =  6  (mod  12) 

1 1  -3  =  -3  (mod  1  2) 


1  2/8  (mod  1  2)  does  not  exist 
64/13  =  4  (mod  12) 


Page  5-16 


before  operating  on  them.  You  can  also  convert  any  number  into  a  ring 
number  by  using  the  function  EXPANDMOD.  For  example, 


EXPANDMOD(l  25)  =  5  (mod  1  2) 
EXPANDMOD(l  7)   =  5  (mod  1  2) 
EXPANDMOD(6)  =  6  (mod  12) 


The  modular  inverse  of  a  number 

Let  a  number  k  belong  to  a  finite  arithmetic  ring  of  modulus  n,  then  the 
modular  inverse  of  k,  i.e.,  1/k  (mod  n),  is  a  number  /,  such  that  j-k  =  7  (mod 
n).  The  modular  inverse  of  a  number  can  be  obtained  by  using  the  function 
INVMOD  in  the  MODULO  sub-menu  of  the  ARITHMETIC  menu.  For  example, 
in  modulus  12  arithmetic: 


1/11  =-1  (mod  12) 
The  MOD  operator 

The  MOD  operator  is  used  to  obtain  the  ring  number  of  a  given  modulus 
corresponding  to  a  given  integer  number.  On  paper  this  operation  is  written 
as  m  mod  n  =  p,  and  is  read  as  "m  modulo  n  is  equal  to  p".  For  example, 
to  calculate  15  mod  8,  enter: 

•  ALG  mode:       CD  CD  MOD  CDS 

•  RPN  mode:       CDCDSCDS  MOD 

The  result  is  7,  i.e.,  1  5  mod  8  =  7.  Try  the  following  exercises: 
1  8  mod  1 1  =  7  23  mod  2  =1  40  mod  1  3  =  1 

23  mod  17  =  6  34  mod  6  =  4 

One  practical  application  of  the  MOD  function  for  programming  purposes  is 
to  determine  when  an  integer  number  is  odd  or  even,  since  n  mod  2  =  0,  if  n 
is  even,  and  n  mode  2  =  1,  if  n  is  odd.  It  can  also  be  used  to  determine 
when  an  integer  m  is  a  multiple  of  another  integer  n,  for  if  that  is  the  case  m 
mod  n  =  0. 


1/6  (mod  1  2)  does  not  exist. 
1/7  =  -5  (mod  12) 


1/5  =  5  (mod  12) 

1/3  (mod  1  2)  does  not  exist. 


Page  5-17 


Note:  Refer  to  the  help  facility  in  the  calculator  for  description  and  examples 
on  other  modular  arithmetic.  Many  of  these  functions  are  applicable  to 
polynomials.  For  information  on  modular  arithmetic  with  polynomials  please 
refer  to  a  textbook  on  number  theory. 


Polynomials 

Polynomials  are  algebraic  expressions  consisting  of  one  or  more  terms 
containing  decreasing  powers  of  a  given  variable.  For  example, 
'XA3+2*XA2-3*X+2'  is  a  third-order  polynomial  in  X,  while  'SIN(X)A2-2'  is  a 
second-order  polynomial  in  SIN(X).   A  listing  of  polynomial-related  functions 
in  the  ARITHMETIC  menu  was  presented  earlier.  Some  general  definitions  on 
polynomials  are  provided  next.    In  these  definitions  A(X),  B(X),  C(X),  P(X), 
Q(X),  U(X),  V(X),  etc.,  are  polynomials. 

•  Polynomial  fraction:  a  fraction  whose  numerator  and  denominator  are 
polynomials,  say,  C(X)  =  A(X)/B(X) 

•  Roots,  or  zeros,  of  a  polynomial:  values  of  X  for  which  P(X)  =  0 

•  Poles  of  a  fraction:  roots  of  the  denominator 

•  Multiplicity  of  roots  or  poles:  the  number  of  times  a  root  shows  up,  e.g., 
P(X)  =  (X+l)2(X-3)  has  roots  {-1,  3}  with  multiplicities  {2,1} 

•  Cyclotomic  polynomial  (Pn(X)):  a  polynomial  of  order  EULER(nj  whose 
roots  are  the  primitive  n-th  roots  of  unity,  e.g.,  P2(X)  =  X+l,  P4(X)  =  X2+l 

•  Bezout's  polynomial  equation:  A(X)  U(X)  +  B(X)V(X)  =  C(X) 
Specific  examples  of  polynomial  applications  are  provided  next. 

Modular  arithmetic  with  polynomials 

The  same  way  that  we  defined  a  finite-arithmetic  ring  for  numbers  in  a 
previous  section,  we  can  define  a  finite-arithmetic  ring  for  polynomials  with  a 
given  polynomial  as  modulo.  For  example,  we  can  write  a  certain  polynomial 
P(X)  as  P(X)  =  X  (mod  X2),  or  another  polynomial  Q(X)  =  X  +  7  (mod  X-2). 

A  polynomial,  P(X)  belongs  to  a  finite  arithmetic  ring  of  polynomial  modulus 
M(X),  if  there  exists  a  third  polynomial  Q(X),  such  that  (P(X)  -  Q(X))  is  a 
multiple  of  M(X).  We  then  would  write:  P(X)  =  Q(X)  (mod  M(X)).  The  later 
expression  is  interpreted  as  "P(X)  is  congruent  to  Q(X),  modulo  M(X)". 


Page  5-18 


The  CHINREM  function 

CHINREM  stands  for  CHINese  REMainder.  The  operation  coded  in  this 
command  solves  a  system  of  two  congruences  using  the  Chinese  Remainder 
Theorem.  This  command  can  be  used  with  polynomials,  as  well  as  with 
integer  numbers  (function  ICHINREM).  The  input  consists  of  two  vectors 
[expression^,  modulo^]  and  [expression_2,  modulo_2].  The  output  is  a 
vector  containing  [expression_3/  modulo_3],  where  modulo_3  is  related  to  the 
product    (modulo_l).(modulo_2).  Example:    CHINREM(['X+1 ',  'XA2- 

r],[/x+r/xA2/])  =  ['x+r,-(xA4-xA2)] 

Statement  of  the  Chinese  Remainder  Theorem  for  integers 

If  m7/  m2/...,mr  are  natural  numbers  every  pair  of  which  are  relatively  prime, 
and  a7/  a2,       ar  are  any  integers,  then  there  is  an  integer  x  that 
simultaneously  satisfies  the  congruences:  x  =o}  (mod  mj,  x  =o2  (mod 
m2),        x  =ar  (mod  mr).  Additionally,  if  x  =  a  is  any  solution  then  all  other 
solutions  are  congruent  to  a  modulo  equal  to  the  product  m7-m2-  ...  mr 

The  EGCD  function 

EGCD  stands  for  Extended  Greatest  Common  Divisor.  Given  two  polynomials, 
A(X)  and  B(X),  function  EGCD  produces  the  polynomials  C(X),  U(X),  and  V(X), 
so  that  C(X)  =  U(X)*A(X)  +  V(X)*B(X).  For  example,  for  A(X)  =  XA2+1 ,  B(X)  = 
XA2-1,  EGCD(A(X),B(X))  =  {2,  1,  -1}.  i.e.,  2  =  1*(  XA2+T)-1*(  XA2-1).  Also, 
EGCD('XA3-2*X+5','X')  =  {  5,  '-(XA2-2)',  i},  j.e.,  5  =  -  (XA2-2)*X  +  1  *(XA3- 
2*X+5). 

The  GCD  function 

The  function  GCD  (Greatest  Common  Denominator)  can  be  used  to  obtain  the 
greatest  common  denominator  of  two  polynomials  or  of  two  lists  of 
polynomials  of  the  same  length.  The  two  polynomials  or  lists  of  polynomials 
will  be  placed  in  stack  levels  2  and  1  before  using  GCD.  The  results  will  be  a 
polynomial  or  a  list  representing  the  greatest  common  denominator  of  the  two 
polynomials  or  of  each  list  of  polynomials.  Examples,  in  RPN  mode,  follow 
(calculator  set  to  Exact  mode): 
'XA  3- 1  '[enter} 'Xa 2- 1  '[enter} GCD  Results  in:  %V 

{'XA2+2*X+1','XA3+XA2'}  [enter}  {/XA3+r,,XA2+V}  [enter}  GCD  results  in 
fX+T  1} 


Page  5-19 


The  HERMITE  function 

The  function  HERMITE  [HERMI]  uses  as  argument  an  integer  number,  k,  and 
returns  the  Hermite  polynomial  of  k-th  degree.  A  Hermite  polynomial,  Hek(x) 
is  defined  as 

He0=l,    Hen(x)  =  (-l)"e*2/2^(e-*2/2),    n  =  l,2,.. 

ax 

An  alternate  definition  of  the  Hermite  polynomials  is 

H*  =  l,    Hn^x)  =  (-iyex2^(e-x2),    n  =  \X- 

ax 

where  dn/dxn  =  n-th  derivative  with  respect  to  x.  This  is  the  definition  used  in 
the  calculator. 

Examples:  The  Hermite  polynomials  of  orders  3  and  5  are  given  by: 

HERMITE(3)  =  /8*XA3-12*X' 
And  HERMITE(5)  =  '32*xA5-l  60*XA3+1  20*X'. 

The  HORNER  function 

The  function  HORNER  produces  the  Horner  division,  or  synthetic  division,  of  a 
polynomial  P(X)  by  the  factor  (X-a).  The  input  to  the  function  are  the 
polynomial  P(X)  and  the  number  a.  The  function  returns  the  quotient 
polynomial  Q(X)  that  results  from  dividing  P(X)  by  (X-a),  the  value  of  a,  and  the 
value  of  P(a),  in  that  order.  In  other  words,  P(X)  =  Q(X)(X-a)+P(a).  For 
example,  HORNER('XA3+2*XA2-3*X+r,2)  =  {'XA2+4*X+5',  2,  11}.  We 
could,  therefore,  write  X3+2X2-3X+1  =  (X2+4X+5)(X-2)+l  1 .  A  second 
example:  HORNER('XA6-l  ,,-5)=  {'XA5-5*XA4+25*XA3-1  25*XA2+625*X-31 25',-5, 
15624}  i.e.,     X6-l  =  (X5-5*X4+25X3-1  25X2+625X-31  25)(X+5)+l  5624. 

The  variable  VX 

A  variable  called  VX  exists  in  the  calculator's  {HOME  CASDIR}  directory  that 
takes,  by  default,  the  value  of  'X'.  This  is  the  name  of  the  preferred 
independent  variable  for  algebraic  and  calculus  applications.  Avoid  using 
the  variable  VX  in  your  programs  or  equations,  so  as  to  not  get  it  confused 
with  the  CAS'  VX.  If  you  need  to  refer  to  the  x-component  of  velocity,  for 
example,  you  can  use  vx  or  Vx.  For  additional  information  on  the  CAS 
variable  see  Appendix  C. 


Page  5-20 


The  LAGRANGE  function 

The  function  LAGRANGE  requires  as  input  a  matrix  having  two  rows  and  n 
columns.  The  matrix  stores  data  points  of  the  form  [[x1/x2/       xn]  [yl7  y2/ 
yn]].  Application  of  the  function   LAGRANGE   produces  the  polynomial 
expanded  from 

n 

pn-Ax)  =  2^—^  yr 

For  example,  for  n  =  2,  we  will  write: 

p^=x~x2  y  \x~*i  y2  =  (yi-y2)'x+(y2'xi-yrx2) 

OCy  "^2  X-y  OCy  "^2 


Check  this  result  with  your  calculator: 

LAGRANGE([[  xl  ,x2],[yl  ,y2]])  =  '((yl-y2)*X+(y2*xl-yl  *x2))/(xl-x2)'. 

Other  examples:  LAGRANGE([[1 ,  2,  3][2,  8,  15]])  =  '(XA2+9*X-6)/2' 
LAGRANGE([[0.5,1.5,2.5,3.5,4.5][1 2. 2, 13.5, 19.2,27.3,32.5]])  = 
'-(.1 375*XA4+  -.7666666666667*XA3+  -  .74375*XA2  = 
1 .991 666666667*X-1 2.92265625)'. 


Note:  Matrices  are  introduced  in  Chapter  1 0. 
The  LCM  function 

The  function  LCM  (Least  Common  Multiple)  obtains  the  least  common  multiple 
of  two  polynomials  or  of  lists  of  polynomials  of  the  same  length.  Examples: 

LCM('2*XA2+4*X+2'  ,'XA2-1' )  =  '(2*XA2+4*X+2)*(X-1 )'. 
LCM('XA3-1  ','XA2+2*X')  =  '(XA3-1  )*(  XA2+2*X)' 


Page  5-21 


The  LEGENDRE  function 

A  Legendre  polynomial  of  order  n  is  a  polynomial  function  that  solves  the 

differential  equation  (1  - x2)  •  — y-  -  2  •  x  •  —  +  n-(n  +  l)-y  =  0 

dx  dx 

To  obtain  the  n-th  order  Legendre  polynomial,  use  LEGENDRE(n),  e.g., 

LEGENDRE(3)  =  '(5*XA3-3*X)/2' 
LEGENDRE(5)  =  '(63*X  A5-70*XA3+1 5*X)/8' 

The  PCOEF  function 

Given  an  array  containing  the  roots  of  a  polynomial,  the  function  PCOEF 
generates  an  array  containing  the  coefficients  of  the  corresponding 
polynomial.  The  coefficients  correspond  to  decreasing  order  of  the 
independent  variable.  For  example:  PCOEF([-2,-l,0,l,l,2])  =  [1.  -1 .  -5.  5. 
4.  -4.  0.],  which  represents  the  polynomial  X6-X5-5X4+5X3+4X2-4X. 

The  PROOT  function 

Given  an  array  containing  the  coefficients  of  a  polynomial,  in  decreasing 
order,  the  function  PROOT  provides  the  roots  of  the  polynomial.  Example, 
from  X2+5X-6  =0,  PROOT([l,  -5,  6])  =  [2.  3.]. 

The  PTAYL  function 

Given  a  polynomial  P(X)  and  a  number  a,  the  function  PTAYL  is  used  to  obtain 
an  expression  Q(X-a)  =  P(X),  i.e.,  to  develop  a  polynomial  in  powers  of  (X-  a). 
This  is  also  known  as  a  Taylor  polynomial,  from  which  the  name  of  the 
function,  Polynomial  &  TAYLor,  follow: 

For  example,  PTAYL('XA3-2*X+2',2)  =  'XA3+6*XA2+1 0*X+6'. 

In  actuality,  you  should  interpret  this  result  to  mean 

'(X-2)  A3+6*(X-2)  A2+10*(X-2)  +6'. 

Let's  check  by  using  the  substitution:  'X  =  x  -  2'.  We  recover  the  original 
polynomial,  but  in  terms  of  lower-case  x  rather  than  upper-case  x. 


Page  5-22 


The  QUOT  and  REMAINDER  functions 

The  functions  QUOT  and  REMAINDER  provide,  respectively,  the  quotient  Q(X) 
and  the  remainder  R(X),  resulting  from  dividing  two  polynomials,  P^X)  and 
P2(X).  In  other  words,  they  provide  the  values  of  Q(X)  and  R(X)  from 
Pi(X)/P2(X)  =  Q(X)  +  R(X)/P2(X).  For  example, 

QUOT(XA3-2*X+2,  X-l)  =  XA2+X-1 
REMAINDER(XA3-2*X+2,  X-l)  =  1. 

Thus,  we  can  write:    (X3-2X+2)/(X-l )  =  X2+X-l  +  1/(X-1 ). 


Note:  you  could  get  the  latter  result  by  using  PROPFRAC: 

PROPFRAC('(XA3-2*X+2)/(X-l)')  =  'XA2+X-1  +  1/(X-1)'. 


The  EPSXO  function  and  the  CAS  variable  EPS 

The  variable  s  (epsilon)  is  typically  used  in  mathematical  textbooks  to 
represent  a  very  small  number.  The  calculator's  CAS  creates  a  variable  EPS, 
with  default  value  0.0000000001  =  1010,  when  you  use  the  EPSXO  function. 
You  can  change  this  value,  once  created,  if  you  prefer  a  different  value  for 
EPS.  The  function  EPSXO,  when  applied  to  a  polynomial,  will  replace  all 
coefficients  whose  absolute  value  is  less  than  EPS  with  a  zero.  Function 
EPSXO  is  not  available  in  the  ARITHMETIC  menu,  it  must  be  accessed  from  the 
function  catalog  (N).  Example: 

EPSX0('XA3-1 .2E-1 2*XA2+1 .2E-6*X+6.2E-1 1  )= 

/XA3-0*XA2+.000001 2*X+0'. 

With  GmT):  'Xa3+.0000012*X'. 

The  PEVAL  function 

The  functions  PEVAL  (Polynomial  EVALuation)  can  be  used  to  evaluate  a 
polynomial  p(x)  =  on-xn+on.rx  n']+  ...+  o2-x2+arx+  a0,  given  an  array  of 
coefficients  [an,  on.u  ...  a2/  a7/  a0]  and  a  value  of  x0.  The  result  is  the 
evaluation  p(x0).  Function  PEVAL  is  not  available  in  the  ARITHMETIC  menu,  it 
must  be  accessed  from  the  function  catalog  (,N).  Example: 

PEVAL([1,5,6,1],5)  =  281. 


Page  5-23 


The  TCHEBYCHEFF  function 

The  function  TCHEBYCHEFF(n)  generates  the  Tchebycheff  (or  Chebyshev) 
polynomial  of  the  first  kind,  order  n,  defined  as  Tn(X)  =  cos(n-arccos(X)).  If  the 
integer  n  is  negative  (n  <  0),  the  function  TCHEBYCHEFF(n)  generates  the 
Tchebycheff  polynomial  of  the  second  kind,  order  n,  defined  as  Tn(X)  = 
sin(n-arccos(X))/sin(orccos(X)).  Examples: 

TCHEBYCHEFF(3)  =  4*XA3-3*X 
TCHEBYCHEFF(-3)  =  4*XA2-1 

Fractions 

Fractions  can  be  expanded  and  factored  by  using  functions  EXPAND  and 
FACTOR,  from  the  ALG  menu  (,x).  For  example: 

EXPAND('(l+X)A3/((X-l)(X+3))')  =  '(XA3+3*XA2+3*X+1  )/(XA2+2*X-3)' 
EXPAND('(XA2*(X+Y)/(2*X-XA2)A2')  =  '(X+Y)/(XA2-4*X+4)' 
EXPANDp<*(X+Y)/(XA2-l)')  =  ,(XA2+Y*X)/(XA2-1 )' 
EXPAND('4+2*(X-1  )+3/((X-2)*(X+3))-5/XA2')  = 

/(2*XA5+4*XA4-10*XA3-14*XA2-5*X)/(XA4+XA3-6*XA2), 

FACTOR('(3*XA3-2*XA2)/(XA2-5*X+6)')  =  'XA2*(3*X-2)/((X-2)*(X-3))/ 
FACTOR('(XA3-9*X)/(XA2-5*X+6)' )  =  'X*(X+3)/(X-2)' 
FACTOR('(XA2-l)/(XA3*Y-Y)')  =  '(X+l  )/((XA2+X+l  )*Y)' 

TheSIMP2  function 

Functions  SIMP2  and  PROPFRAC  are  used  to  simplify  a  fraction  and  to 
produce  a  proper  fraction,  respectively.  Function  SIMP2  takes  as  arguments 
two  numbers  or  polynomials,  representing  the  numerator  and  denominator  of 
a  rational  fraction,  and  returns  the  simplified  numerator  and  denominator.  For 
example:  SIMP2(/XA3-r/XA2-4*X+3/)  =  {  /XA2+X+r,/X-3/}. 

The  PROPFRAC  function 

The  function  PROPFRAC  converts  a  rational  fraction  into  a  "proper"  fraction, 
i.e.,  an  integer  part  added  to  a  fractional  part,  if  such  decomposition  is 
possible.  For  example: 


Page  5-24 


PROPFRAC('5/4')  = '1+1/4' 
PROPFRAC('(xA2+l)/xA2')  =  'l+l/xA2' 

The  PARTFRAC  function 

The  function  PARTFRAC  decomposes  a  rational  fraction  into  the  partial 
fractions  that  produce  the  original  fraction.  For  example: 

PARTFRAC(/(2*XA6-14*XA5+29*XA4-37*XA3+41*XA2-16*X+5)/(XA5- 
7*XA4+1 1  *XA3-7*XA2+1 0*X)')  = 

'2  *X+(  1  /2/(X-2)+5/(X-5)+ 1  /2/X+X/(XA  2+ 1 ))' 

This  technique  is  useful  in  calculating  integrals  (see  chapter  on  calculus)  of 
rational  fractions. 

If  you  have  the  Complex  mode  active,  the  result  will  be: 

'2  *X+(  1  /2/(X+i)+ 1  /2/(X-2)+5/(X-5)+ 1  /2/X+ 1  /2/(X-i))' 
The  FCOEF  function 

The  function  FCOEF  is  used  to  obtain  a  rational  fraction,  given  the  roots  and 
poles  of  the  fraction. 


Note:  If  a  rational  fraction  is  given  as  F(X)  =  N(X)/D(X),  the  roots  of  the 
fraction  result  from  solving  the  equation  N(X)  =  0,  while  the  poles  result  from 
solving  the  equation  D(X)  =  0. 


The  input  for  the  function  is  a  vector  listing  the  roots  followed  by  their 
multiplicity  (i.e.,  how  many  times  a  given  root  is  repeated),  and  the  poles 
followed  by  their  multiplicity  represented  as  a  negative  number.  For  example, 
if  we  want  to  create  a  fraction  having  roots  2  with  multiplicity  1,  0  with 
multiplicity  3,  and  -5  with  multiplicity  2,  and  poles  1  with  multiplicity  2  and  -3 
with  multiplicity  5,  use: 

FCOEF([2  10  3  -5  21-2-3  -5])  =  /(X-5)A2*XA3*(X-2)/(X-3)A5*(X-l )A2' 


Page  5-25 


If  you  press  Ieval)  you  will  get: 

/(XA6+8*XA5+5*XA4-50*XA3)/(XA7+13*XA6+61*XA5+105*XA4-45*XA3- 
297*XA2-81*X+243)' 

The  FROOTS  function 

The  function  FROOTS  obtains  the  roots  and  poles  of  a  fraction.   As  an 
example,  applying  function  FROOTS  to  the  result  produced  above,  will  result 
in:  [1  -2.  -3  -5.  0  3.2  1.-5  2.].  The  result  shows  poles  followed  by  their 
multiplicity  as  a  negative  number,  and  roots  followed  by  their  multiplicity  as  a 
positive  number.  In  this  case,  the  poles  are  (1,  -3)  with  multiplicities  (2,5) 
respectively,  and  the  roots  are  (0,  2,  -5)  with  multiplicities  (3,  1,  2), 
respectively. 

Another  example  is:  FROOTS('(XA2-5*X+6)/(XA5-XA2)>  [0  -2.1  -1 .3  1 .2 
1 .].  i.e.,  poles  =  0  (2),  1  (1 ),  and  roots  =  3(1 ),  2(1 ).  If  you  have  had  Complex 
mode  selected,  then  the  results  would  be: 

[0-2.  1  -1.  '-((l+i*V3)/2'-l.  '-((l-i*V3)/2'-l.]. 

Step-by-step  operations  with  polynomials  and  fractions 

By  setting  the  CAS  modes  to  Step/step  the  calculator  will  show  simplifications 
of  fractions  or  operations  with  polynomials  in  a  step-by-step  fashion.  This  is 
very  useful  to  see  the  steps  of  a  synthetic  division.  The  example  of  dividing 

X3 -5X2  +3X-2 


X-2 

is  shown  in  detail  in  Appendix  C.  The  following  example  shows  a  lengthier 
synthetic  division: 

X9-l 
X2-l 


j  I  V2<XA9-1?XA2-1) 
■l^i'utiiMiiiittti^i'M^i^uiiiaiii 


Division  H=BQ+R 

R:   ::  1 , 0, 9,  U,  9,  U,  G,  U,  G, 

B:  a,e,-i> 

R:  £0? 1,0,0,0,0,0,0,-1 
^ress  a  key  to  90  on 
■l^l'UtliMllilWtl^I'M^l^tllliaili 


Page  5-26 


^iuision  fl=BQ+R 

h:  ■:: i , u? y, u, q, q, i-j, q, g, 

B:  0,0,-1} 
Q:  U,8> 

r:  tiPes0,0,e,0,eP-i> 

Press  a  ke^j  to  90  on 


fl:    t 1?0,0?0?0?0?0?0?0? 

B:  £1,8,-1} 

^  £1,0,0,0,0,-1} 

Press  a  key  to  90  on 


Muision  fl=BQ+R 

=i:  £1,0,0,0,0,0,0,0,0, 

B:  £1,0,-1} 
Q:  £1,0,1} 

R:  £0,1,0,0,0,0,-1} 

Press  a  key  to  90  on 


pyi^M!Mnai^mpwc|iMiai«Miio  pyii^M;Mnai^M|nwc|iMiai«Miio 

Myision  H=BQ+R  division  H=BQ+R 


wision  H=BQ+ 
R:  £1,0,0,0,0,0,0,0,0, 

B:  £1,0,-1} 
Q:  £1,0,1,0,1} 
R:  £0,1,0,0,-1} 
Press  a  key  to  90  on 


A  E:  ■!  U  V I C  H I  Ti  F;  I C  V  C  L  ■:■  |  DI'.'3  |  EGCD  FACTO  MABCUVCHinRCYCLO  DI'.'2  I  EG  CD  IFiiCTu 


^lyision  R=BQ+R 
=i:  £1,0,0,0,0,0,0,0,0, 

B:  £1,0,-1} 
Q:  £1,0,1,0,1,0,1} 


Jiuision  R=BQ+R 

£1,0,0,0,0,0,0,0,0, 

B:  £1,0, -1} 
Q:  £1,0, 1,0, 1,0} 

£1,0,0,-1} 
^ress  a  key  to  90  on 

ii:iil^tiir,iiiiitttwim-tl^i'igiH^ 


i vision  H=BQ+ 

=|:  £1,0,0,0,0,0,0,0,0, 

B:  £1,0,-1} 
Q:  £1,0, 1,0, 1,0, 1,0} 
R:  £1,-1} 

^ress  a  key  to  90  on 

siMBHaiiiaHBBEsai^aBHa 


W  ■      '.1  lUj  1  lUj 

R:  £0,1,-1} 
^ress  a  key  to  90  on 


:DIV2(x9-lJX2-l) 

{q:(x7+X5+X3+x)  R:CX-1)) 
BMBHEiiaHBBESai^aBHE 


The  CONVERT  Menu  and  algebraic  operations 

The  CONVERT  menu  is  activated  by  using  HT1  convert  key  (the  CS  key).  This 
menu  summarizes  all  conversion  menus  in  the  calculator.  The  list  of  these 
menus  is  shown  next: 


■: ■:■  n ■■■ert  hehu 


3.  EASE.. 

3. TRIG  COnV.. 

H .  REHRITE.. 

S .  MhTRIX  C0FWERT.. 


The  functions  available  in  each  of  the  sub-menus  are  shown  next. 


UNITS  convert  menu  (Option  1) 

This  menu  is  the  same  as  the  UNITS  menu  obtained  by  using  L  r>  j  UNITSs  .  The 
applications  of  this  menu  are  discussed  in  detail  in  Chapter  3. 


Page  5-27 


BASE  convert  menu  (Option  2) 

This  menu  is  the  same  as  the  UNITS  menu  obtained  by  using  (j^JJ^  .  The 
applications  of  this  menu  are  discussed  in  detail  in  Chapter  19. 

TRIGONOMETRIC  convert  menu  (Option  3) 

This  menu  is  the  same  as  the  TRIG  menu  obtained  by  using  L  r>  j  trig  .  The 
applications  of  this  menu  are  discussed  in  detail  in  this  Chapter. 


MATRICES  convert  menu  (Option  5) 


HRTRIK  CORYERT  HERU 

^^^^^^ 

r.GKR 

H.SYST3HRT 
5 .  CORYERT.. 

These  functions  are  discussed  in  detail  in  Chapter  10. 


REWRITE  convert  menu  (Option  4) 


REHRITE  HERU 

3.EKPLR 

3.EKP3P0H 

H.FDISTRIE 

5.I-»R 

e.LIR 

REHRITE  HERU 


? . LRCOLLECT 
E .  POHEXPRRD 
5 . SIHPLIFY 
10.-H1UH 


REHRITE  HERU 


3. SIHPLIFY 

lLl.-*flUH 

ii. 

ia.-^n 
13.R-H 


Functions  l->R  and  R->l  are  used  to  convert  a  number  from  integer  (I)  to  real 
(R),  or  vice  versa.  Integer  numbers  are  shown  without  trailing  decimal  points, 
while  real  numbers  representing  integers  will  have  a  trailing  decimal  point, 
e.g., 


Page  5-28 


5. 

:R+I(12.:i 

12 

Function  ->NUM  has  the  same  effect  as  the  keystroke  combination  Lr^J^M^f 
(associated  with  the  (enter}  key).  Function  ->NUM  converts  a  symbolic  result 
into  its  floating-point  value.    Function  ->Q  converts  a  floating-point  value  into 
a  fraction.  Function  ->G)7i  converts  a  floating-point  value  into  a  fraction  of  n, 
if  a  fraction  of  n  can  be  found  for  the  number;  otherwise,  it  converts  the 
number  to  a  fraction.    Examples  are  of  these  three  functions  are  shown  next. 


:+Q(£.5533) 


■B660254037B5 
25533 

10000 


:+Qtt(.7586) 


3793 

5000 

+Qtt(2.  09439510239) 

M 

BitiiiHitiiEC'HII'HO'HllEia 


Out  of  the  functions  in  the  REWRITE  menu,  functions  DISTRIB,  EXPLN, 
EXP2POW,  FDISTRIB,  LIN,  LNCOLLECT,  POWEREXPAND,  and  SIMPLIFY 
apply  to  algebraic  expressions.  Many  of  these  functions  are  presented  in  this 
Chapter.  However,  for  the  sake  of  completeness  we  present  here  the  help- 
facility  entries  for  these  functions. 


DISTRIB  EXPLN 

EXPLN: 

Rewrites  transcendent- 
functions  in  terms  of 
EXP  and  LH 

explh<:cos<:x>:> 

<EXP<i*X>  +  l^EXP<i*X>>„, 
See:  SI NCOS  EXP2HYP 


:TSTRTF: 
5t  ep-^st  ep  d  i  st  r  i  but  i  on 
nf  *  and  s  oyer  +and  - 

)istrib<<x+y)*<z+1>> 
x*<z+i>+y*<z+i> 

See:  FDISTRIB 


EXP2POW 


EXP2P0N: 

?ewr i  t  e  exp  <!  a*Ln  £  b  >  > 
as  b--a 

EXP2P0N  < EXP  < X*LN  < Y  > )  ) 
See: 


FDISTRIB 


FDISTRIB: 

Full  distribution  of  * 
and  s  oyer  +  and  - 
FDISTRIB<<X+Y>*<Z+1>) 
Z*X+1*X+Z*Y+1*M 

See:  DISTRIB 


Page  5-29 


IN 


Linearization  of 
exponent  ials 
LIN<EXP<X>A2> 

See :  TEXFRHD  TLIH  


LNCOLLECT 


UNCOLLECT: 

Coll ect  s  1 ogar  i  t  hms 

LHCOLLECT  (.  LH  (.  X  >  +LH  (.  Y 

LHt!X*Y> 


See:  TEXFRHD  

EaiE5mE3aE3aE3aiinil 


POWEREXPAND 
PONEXPRHD:  I 
Step^step  expansion  of 
cowers 

30WEXPfMK<X+Y>A2> 

«+Y>*t:X+Y> 

See:  


SIMPLIFY 

SIMPLIFY:  I 
Attempts  to  simplify 
an  expression 
SIMPLIFY<SIH<3X>^SIH<H 

>) 

4*C0S<X>A2-1 
See :  EXFRHD  COLLECT 


Page  5-30 


Chapter  6 

Solution  to  single  equations 

In  this  chapter  we  feature  those  functions  that  the  calculator  provides  for 
solving  single  equations  of  the  form  f(X)  =  0.   Associated  with  the  L  7  J  key 
there  are  two  menus  of  equation-solving  functions,  the  Symbolic  SOLVer 

J5zv_  ),  and  the  NUMerical  SoLVer  (©»  ).  Following,  we  present 
some  of  the  functions  contained  in  these  menus.  Change  CAS  mode  to 
Complex  for  these  exercises  (see  Chapter  2). 


Symbolic  solution  of  algebraic  equations 

Here  we  describe  some  of  the  functions  from  the  Symbolic  Solver  menu. 
Activate  the  menu  by  using  the  keystroke  combination  .  With  system  flag  1  1  7 


s.slv  HEnu 


a.isoL 

3. LDEC 
H.LinSOLVE 
S . SOLVEVK 
6 . SOLVE 


S.SLV  HEnu 

a.isoL 

?■ .  LDEC 
H.LinSOLVE 
5 . SOLVEVK 
6. SOLVE 

Functions  DESOLVE  and  LDEC  are  used  for  the  solution  of  differential 
equations,  the  subject  of  a  different  chapter,  and  therefore  will  not  be 
presented  here.  Similarly,  function  LINSOLVE  relates  to  the  solution  of  multiple 
linear  equations,  and  it  will  be  presented  in  a  different  chapter.  Functions 
ISOL  and  SOLVE  can  be  used  to  solve  for  any  unknown  in  a  polynomial 
equation.  Function  SOLVEVX  solves  a  polynomial  equation  where  the 
unknown  is  the  default  CAS  variable  VX  (typically  set  to  X).  Finally,  function 
ZEROS  provides  the  zeros,  or  roots,  of  a  polynomial.  Entries  for  all  the 
functions  in  the  S.SLV  menu,  except  ISOL,  are  available  through  the  CAS  help 
facility  ((^(W)III1H  ). 


Function  ISOL 

Function  ISOL(Equation,  variable)  will  produce  the  solution(s)  to  Equation  by 
isolating  variable.  For  example,  with  the  calculator  set  to  ALG  mode,  to  solve 
for  t  in  the  equation  at3-bt  =  0  we  can  use  the  following: 


Page  6-1 


Using  the  RPN  mode,  the  solution  is  accomplished  by  entering  the  equation  in 
the  stack,  followed  by  the  variable,  before  entering  function  ISOL.  Right 
before  the  execution  of  ISOL,  the  RPN  stack  should  look  as  in  the  figure  to  the 
left.  After  applying  ISOL,  the  result  is  shown  in  the  figure  to  the  right: 


3: 

a-t  -b-t 

l:  't 1 

MitiiffltiCE'nirnqi'MMtuaj 


The  first  argument  in  ISOL  can  be  an  expression,  as  shown  above,  or  an 
equation.  For  example,  in  ALG  mode,  try: 


CflS CH|  HELP 


Note:  To  type  the  equal  sign  (=)  in  an  equation,  use   =  (associated 

with  the  (JtJ  key). 

The  same  problem  can  be  solved  in  RPN  mode  as  illustrated  below  (figures 
show  the  RPN  stack  before  and  after  the  application  of  function  ISOL): 


3: 
2: 


CfiSCH  HELP 


2  2 


2: 


1=  I.  _  bl+j5k  ,_tl+J5)j! 
L:""        2        X"  2 


Function  SOLVE 

Function  SOLVE  has  the  same  syntax  as  function  ISOL,  except  that  SOLVE  can 
also  be  used  to  solve  a  set  of  polynomial  equations.  The  help-facility  entry  for 
function  SOLVE,  with  the  solution  to  equation  XA4  -  1  =  3  ,  is  shown  next: 

EOLVE:  I 
Solves  a  tor  a  set  of) 
po 1 y nom i  a 1  equat  i  on 
S0LVE<XA4-l=3iX> 

CA=T2  X=-T2  J- 

See:  LIHSOLVE  SOLVEVX 
fri!¥iira!PlM*ww.iM*«i;r™^ 


Page  6-2 


The  following  examples  show  the  use  of  function  SOLVE  in  ALG  and  RPN 
modes:     

isoi-veCp^s-p^s'/p1) 


S0LVE('p4-5-p=6,;pi 
3=-ip=2  p=-l±^E3 
ta[fiiH[fiEEgiiigiqNi«tii^l 


The  screen  shot  shown  above  displays  two  solutions.  In  the  first  one,  p4-5p 
=  1  25,  SOLVE  produces  no  solutions  {  }.  In  the  second  one,  p4  -  5p  =  6, 
SOLVE  produces  four  solutions,  shown  in  the  last  output  line.    The  very  last 
solution  is  not  visible  because  the  result  occupies  more  characters  than  the 
width  of  the  calculator's  screen.  However,  you  can  still  see  all  the  solutions  by 
using  the  down  arrow  key  fs^?),  which  triggers  the  line  editor  (this  operation 
can  be  used  to  access  any  output  line  that  is  wider  than  the  calculator's 
screen): 


:  S0LVEl>tfr-5*p=6";§i. 

[p=-l  p=2  p=-^El  p=-^ 

Cp=-l,p=2?p=-<:  a  +  i+T 
ll>^2?,p=-<<l-i*J"ll  V 


The  corresponding  RPN  screens  for  these  two  examples,  before  and  after  the 
application  of  function  SOLVE,  are  shown  next: 


3: 
2: 


p  -5-p=125 
«p« 


l:  'p 


4: 
3: 
2: 
l: 


{p=-ip=2  p=-A±^EU 
tamiHiHEEgiiigiqigiMtiiad 


Use  of  the  down-arrow  key  (<\3?)  in  this  mode  will  launch  the  line  editor: 


■6=2'  'p=-<  ■:: 
■2V   'p=-(  a- 


l+i*Tll)/2 
i*-J"llV2V  > 


Page  6-3 


Function  SOLVEVX 

The  function  SOLVEVX  solves  an  equation  for  the  default  CAS  variable 
contained  in  the  reserved  variable  name  VX.  By  default,  this  variable  is  set  to 
'X'.  Examples,  using  the  ALG  mode  with  VX  =  'X',  are  shown  below: 


S0LVEVx(x5-a-X=b) 

SOLVEVx(x5-6-X=20) 

X=2 


In  the  first  case  SOLVEVX  could  not  find  a  solution.  In  the  second  case, 
SOLVEVX  found  a  single  solution,  X  =  2. 


The  following  screens  show  the  RPN  stack  for  solving  the  two  examples  shown 
above  (before  and  after  application  of  SOLVEVX):  


The  equation  used  as  argument  for  function  SOLVEVX  must  be  reducible  to  a 
rational  expression.  For  example,  the  following  equation  will  not  processed 
by  SOLVEVX:  


iSOLVEVXtJX^JX  +  l) 
"Hot  reducible  to  a  r... 


A  SOLVEVX 
Error: 

Hot  reducible 
to  a  rational 
expression 

"Hot  reducible  to  a 

i  r„, 

Function  ZEROS 

The  function  ZEROS  finds  the  solutions  of  a  polynomial  equation,  without 
showing  their  multiplicity.  The  function  requires  having  as  input  the 
expression  for  the  equation  and  the  name  of  the  variable  to  solve  for. 
Examples  in  ALG  mode  are  shown  next: 


:ZER0s(k5-k^k) 
(01-!±U3 


l-i-J3 


)  ( 


:  ZER0slmD=32,mJ 

2'j'TT  4'1'TT 


SKIPH  -KjEL   DEL-*  DEL  L  ins  i 


  __  6'j'TT 

2-e   5    2-e   5    2-e  5 
aiHJHiHCE'HII'HO-HllIiigJ 


Page  6-4 


To  use  function  ZEROS  in  RPN  mode,  enter  first  the  polynomial  expression, 
then  the  variable  to  solve  for,  and  then  function  ZEROS.    The  following  screen 
shots  show  the  RPN  stack  before  and  after  the  application  of  ZEROS  to  the 
two  examples  above: 


2: 
l: 


0  1  - 


l  +  i-J3  l-i-J3 


2  2 
^if1iPJif1CE'J1ll'J1^JI'jmIIig 


2-j-TT 


6-i 


2-e   5    2-e   5  2-e 

aiHJHiHCE'Hii'Hai'Hiaiijg] 


The  Symbolic  Solver  functions  presented  above  produce  solutions  to  rational 
equations  (mainly,  polynomial  equations).  If  the  equation  to  be  solved  for  has 
all  numerical  coefficients,  a  numerical  solution  is  possible  through  the  use  of 
the  Numerical  Solver  features  of  the  calculator. 


Numerical  solver  menu 

The  calculator  provides  a  very  powerful  environment  for  the  solution  of  single 
algebraic  or  transcendental  equations.    To  access  this  environment  we  start 
the  numerical  solver  (NUM.SLV)  by  using  L  r>  Jnumslv  .  This  produces  a  drop- 
down menu  that  includes  the  following  options: 


2 

Solus 

diFF  sg.. 

3 

Solus 

poly.. 

H 

Solus 

lin  fyf.. 

5 

Solus 

financs.. 

6 

HSLY 

Item  2.  Solve  cliff  eq..  is  to  be  discussed  in  a  later  chapter  on  differential 
equations.  Item  4.  Solve  lin  sys..  will  be  discussed  in  a  later  Chapter  on 
matrices.  Item  6.  MSL V  (Multiple  equation  SoLVer)  will  be  presented  in  the 
next  chapter.    Following,  we  present  applications  of  items  3.  Solve  poly..,  5. 
Solve  finance,  and  7.  Solve  equation..,  in  that  order.  Appendix  1-A,  at  the 
end  of  Chapter  1,  contains  instructions  on  how  to  use  input  forms  with 
examples  for  the  numerical  solver  applications. 


Page  6-5 


Notes: 

1 .  Whenever  you  solve  for  a  value  in  the  NUM.SLV  applications,  the  value 
solved  for  will  be  placed  in  the  stack.  This  is  useful  if  you  need  to  keep  that 
value  available  for  other  operations. 

2.  There  will  be  one  or  more  variables  created  whenever  you  activate  some  of 
the  applications  in  the  NUM.SLV  menu. 


Polynomial  Equations 

Using  the  Solve  poly...  option  in  the  calculator's  SOLVE  environment  you  can: 

(1 )  find  the  solutions  to  a  polynomial  equation; 

(2)  obtain  the  coefficients  of  the  polynomial  having  a  number  of  given  roots; 

(3)  obtain  an  algebraic  expression  for  the  polynomial  as  a  function  of  X. 

Finding  the  solutions  to  a  polynomial  equation 

A  polynomial  equation  is  an  equation  of  the  form:  onxn  +  an.7xn"7  +  ...+  o7x  + 
o0  =  0.  The  fundamental  theorem  of  algebra  indicates  that  there  are  n 
solutions  to  any  polynomial  equation  of  order  n.  Some  of  the  solutions  could 
be  complex  numbers,  nevertheless.  As  an  example,  solve  the  equation:  3s4  + 
2s3  -  s  +  1  =  0. 


We  want  to  place  the  coefficients  of  the  equation  in  a  vector  [0^0^,01  a0]. 
For  this  example,  let's  use  the  vector  [3,2,0,-1,1].  To  solve  for  this  polynomial 
equation  using  the  calculator,  try  the  following: 
{j^hiMsiv  mm  Select  Solve  poly. . . 

EDS— QDO  'CX3CB  >  CD 

CD  >  CD  CD  CD  l  CD  EH  Enter  vector  of  coefficients 

lliHIIIl  Solve  equation 


The  screen  will  show  the  solution  as  follows: 


SOLVE  fln-KAn+..+fli  K+flD 
Coefficients  [  an  ..  oi       ]  = 
[3. ,2. j0. ? 1. ] 

Enter  roots  or  press  SOLVE  


Page  6-6 


Press  [enter]  to  return  to  stack.  The  stack  will  show  the  following  results  in  ALG 
mode  (the  same  result  would  be  shown  in  RPN  mode): 

Roots:  [(,  432194894623,  * 

To  see  all  the  solutions,  press  the  down-arrow  key  (<\3?)  to  trigger  the  line 
editor: 

Roots:  [(.  432194094623,  > 
: Roots: 

[  < .  432194B94623,-.3S9... 

All  the  solutions  are  complex  numbers:  (0.432,-0.389),  (0.432,0.389),  (- 
0.766,  0.632),  (-0.766,  -0.632). 


Note:  Recall  that  complex  numbers  in  the  calculator  are  represented  as 
ordered  pairs,  with  the  first  number  in  the  pair  being  the  real  part,  and  the 
second  number,  the  imaginary  part.  For  example,  the  number  (0.432,-0.389), 
a  complex  number,  will  be  written  normally  as  0.432  -  0.389/,  where  /  is  the 
imaginary  unit,  i.e.,  i2  =  -1 . 

Note:  The  fundamental  theorem  of  algebra  indicates  that  there  are  n  solutions 
for  any  polynomial  equation  of  order  n.  There  is  another  theorem  of  algebra 
that  indicates  that  if  one  of  the  solutions  to  a  polynomial  equation  with  real 
coefficients  is  a  complex  number,  then  the  conjugate  of  that  number  is  also  a 
solution.  In  other  words,  complex  solutions  to  a  polynomial  equation  with  real 
coefficients  come  in  pairs.  That  means  that  polynomial  equations  with  real 
coefficients  of  odd  order  will  have  at  least  one  real  solution. 


Generating  polynomial  coefficients  given  the  polynomial's  roots 

Suppose  you  want  to  generate  the  polynomial  whose  roots  are  the  numbers 
[1,5,  -2,  4].  To  use  the  calculator  for  this  purpose,  follow  these  steps: 


{jfjmw  ^7  ^7  lilEHS 

CB  >  CD  (JtJ  CB  >  GD  liiil 

missa 


Select  Solve  poly... 

Enter  vector  of  roots 
Solve  for  coefficients 


Page  6-7 


Press  [enter]  to  return  to  stack,  the  coefficients  will  be  shown  in  the  stack. 


Coefficient;  [  an  ..  ol  afl  ] : 

■CI. i-B. i9. i3B 

.  ,-40. ]| 

[  1  ■  -  5  -  -  -2- -  4- 

Enter  coefficients  or 

] 

press  SOLVE 

EDIT  |         |  ! 

Press  ^3?  to  trigger  the  line  editor  to  see  all  the  coefficients. 
Note:  If  you  want  to  get  a  polynomial  with  real  coefficients,  but  having 
complex  roots,  you  must  include  the  complex  roots  in  pairs  of  conjugate 
numbers.  To  illustrate  the  point,  generate  a  polynomial  having  the  roots  [1 
(1,2)  (1,-2)].  Verify  that  the  resulting  polynomial  has  only  real  coefficients. 
Also,  try  generating  a  polynomial  with  roots  [1  (1,2)  (-1,2)],  and  verify  that 
the  resulting  polynomial  will  have  complex  coefficients. 


Generating  an  algebraic  expression  for  the  polynomial 

You  can  use  the  calculator  to  generate  an  algebraic  expression  for  a 
polynomial  given  the  coefficients  or  the  roots  of  the  polynomial.  The  resulting 
expression  will  be  given  in  terms  of  the  default  CAS  variable  X.  (The 
examples  below  shows  how  you  can  replace  X  with  any  other  variable  by 
using  the  function  | .) 

To  generate  the  algebraic  expression  using  the  coefficients,  try  the  following 
example.  Assume  that  the  polynomial  coefficients  are  [1,5,-2,4].  Use  the 
following  keystrokes: 


{J+Jnumslv  111!! 

EDO— CDCB  '(X) 

C3  >  CD  (jlD  O  >  GD IIEI 

C±^>  ::::;::.::.:.i.:;:h 

(enter} 


Select  Solve  poly... 
Enter  vector  of  coefficients 

Generate  symbolic  expression 
Return  to  stack. 


The  expression  thus  generated  is  shown  in  the  stack  as: 

'XA3+5*XA2+-2*X+4'. 


Page  6-8 


To  generate  the  algebraic  expression  using  the  roots,  try  the  following 
example.  Assume  that  the  polynomial  roots  are  [1,3,-2,1].  Use  the  following 
keystrokes: 


C3  >  (X)      CB  >  CD  E™ 

^3?  iiini 


[Qmw  ^7  ^7  Ij:[|S! 


Select  Solve  poly... 
Enter  vector  of  roots 


Generate  symbolic  expression 
Return  to  stack. 


The  expression  thus  generated  is  shown  in  the  stack  as:'(X-l  )*(X-3)*(X+2)*(X-1 )'. 
To  expand  the  products,  you  can  use  the  EXPAND  command.  The  resulting 
expression  is:  'XA4+-3*XA3+  -3*XA2+1 1  *X-6'. 

A  different  approach  to  obtaining  an  expression  for  the  polynomial  is  to 
generate  the  coefficients  first,  then  generate  the  algebraic  expression  with  the 
coefficients  highlighted.  For  example,  for  this  case  try: 

{J^nww           mm  Select  Solve  poly. . . 

^3?CED£  CDO  'CD  Enter  vector  of  roots 


The  expression  thus  generated  is  shown  in  the  stack  as:  'XA4+-3*XA3+  - 
3*XA2+1 1  *X+-6*XA0\  The  coefficients  are  listed  in  stack  level  2. 

Financial  calculations 

The  calculations  in  item  5.  Solve  finance.,  in  the  Numerical  Solver  (NUM.SLV) 
are  used  for  calculations  of  time  value  of  money  of  interest  in  the  discipline  of 
engineering  economics  and  other  financial  applications.  This  application  can 
also  be  started  by  using  the  keystroke  combination  [jij(s\ finance  (associated 
with  the  L  9  j  key).  Before  discussing  in  detail  the  operation  of  this  solving 
environment,  we  present  some  definitions  needed  to  understand  financial 
operations  in  the  calculator. 


Solve  for  coefficients 
Generate  symbolic  expression 
Return  to  stack. 


Page  6-9 


Definitions 

Often,  to  develop  projects,  it  is  necessary  to  borrow  money  from  a  financial 
institution  or  from  public  funds.    The  amount  of  money  borrowed  is  referred  to 
as  the  Present  Value  (PV).  This  money  is  to  be  repaid  through  n  periods 
(typically  multiples  or  sub-multiples  of  a  month)  subject  to  an  annual  interest 
rate  of  l%YR.  The  number  of  periods  per  year  (P/YR)  is  an  integer  number  of 
periods  in  which  the  year  will  be  divided  for  the  purpose  of  repaying  the  loan 
money.  Typical  values  of  P/YR  are  1  2  (one  payment  per  month),  24 
(payment  twice  a  month),  or  52  (weekly  payments).    The  payment(PI\Al)  is 
the  amount  that  the  borrower  must  pay  to  the  lender  at  the  beginning  or  end 
of  each  of  the  n  periods  of  the  loan.    The  future  value  of  the  money  (FV)  is  the 
value  that  the  borrowed  amount  of  money  will  be  worth  at  the  end  of  n 
periods.    Typically  payment  occurs  at  the  end  of  each  period,  so  that  the 
borrower  starts  paying  at  the  end  of  the  first  period,  and  pays  the  same  fixed 
amount  at  the  end  of  the  second,  third,  etc.,  up  to  the  end  of  the  n-th  period. 

Example  1  -  Calculating  payment  on  a  loan 

If  $2  million  are  borrowed  at  an  annual  interest  rate  of  6.5%  to  be  repaid  in 
60  monthly  payments,  what  should  be  the  monthly  payment?  For  the  debt  to 
be  totally  repaid  in  60  months,  the  future  values  of  the  loan  should  be  zero. 
So,  for  the  purpose  of  using  the  financial  calculation  feature  of  the  calculator 
we  will  use  the  following  values:  n  =  60,  l%YR  =  6.5,  PV  =  2000000,  FV  = 
0,  P/YR  =12.  To  enter  the  data  and  solve  for  the  payment,  PMT,  use: 
B  J  finance  Start  the  financial  calculation  input  form 

68  MM  Enter  n  =  60 


6 « 5  EES 
2888880  IB11 

^  cd  rasa 


Enter  l%YR  =  6.5  % 

Enter  PV  =  2,000,000  US$ 

Skip  PMT,  since  we  will  be  solving  for  it 

Enter  FV  =  0,  the  option  End  is  highlighted 

Highlight  PMT  and  solve  for  it 


The  solution  screen  will  look  like  this: 


j$H»TIHE  VALUE  OF  HOriEV^^ 
n=      60  IXVR:  6. 5 

PV:  2000000-00 


fv:  0.00  End 


Enter  pa^Hgnt  OHOunt  or_  SOLV E 


Page  6-10 


The  screen  now  shows  the  value  of  PMT  as  -39,1  32.30,  i.e.,  the  borrower 
must  pay  the  lender  US  $  39,1  32.30  at  the  end  of  each  month  for  the  next 
60  months  to  repay  the  entire  amount.    The  reason  why  the  value  of  PMT 
turned  out  to  be  negative  is  because  the  calculator  is  looking  at  the  money 
amounts  from  the  point  of  view  of  the  borrower.  The  borrower  has  +  US  $ 
2,000,000.00  at  time  period  t  =  0,  then  he  starts  paying,  i.e.,  adding  -US  $ 
391  32.30  at  times  t  =  1 ,  2,      60.  At  t  =  60,  the  net  value  in  the  hands  of 
the  borrower  is  zero.    Now,  if  you  take  the  value  US  $  39,1  32.30  and 
multiply  it  by  the  60  payments,  the  total  paid  back  by  the  borrower  is  US  $ 
2,347,937.79.  Thus,  the  lender  makes  a  net  profit  of  $  347,937.79  in  the  5 
years  that  his  money  is  used  to  finance  the  borrower's  project. 

Example  2  -  Calculating  amortization  of  a  loan 

The  same  solution  to  the  problem  in  Example  1  can  be  found  by  pressing 
13103,  which  is  stands  for  AMORTIZATION.  This  option  is  used  to  calculate 
how  much  of  the  loan  has  been  amortized  at  the  end  of  a  certain  number  of 
payments.    Suppose  that  we  use  24  periods  in  the  first  line  of  the 
amortization  screen,  i.e.,  L2JLiJ  EM.  Then,  press  EIHIII.    You  will  get  the 
following  result:  mmmmmmm  ™™™™ 


§§§  AHORTIZE  §§§§§§§§§§§§§§§§§§§§§§§ 

Parents  = 

24 

Principal.: 

Interest  = 

-215963. 68 

Balance: 

1276788.57 

This  screen  is  interpreted  as  indicating  that  after  24  months  of  paying  back 
the  debt,  the  borrower  has  paid  up  US  $  723,21  1 .43  into  the  principal 
amount  borrowed,  and  US  $  215,963.68  of  interest.    The  borrower  still  has 
to  pay  a  balance  of  US  $1,276,788.57  in  the  next  36  months. 

Check  what  happens  if  you  replace  60  in  the  Payments:  entry  in  the 
amortization  screen,  then  press  111:1  llllllllIElLlllllil. The  screen  now  looks  like  this: 


Parents  = 

60  

Principal 

-2000000.00  M 

Interest: 

-347937.79 

Balance: 

-3. 16E-6 

EDIT  1         1         !         1  P-*f  V  1  flHOR 

Page  6-1 1 


This  means  that  at  the  end  of  60  months  the  US  $  2,000,000.00  principal 
amount  has  been  paid,  together  with  US  $  347,937.79  of  interest,  with  the 
balance  being  that  the  lender  owes  the  borrower  US  $  0.00031  6.  Of 
course,  the  balance  should  be  zero.  The  value  shown  in  the  screen  above  is 
simply  round-off  error  resulting  from  the  numerical  solution. 

Press  jQN  jor  [enter]  f  twice,  to  return  to  normal  calculator  display. 

Example  3  -  Calculating  payment  with  payments  at  beginning  of  period 

Let's  solve  the  same  problem  as  in  Examples  1  and  2,  but  using  the  option 
that  payment  occurs  at  the  beginning  of  the  payment  period.  Use: 

(jjjj finance  Start  the  financial  calculation  input  form 

60  lEI  Enter  n  =  60 

6.5  IIIII  Enter  l%YR  =  6.5  % 

2000009  lililii  Enter  PV  =  2,000,000  US$ 

^▼7  Skip  PMT,  since  we  will  be  solving  for  it 

9  III!  Enter  FV  =  0,  the  option  End  is  highlighted 

SilOiySlll^^ IIIIIIIIIIIIII  Change  payment  option  to  Begin 

^  CD  BSSa  Highlight  PMT  and  solve  for  it 

The  screen  now  shows  the  value  of  PMT  as  -38,921 .47,  i.e.,  the  borrower 
must  pay  the  lender  US  $  38,921 .48  at  the  beginning  of  each  month  for  the 
next  60  months  to  repay  the  entire  amount.    Notice  that  the  amount  the 
borrower  pays  monthly,  if  paying  at  the  beginning  of  each  payment  period,  is 
slightly  smaller  than  that  paid  at  the  end  of  each  payment  period.  The  reason 
for  that  difference  is  that  the  lender  gets  interest  earnings  from  the  payments 
from  the  beginning  of  the  period,  thus  alleviating  the  burden  on  the  lender. 

Notes: 

1 .  The  financial  calculator  environment  allows  you  to  solve  for  any  of  the 
terms  involved,  i.e.,  n,  l%YR,  PV,  FV,  P/Y,  given  the  remaining  terms  in  the 
loan  calculation.   Just  highlight  the  value  you  want  to  solve  for,  and  press 
IIISI33.    The  result  will  be  shown  in  the  highlighted  field. 


Page  6-12 


2.  The  values  calculated  in  the  financial  calculator  environment  are  copied  to 
the  stack  with  their  corresponding  tag  (identifying  label). 


Deleting  the  variables 

When  you  use  the  financial  calculator  environment  for  the  first  time  within  the 
HOME  directory,  or  any  sub-directory,  it  will  generate  the  variables  iiiiiil  lilll 


llili  IIIIIIIIII 11113391 IIIIIIII  to  store  the  corresponding  terms  in  the  calculations.. 
You  can  see  the  contents  of  these  variables  by  using: 

(      v    1  -I;:  :■"::::::  (      ,    ^  i::":"":"™:::  (  ^  (      s,   1  ::::"!:"":":::::  (  ^  (      ■■    ^>  ::::"n"sE 

LClJ  iiii iiiilili  Lj^LJ  IIL:I:iSILILi:i!I  ISZ_J  I!!!.:!!:!:!!!!!  LCLJ  Sa-sis-s-sLSSi!  IXlJ  I!!!.:!!!.!!.:"!!!  LCLJ  ss-siiisii • 

You  can  either  keep  these  variables  for  future  use,  or  use  the  PURGE  function 
to  erase  them  from  your  directory.  To  erase  all  of  the  variables  at  once,  if 
using  ALG  mode,  try  the  following: 

Enter  PURGE,  prepare  list  of  variables 
Enter  name  of  variable  N 
Enter  a  comma 
Enter  name  of  variable  l%YR 
Enter  a  comma 
Enter  name  of  variable  PV 
Enter  a  comma 
Enter  name  of  variable  PMT 
Enter  a  comma 
Enter  name  of  variable  PYR 
Enter  a  comma 
Enter  name  of  variable  FV 
Execute  PURGE  command 


CD  C3 
HE 
CD  C3  i 

cd  cb™ 
cd  ra  i 

CD  C3 
CD  CB  > 

CD  CB  \ 

CD  C3I1I. 

[enter] 


The  following  two  screen  shots  show  the  PURGE  command  for  purging  all  the 
variables  in  the  directory,  and  the  result  after  executing  the  command. 


PURGE <£ 'N' , ' I*YR' , 
1 PV 1 j  1 PMT 1 ?  1 PYR 1 ? 

gaitiiHitiieEgiiigiqi'JiMtijaj 


PURGEK'N'  'I^YR'  "PV  'PMT* 
HOVRL 

taitiiHitiieEgiiigiqi'JiMtuaj 


In  RPN  mode,  the  command  is  executed  by  using: 


Page  6-13 


!!!!.•!!.:.!!.!!!!!! 


(jmJ  UnJU   Prepare  a  list  of  variables  to  be  purged 

ill!  Enter  name  of  variable  N 

Enter  name  of  variable  l%YR 
Enter  name  of  variable  PV 
Enter  name  of  variable  PMT 
Enter  name  of  variable  PYR 
Enter  name  of  variable  FV 
(enter}  Enter  list  of  variables  in  stack 

[too) HIDES  Purge  variables  in  list 

Before  the  command  PURGE  is  entered,  the  RPN  stack  will  look  like  this: 


ff:   <H  I*YR  PV  PMT  PYR  FV! 


Solving  equations  with  one  unknown  through  NUM.SLV 

The  calculator's  NUM.SLV  menu  provides  item  1.  Solve  equation.,  solve 
different  types  of  equations  in  a  single  variable,  including  non-linear  algebraic 
and  transcendental  equations.       For  example,  let's  solve  the  equation:  ex- 

sin(7rx/3)  =  0. 

Simply  enter  the  expression  as  an  algebraic  object  and  store  it  into  variable 
EQ.  The  required  keystrokes  in  ALG  mode  are  the  following: 

CD  ED  £—  ^  ED  (3  CD  CE3  C*D  ED  5_ 
ED(^ED(3ED(X}®C3^  QDCD 

[stov]  [alpha}  (T\  [alpha}  (q\  [enter] 


Function  STEQ 

Function  STEQ,  available  through  the  command  catalog,  CS_^r  /  will  store 
its  argument  into  variable  EQ,  e.g.,  in  ALG  mode: 


:STEQ[eX-SIH(^]=0] 
.   HOVRL 

■MiBaffiiiniiiMniici^CBaB 

In  RPN  mode,  enter  the  equation  between  apostrophes  and  activate  command 
STEQ.  Thus,  function  STEQ  can  be  used  as  a  shortcut  to  store  an  expression 
into  variable  EQ. 


Page  6-14 


Press  jvAR  j  to  see  the  newly  created  EQ  variable 


Then,  enter  the  SOLVE  environment  and  select  Solve  equation...,  by  using: 
[j^jnumslv  Ililllll,  The  corresponding  screen  y/jjlbeshown  as: 


^SOLVE  ECUflTIOn^^™ 


Enter  Function  to  sol.u« 


The  equation  we  stored  in  variable  EQ  is  already  loaded  in  the  Eq  field  in  the 
SOLVE  EQUATION  input  form.  Also,  a  field  labeled  x  is  provided.  To  solve 
the  equation  all  you  need  to  do  is  highlight  the  field  in  front  of  X:  by  using 
^▼7,  and  press  MMM.  The  solution  shown  is  X:  4.5006E-2: 


SOLVE  EGUflTIOn 
Eg  =  EXP  (  x  >  -S I H  (  TT*X^3  >  =0 


4.50061385902E-2 


Enter  uglm  or  prgjj  SOLVE 


This,  however,  is  not  the  only  possible  solution  for  this  equation.  To  obtain  a 
negative  solution,  for  example,  enter  a  negative  number  in  the  X:  field  before 
solving  the  equation.  Try  LU l_!hJ KWs^? ii:;:i!i!i]!hili.  The  solution  is  now  X:  - 
3.045. 


Solution  procedure  for  Equation  Solve... 

The  numerical  solver  for  single-unknown  equations  works  as  follows: 

•  It  lets  the  user  type  in  or  EDIilll  an  equation  to  solve. 

•  It  creates  an  input  form  with  input  fields  corresponding  to  all  variables 
involved  in  equation  stored  in  variable  EQ. 

•  The  user  needs  to  enter  values  for  all  variables  involved,  except  one. 


Page  6-15 


•  The  user  then  highlights  the  field  corresponding  to  the  unknown  for 
which  to  solve  the  equation,  and  presses  iEilili 

•  The  user  may  force  a  solution  by  providing  an  initial  guess  for  the 
solution  in  the  appropriate  input  field  before  solving  the  equation. 

The  calculator  uses  a  search  algorithm  to  pinpoint  an  interval  for  which  the 
function  changes  sign,  which  indicates  the  existence  of  a  root  or  solution.  It 
then  utilizes  a  numerical  method  to  converge  into  the  solution. 

The  solution  the  calculator  seeks  is  determined  by  the  initial  value  present  in 
the  unknown  input  field.  If  no  value  is  present,  the  calculator  uses  a  default 
value  of  zero.  Thus,  you  can  search  for  more  than  one  solution  to  an 
equation  by  changing  the  initial  value  in  the  unknown  input  field.  Examples 
of  the  equations  solutions  are  shown  following. 

Example  1  -  Hooke's  law  for  stress  and  strain 

The  equation  to  use  is  Hooke's  law  for  the  normal  strain  in  the  x-direction  for  a 
solid  particle  subjected  to  a  state  of  stress  given  by 


°xx 

<J 

°yy 

(7 

Gzx 

The  equation  is  exx  =  — [<JXX  -  n  •  (a    +  <JZZ)]  +  OC  •  AT,  here  exx  is  the  unit 

strain  in  the  x-direction,  <jxx/  <yyyf  and  <rzz,  are  the  normal  stresses  on  the 
particle  in  the  directions  of  the  x-,  y-,  and  z-axes,  E  is  Young's  modulus  or 
modulus  of  elasticity  of  the  material,  n  is  the  Poisson  ratio  of  the  material,  a  is 
the  thermal  expansion  coefficient  of  the  material,  and  zlTis  a  temperature 
increase. 

Suppose  that  you  are  given  the  following  data:  <jxx=  2500  psi,  cryy  =  l  200  psi, 
and  azz  =  500  psi,  E  =  1200000  psi,  n  =  0.15,  a  =  0.0000 1/°F,  AT  =  60  °F. 
To  calculate  the  strain  exx  use  the  following: 

L  r>  J num.slv  Access  numerical  solver  to  solve  equations 

[  r>  j  eqw  Access  the  equation  writer  to  enter  equation 


Page  6-16 


At  this  point  follow  the  instructions  from  Chapter  2  on  how  to  use  the  Equation 
Writer  to  build  an  equation.  The  equation  to  enter  in  the  Eq  field  should  look 
like  this  (notice  that  we  use  only  one  sub-index  to  refer  to  the  variables,  i.e., 
exx  is  translated  as  ex,  etc.  --  this  is  done  to  save  typing  time): 


?X=^i((JX-ni(cry+crz))+ai^T 

Use  the  following  shortcuts  for  special  characters: 

a:         (alpha}[^J(s]         a:         (alpha}[W)(a]         A:  (alpha}[^J(c] 


and  recall  that  lower-case  letters  are  entered  by  using  [alpha)  j<-\  j  before  the 
letter  key,  thus,  x  is  typed  as  (^[jT}g] . 

Press  Lenter) to  return  to  the  solver  screen.  Enter  the  values  proposed  above  into 
the  corresponding  fields,  so  that  the  solver  screen  looks  like  this: 


SOLVE  EGUATIOni 

Eg  =  ex=l^E#  <  ux-n*  ( try+tr... 
*x-  E:  120...  m=  25... 

n=   .15  12...   -n:  500 

«:    -00...    fiT=  60 

Enter  ualm  or  prjjg  SOLVE 


With  the  ex:  field  highlighted,  press  to  solve  for  ex: 


SOLVE  EGUATIOn 
Eg :  ex=l/E* <. r/x-n* ( ay+tT... 

«X=    ail    E=    120...    m=  25... 

n=  .15        =    12...  -n:  500 

«=    .00...    *T:  60 

Enter  uglm  or  prjjj  SOLVE 


Page  6-17 


The  solution  can  be  seen  from  within  the  SOLVE  EQUATION  input  form  by 
pressing  EES]  while  the  ex:  field  is  highlighted.  The  resulting  value  is 
2.470833333333E-3.  Press  MM  to  exit  the  EDIT  feature. 

Suppose  that  you  now,  want  to  determine  the  Young's  modulus  that  will 
produce  a  strain  of  exx  =  0.005  under  the  same  state  of  stress,  neglecting 
thermal  expansion.  In  this  case,  you  should  enter  a  value  of  0.005  in  the  ex: 
field,  and  a  zero  in  the  AT:  field  (with  AT  =  0,  no  thermal  effects  are 
included).  To  solve  for  E,  highlight  the  E:  field  and  press  BHU.  The  result, 
seeing  with  the  EMi  feature  is,  E  =  449000  psi.    Press  mBM  [enter]  to  return 
to  normal  display. 

Notice  that  the  results  of  the  calculations  performed  within  the  numerical 
solver  environment  have  been  copied  to  the  stack: 


ex:2.47083333333E-3 
E: 449003. 


Also,  you  will  see  in  your  soft-menu  key  labels  variables  corresponding  to 
those  variables  in  the  equation  stored  in  EQ  (press  jNxrj  to  see  all  variables  in 
your  directory),  i.e.,  variables  ex,  AT,  a,  oz,  ay,  n,  ax,  and  E. 

Example  2  -  Specific  energy  in  open  channel  flow 

Specific  energy  in  an  open  channel  is  defined  as  the  energy  per  unit  weight 
measured  with  respect  to  the  channel  bottom.  Let  E  =  specific  energy,  y  = 
channel  depth,  V  =  flow  velocity,  g  =  acceleration  of  gravity,  then  we  write 

v2 

E  =  y  +  . 

The  flow  velocity,  in  turn,  is  given  by  V  =  Q/A,  where  Q  =  water  discharge, 
A  =  cross-sectional  area.    The  area  depends  on  the  cross-section  used,  for 
example,  for  a  trapezoidal  cross-section,  as  shown  in  the  figure  below,  A  = 
(b+m-y)  y,  where  b  =  bottom  width,  and  m  =  side  slope  of  cross  section. 


Page  6-18 


We  can  type  in  the  equation  for  E  as  shown  above  and  use  auxiliary 
variables  for  A  and  V,  so  that  the  resulting  input  form  will  have  fields  for  the 
fundamental  variables  y,  Q,  g,  m,  and  b,  as  follows: 

•     First,  create  a  sub-directory  called  SPEN  (SPecific  ENergy)  and  work 
within  that  sub-directory. 


Q 
A 

A. 

H 

:  E=y+^  ►EQ 

Launch  the  numerical  solver  for  solving  equations:  IIIEI1III. 
Notice  that  the  input  form  contains  entries  for  the  variables  y,  Q,  b,  m, 
and  g: 


b  = 

Enter  Function 

to '  folm 

Try  the  following  input  data:  E  =  1  0  ft,  Q  = 
second),  b  =  2.5ft^ri^  1 .0,  g  =  32.2ft/sfr 


0  cfs  (cubic  feet  per 


^  EtUflTIOn  §§§§§§§§§§§§§§§§§ 
Eg:E=y+V-A-2^^2*a> 

E=  10  y 
G  =  10  I 
1  3 
Enter  ualm  or  prjjg  SOLVE 


Solve  for  y. 


Page  6-19 


^  EtUflTIOl"!  §§§§§§§§§§§§§§§§§ 
Eg:  E=y+V-A-2^2*g>  

10  b=  2.5 

H=    1  3=  32.2 

Enter  ualm  or  prgss  SOLVE  


The  result  is  0.149836..,  i.e.,  y  =  0.149836. 
It  is  known,  however,  that  there  are  actually  two  solutions  available 
for  y  in  the  specific  energy  equation.  The  solution  we  just  found 
corresponds  to  a  numerical  solution  with  an  initial  value  of  0  (the 
default  value  for  y,  i.e.,  whenever  the  solution  field  is  empty,  the 
initial  value  is  zero).  To  find  the  other  solution,  we  need  to  enter  a 
larger  value  of  y,  say  15,  highlight  the  y  input  field  and  solve  for  y 
once  more: 


SOLVE  EGUATIOn 

Eq:E=y+VA2^<2*a> 

E=  10  y 
G  =  10  b 
H=    1  3 


2.5 
32.2 


Enter  ugluc  or  press  SOLVE 


The  result  is  now  9.99990,  i.e.,  y  =  9.99990  ft. 

This  example  illustrates  the  use  of  auxiliary  variables  to  write  complicated 
equations.  When  NUM.SLV  is  activated,  the  substitutions  implied  by  the 
auxiliary  variables  are  implemented,  and  the  input  screen  for  the  equation 
provides  input  field  for  the  primitive  or  fundamental  variables  resulting  from 
the  substitutions.  The  example  also  illustrates  an  equation  that  has  more  than 
one  solution,  and  how  choosing  the  initial  guess  for  the  solution  may  produce 
those  different  solutions. 


In  the  next  example  we  will  use  the  DARCY  function  for  finding  friction  factors 
in  pipelines.  Thus,  we  define  the  function  in  the  following  frame. 

Special  function  for  pipe  flow:  DARCY  (s/D,Re) 

The  Darcy-Weisbach  equation  is  used  to  calculate  the  energy  loss  (per  unit 
weight),  hf,  in  a  pipe  flow  through  a  pipe  of  diameter  D,  absolute  roughness 
c,  and  length  L,  when  the  flow  velocity  in  the  pipe  is  V.    The  equation  is 


Page  6-20 


L  V 

written  as  hf  =  f  .  The  quantity  f  is  known  as  the  friction  factor  of 

D  2g 

the  flow  and  it  has  been  found  to  be  a  function  of  the  relative  roughness  of  the 
pipe,  s/D,  and  a  (dimensionless)  Reynolds  number,  Re.  The  Reynolds  number 
is  defined  as  Re  =  pVD/ja  =  VD/v,  where  p  and  u.  are  the  density  and 
dynamic  viscosity  of  the  fluid,  respectively,  and  v  =  uVp  is  the  kinematic 
viscosity  of  the  fluid. 

The  calculator  provides  a  function  called  DARCY  that  uses  as  input  the  relative 
roughness  s/D  and  the  Reynolds  number,  in  that  order,  to  calculate  the  friction 


CATALOG :  F63  COHHflnDS 

DATE+ 

DBUG 

DDHVL" 

DEC 

For  example,  for  e/D  =  0.0001 ,  Re  =  1 000000,  you  can  find  the  friction 
factor  by  using:  DARCY(0. 000 1,1 000000).  In  the  following  screen,  the 


function  ->NUM  ()  was  used  to  obtain  a  numerical  va 


DflRC YC  -  000 1 , 1 0000003 

DflRCYC-  0001,10000005 
■*HUM(flHS(lH 

1.34414320724E-2 


ue  of  the  function: 


The  result  is  f  =  DARCY(0.0001 , 1 000000)  =  0.01  341 . 


The  function  FANNING(s/D,Re) 

In  aerodynamics  applications  a  different  friction  factor,  the  Fanning  friction 
factor,  is  used.  The  Fanning  friction  factor,  fF,  is  defined  as  4  times  the  Darcy- 
Weisbach  friction  factor,  f.  The  calculator  also  provides  a  function  called 
FANNING  that  uses  the  same  input  as  DARCY,  i.e.,  s/D  and  Re,  and 
provides  the  FANNING  friction  factor.    Check  that 
FANNING(0.0001, 1000000)  =  0.0033603589181s. 


Page  6-21 


DflRCY(.000i,i000000]| 
:+HUM(flHS(lH 

1.34414320724E-2 
:  FflHH  I HGC .  000 1 , 1 000000) 
FRHHIHGC.  0001,1000000) 
:+NUM(flNS(lH 

3.36035S01S1E-3 


Example  3  -  Flow  in  a  pipe 

You  may  want  to  create  a  separate  sub-directory  (PIPES)  to  try  this  example. 
The  main  equation  governing  flow  in  a  pipe  is,  of  course,  the  Darcy-Weisbach 


equation.  Thus,  type  in  the  following  equation  into  EQ: 


Also,  enter  the  following  variables  (f,  A,  V,  Re): 


hf  = 


f  'V  'L 


: DRRCY 


2-g-D 


DflRCY^Re 


TT-CT 

4 

Q 

fl 

In  this  case  we  stored  the  main  equation  (Darcy-Weisbach  equation)  into  EQ, 
and  then  replaced  several  of  its  variables  by  other  expressions  through  the 
definition  of  variables  f,  A,  V,  and  Re.  To  see  the  combined  equation,  use 
EVAL(EQ).  In  this  example  we  changed  the  display  setting  so  that  we  can  see 
the  entire  equation  in  the  screen: 


:EYAL(EGJ 


S-G  -L-DARCY 


D 


nu 


Page  6-22 


Thus,  the  equation  we  are  solving,  after  combining  the  different  variables  in 
the  directory,  is: 

f        QD  ^ 


h  =MJl..Darcy 

f  *2gD5 


D '  Nu 


The  combined  equation  has  primitive  variables:  hf,  Q,  L,  g,  D,  s,  and  Nu. 
Launch  the  numerical  solver  (\j+jnumslv  IjEGIl)  to  see  the  primitive  variables 
listed  in  the  SOLVE  EQUATION  input  form: 


SOLVE  ECUfiTIOn 


Eg 

y:  nu: 

Enter  Function  to  sol.u« 


[i: 
L  = 


Suppose  that  we  use  the  values  hf  =  2  m,  s  =  0.00001  m,  Q  =  0.05  m3/s, 
Nu  =  0.000001  m2/s,  L  =  20  m,  and  g  =  9.806  m/s2,  find  the  diameter  D. 
Enter  the  input  values,  and  solve  for  D,  The  solution  is:  0.1  2,  i.e.,  D  =  0.1  2 
m. 


SOLVE  EQUATION 

Eg  =  hf  =f  *V'-2*Ls  <.  2*g*D  ) 

hf=     2  .00,,,    D=  IffHW 

t=    .05       nu=     .0,,,    L:  £0 
3=  9.S... 

Enter  ujl.ug  or  press  SOLVE 


If  the  equation  is  dimensionally  consistent,  you  can  add  units  to  the  input 
values,  as  shown  in  the  figure  below.  However,  you  must  add  those  units  to 
the  initial  guess  in  the  solution.  Thus,  in  the  example  below  we  place  0_m  in 
the  D:  field  before  solving  the  problem.  The  solution  is  shown  in  the  screen  to 
the  right: 


SOLVE  EGUATIOn; 

Eg:hf=f*VA2*L^2*g*D) 
hf:  2_m  e=  .00,,,  d=  TS8BM 

G  =    .  5_...    nu:     .0,,,    L:    1 0_m 

s=  9.8... 

Enter  ualue  or  press  SOLVE 


»b»b  solve  EtuflTior.g 

Eg 

hf 

2_ru   e=   .00,,,   D : 

y: 

.  5_...    nu=     .0,,,  L: 

10_m 

3: 

9.3... 

1 

259210489698_ru' 

Press  [fi^  to  return  to  normal  calculator  display.  The  solution  for  D  will  be 
listed  in  the  stack. 


Page  6-23 


Example  4  -  Universal  gravitation 

Newton's  law  of  universal  gravitation  indicates  that  the  magnitude  of  the 
attractive  force  between  two  bodies  of  masses  m7  and  m2  separated  by  a 

.           .    .      ,     ,          .     -    ^  M,-M2 
distance  r  is  given  by  the  equation  t  =  Lr  -  . 

r 

Here,  G  is  the  universal  gravitational  constant,  whose  value  can  be  obtained 
through  the  use  of  the  function  CONST  in  the  calculator  by  using: 


COHSTCG) 


6.67259E-11_ 


2. 
s  -kg 


We  can  solve  for  any  term  in  the  equation  (except  G)  by  entering  the 
equation  as: 


F=CONST(G> 


M1-M2 
24 

r 


EaaEnaBgiBaiiMHiHEBini 

This  equation  is  then  stored  in  EQ: 


6.67259E-11_- 


2. 
s  -kg 

F=C0HSTCG3■■!:^:^^EQ 


Launching  the  numerical  solver  for  this  equation  results  in  an  input  form 
containing  input  fields  for  F,  G,  ml,  m2,  and  r. 


SOLVE  EGUATIOn 


HZ: 


Enter  Function  to  so  Luc 


Let's  solve  this  problem  using  units  with  the  following  values  for  the  known 
variables  ml  =  1.0xl06kg,  m2  =  1.0xl012kg,  r  =  l.OxlO11  m.  Also,  enter 
a  value  of  0_N  in  field  F  to  ensure  the  proper  solution  using  units  in  the 
calculator: 


Page  6-24 


SOLVE  EGUATIOng 

Eg  =  F=(6.67259E- 1 1  _mA3„ 

Hi:  100000,,,  f:  1000000,, 
Enter  ualm  or  prjjg  SOLVE 


Solve  for  F,  and  press  to  return  to  normal  calculator  display.  The  solution  is  F  : 
6.67259E-1 5_N,  or  F  =  6.67259x1 0"15  N. 

Note:  When  using  units  in  the  numerical  solver  make  sure  that  all  the 
variables  have  the  proper  units,  that  the  units  are  compatible,  and  that  the 
equation  is  dimensionally  homogeneous. 


Different  ways  to  enter  equations  into  EQ 

In  all  the  examples  shown  above  we  have  entered  the  equation  to  be  solved 
directly  into  variable  EQ  before  activating  the  numerical  solver.  You  can 
actually  type  the  equation  to  be  solved  directly  into  the  solver  after  activating 
it  by  editing  the  contents  of  the  EQ  field  in  the  numerical  solver  input  form.  If 
variable  EQ  has  not  been  defined  previously,  when  you  launch  the  numerical 
solver  (CS^IIlli),  the  EQ  field  will  be  highlighted: 


Enter  Function 

to  so  Lug 

At  this  point  you  can  either  type  a  new  equation  by  pressing  BED.  You  will 
be  provided  with  a  set  of  apostrophes  so  that  you  can  type  the  expression 
between  them: 


Type  an  equation,  say  XA2  -  1  25  =  0,  directly  on  the  stack,  and  press  I 


Page  6-25 


E*UATi.:.r.:;« 

Eq:X--2-12!5= 

=0 

Enter  ual.u«  or 

pfjJJ  SOLVE 

'X^2- 125=0* 
At  this  point  the  equation  is  ready  for  solution 


Alternatively,  you  can  activate  the  equation  writer  after  pressing  lllliiilil  to  enter 
your  equation.  Press  [enter]  to  return  to  the  numerical  solver  screen. 

Another  way  to  enter  an  equation  into  the  EQ  variable  is  to  select  a  variable 
already  existing  in  your  directory  to  be  entered  into  EQ.  This  means  that  your 
equation  would  have  to  have  been  stored  in  a  variable  name  previously  to 
activating  the  numerical  solver.  For  example,  suppose  that  we  have  entered 
the  following  equations  into  variables  EQ1  and  EQ2: 


:X  -5S=0^Q1 

2 


X3-58=0 


a  -a+300=@^Q2 

2 

 _a  -a+30_0=6 


lsts  i  ■:  h  m  n  l  i  f-  ■:■  l  f;  m 


Now,  launch  the  numerical  solver  (C7*1num.slv  mm,  and  highlight  the  EQ  field. 
At  this  point  press  the  HM5  soft  menu  key.  Use  the  up  and  down  arrow  keys 
(^^^3?)  to  select,  say,  variable  EQ1 : 


Eff  EG3 


hLG 


Press  Iliiilll  after  selecting  EQ1  to  load  into  variable  EQ  in  the  solver.  The 
new  equation  is  ready  to  be  solved. 


SOLVE  EGUATIOn 
Eq=XA3.-5S.=0. 

K 


Enter  ual.u«  or  press  SOLVE 


Page  6-26 


The  SOLVE  soft  menu 

The  SOLVE  soft  menu  allows  access  to  some  of  the  numerical  solver  functions 
through  the  soft  menu  keys.  To  access  this  menu  use  in  RPN  mode:  74 
MENU,  or  in  ALG  mode:  MENU(74).  Alternatively,  you  can  use  C3  (hold) 
L_7J  to  activate  the  SOLVE  soft  menu.    The  sub-menus  provided  by  the  SOLVE 
soft  menu  are  the  following: 


P1 
l: 


The  ROOT  sub-menu 

The  ROOT  sub-menu  include  the  followinq  functions  and  sub-menus: 


Function  ROOT 

Function  ROOT  is  used  to  solve  an  equation  for  a  given  variable  with  a 
starting  guess  value.  In  RPN  mode  the  equation  will  be  in  stack  level  3,  while 
the  variable  name  will  be  located  in  level  2,  and  the  initial  guess  in  level  1 . 
The  following  figure  shows  the  RPN  stack  before  and  after  activating  function 


4: 

3: 

TflH(9)=e 

2: 

■e  ■ 

l: 

5 

4: 

3: 

2: 

l: 

7.72525183694 

In  ALG  mode,  you  would  use  ROOT('TAN(e)=e','e',5)  to  activate  function 
ROOT: 


ROOTCTflNCe^e' '9' 5) 

7.72525183694 


Variable  EQ 

The  soft  menu  key  11:11  in  this  sub-menu  is  used  as  a  reference  to  the  variable 
EQ.  Pressing  this  soft  menu  key  is  equivalent  to  using  function  RCEQ  (ReCall 
EQ). 


Page  6-27 


The  SOLVR  sub-menu 

The  SOLVR  sub-menu  activates  the  soft-menu  solver  for  the  equation  currently 
stored  in  EQ.  Some  examples  are  shown  next: 

Example  1  -  Solving  the  equation  t2-5t  =  -4 

For  example,  if  you  store  the  equation  'tA2-5*t=-4'  into  EQ,  and  press  IMI^inMslzLj:!/ 
it  will  activate  the  following  menu: 

r't  \mm   I 

This  result  indicates  that  you  can  solve  for  a  value  of  t  for  the  equation  listed  at 
the  top  of  the  display.  If  you  try,  for  example,  C3D  [   t   ],  it  will  give  you  the 
result  t:  1 .,  after  briefly  flashing  the  message  "Solving  for  t."  There  is  a 
second  root  to  this  equation,  which  can  be  found  by  changing  the  value  of  t, 
before  solving  for  it  again.  Do  the  following:  1 0  [   t   ],  then  press  (JnJ[  t  ]. 
The  result  is  now,  t:  4.0000000003.  To  verify  this  result,  press  the  soft  menu 
key  labeled  IIIEIili,  which  evaluates  the  expression  in  EQ  for  the  current  value 
of  t.  The  results  in  this  case  are: 


J- 

4: 

t:  1. 

3: 

t: 4. 00000000003 

£: 

Left:  (-4.) 

l: 

Right:  (-4] 

-L- 

ll    ll  ll 

To  exit  the  SOLVR  environment,  press  Cjjj j .  The  access  to  the  SOLVE  menu  is 
lost  at  this  point,  so  you  have  to  activate  it  once  more  as  indicated  earlier,  to 
continue  with  the  exercises  below. 

Example  2  -  Solving  the  equation  Q  =  at2+bt 

It  is  possible  to  store  in  EQ,  an  equation  involving  more  than  one  variable, 
say,  'Q  =  atA2  +  bt'.    In  this  case,  after  activating  the  SOLVE  soft  menu,  and 
pressing  E^SEEHEB,  you  will  get  the  following  screen: 

l:  

I  o  II  .i  II  t  II  h  iBflBl  I 

Within  this  SOLVR  environment  you  can  provide  values  for  any  of  the 

variables  listed  by  entering  the  value  in  the  stack  and  pressing  the 

corresponding  soft-menu  keys.  For  example,  say  you  enter  the  values  Q  =  14, 

a  =  2,  and  b  =  3.  You  would  use:  14  [   Q   ],  2  [   a   ],  3  [   b  ]. 


Page  6-28 


As  variables  Q,  a,  and  b,  get  assigned  numerical  values,  the  assignments  are 
listed  in  the  upper  left  corner  of  the  display.  At  this  point  we  can  solve  for  t,  by 
using  LjnJ  [    t   ].  The  result  is  t:  2.    Pressing  lllllllill  s h o ws  the  results: 


4: 

3: 

t:2. 

2: 

Left: 14 

l: 

Right: 14. 

4    II    *  1 

1    t    II    b  lUMJSl 

Example  3  -  Solving  two  simultaneous  equations,  one  at  a  time 
You  can  also  solve  more  than  one  equation  by  solving  one  equation  at  a  time, 
and  repeating  the  process  until  a  solution  is  found.  For  example,  if  you  enter 
the  following  list  of  equations  into  variable  EQ:  {  'a*X+b*Y  =  c',  'k*X*Y=s'}, 
the  keystroke  sequence  C:Ei!!!lj!  ii!:E!EI!:i,  within  the  SOLVE  soft  menu,  will  produce 
the  followinq  screen: 

i 


3  II  ;  ;  II  t-     ■■  II  ■-  mm 

The  first  equation,  namely,  a*X  +  b*Y  =  c,  will  be  listed  in  the  top  part  of  the 
display.  You  can  enter  values  for  the  variables  a,  b,  and  c,  say: 
2[a    ]5[    b   ]  19[c   ].  Also,  since  we  can  only  solve  one  equation  at 
a  time,  let's  enter  a  guess  value  for  Y,  say,  0  [   Y   ],  and  solve  for  X,  by  using 
C3D[   X   ].  This  gives  the  value,  X:  9.4999....  To  check  the  value  of  the 
equation  at  this  point,  press  IIIIIIIII;;!.  The  results  are:     Left:  1  9,  Right:  1  9.  To 
solve  the  next  equation,  press  [nxtJ  EHEH.  The  screen  shows  the  soft  menu  keys  as: 


X: 9. 5000000000^ 

Left: 19 
 Right:  19| 


Say  we  enter  the  values  k  =  2,  s  =  12.  Then  solve  for  Y,  and  press 
IIIIIIIII;;!.  The  results  are  now,  Y: 


X: 


9.50000000002 

Left: 19. 
Right: 19 
.631573547368 
Left: 12. 

  _   Right:  12 

I  h  II  K  II  T  ll  f  IWyBlrtHffl 


We  then  continue  moving  from  the  first  to  the  second  equation,  back  and  forth, 
solving  the  first  equation  for  X  and  the  second  for  Y,  until  the  values  of  X  and 
Y  converge  to  a  solution.    To  move  from  equation  to  equation  use  To 
solve  for  X  and  Y  use  LjnJ  [  X  ],  and  C3D[  Y  ],  respectively.    The  following 
sequence  of  solutions  is  produced: 


Page  6-29 


X:  7. 92105263162) 
V: .757475083056 
X: 7- 60631229237 
Y: .733313519325 
X: 7. 5279537017 
V: .797029343923 
X: 7. 5074266402 


7*  V:.79920S60S69| 

6:  X: 7. 50197347325 

5:  Y: .799739017976 

4:  X: 7. 50052745505 

3:  Y: .799943742032 

2:  X: 7. 5001406443 

l:  Y: .799934993167 


After  solving  the  two  equations,  one  at 
decimal,  X  is  converging  to  a  value  of 
value  o  0.799. 


h  ii  k  ii  v  il  i  Imuflirart 

a  time,  we  notice  that,  up  to  the  third 
7.500,  while  Y  is  converging  to  a 


Using  units  with  the  SOLVR  sub-menu 

These  are  some  rules  on  the  use  of  units  with  the  SOLVR  sub-menu: 

•  Entering  a  guess  with  units  for  a  given  variable,  will  introduce  the  use 
of  those  units  in  the  solution. 

•  If  a  new  guess  is  given  without  units,  the  units  previously  saved  for 
that  particular  variable  are  used. 

•  To  remove  units  enter  a  number  without  units  in  a  list  as  the  new 
guess,  i.e.,  use  the  format  {  number }. 

•  A  list  of  numbers  can  be  given  as  a  guess  for  a  variable.  In  this  case, 
the  units  takes  the  units  used  belong  to  the  last  number  in  the  list.  For 
example,  entering  {  1 .41_ft  l_cm  l_m  }  indicates  that  meters  (m)  will 
be  used  for  that  variable. 

•  The  expression  used  in  the  solution  must  have  consistent  units,  or  an 
error  will  result  when  trying  to  solve  for  a  value. 


The  DIFFE  sub-menu 

The  DIFFE  sub-menu  provides  a  number  of  functions  for  the  numerical  solution 
of  differential  equations.  The  functions  provided  are  the  following: 

ll^MMliflH«iiflfl[raiiamaIMJ 

These  functions  are  presented  in  detail  in  Chapter  16. 


The  POLY  sub-menu 

The  POLY  sub-menu  performs  operations  on  polynomials.  The  functions 
included  are  the  followinc 


PP.00T  FCOEF  PEYflL 


Page  6-30 


Function  PROOT 

This  function  is  used  to  find  the  roots  of  a  polynomial  given  a  vector 
containing  the  polynomial  coefficients  in  decreasing  order  of  the  powers  of 
the  independent  variable.  In  other  words,  if  the  polynomial  is  anxn  +  a^x""1 
+  ...  +  a2x2  +  q}x  +  a0/  the  vector  of  coefficients  should  be  entered  as  [an/  an. 
l7  ...  ,  a2/  O]  ,  a0].  For  example,  the  roots  of  the  polynomial  whose 
coefficients  are  [1,  -5,  6]  are  [2,  3]. 

Function  PCOEF 

This  function  produces  the  coefficients  [an/  an.-|,  ...  ,  a2/  o}  ,  a0]  of  a 
polynomial  anxn  +  a^x""1  +  ...  +  a2x2  +  a^  +  a0/  given  a  vector  of  its  roots 
[r]f  r2/       rn].    For  example,  a  vector  whose  roots  are  given  by 
[-1,  2,  2,  1,  0],  will  produce  the  following  coefficients:  [1,  -4,  3,  4,  -4,  0]. 
The  polynomial  is  x5  -  4x4  +  3x3  +  4x2  -  4x. 

Function  PEVAL 

This  function  evaluates  a  polynomial,  given  a  vector  of  its  coefficients,  [an/  an. 
l7  ...  ,  a2,  ®}  ,  ao]/  and  a  value  x0,  i.e.,  PEVAL  calculates  anx0n  +  a^Xo""1  +  ... 
+  a2x02  +  a^o  +  a0.  For  example,  for  coefficients  [2,  3,  -1 ,  2]  and  a  value  of 
2,  PEVAL  returns  the  value  28. 

The  SYS  sub-menu 

The  SYS  sub-menu  contains  a  listing  of  functions  used  to  solve  linear  systems. 
The  functions  listed  in  this  sub-menu  are: 


These  functions  are  presented  in  detail  in  Chapter  1  1 . 

The  TVM  sub-menu 

The  TVM  sub-menu  contains  functions  for  calculating  Time  Value  of  Money. 
This  is  an  alternative  way  to  solve  FINANCE  problems  (see  Chapter  6).  The 
functions  available  are  shown  next: 

W 

femihiu:i:T.]r:i:H:i|Hini^Ml7llll 


Page  6-31 


The  SOLVR  sub-menu 

The  SOLVR  sub-menu  in  the  TVM  sub-menu  will  launch  the  solver  for  solving 
TVM  problems.  For  example,  pressing  liliiiESIiiEili,  at  this  point,  will  trigger  the 
following  screen: 


As  an  exercise,  try  using  the  values  n  =  1 0,  l%YR  =  5.6,  PV  =  1 0000,  and  FV 
=  0,  and  enter  ED  [  PMT  ]  to  find  PMT  =  -1021.08....  Pressing  ®; 
produces  the  following  screen: 


I'd.  payment s^year 
BEGIN  mode 
5: 
4: 
3: 
£: 

l:  PMT:  (-1021. 08086483] 
™  I""  


Press  jvAR  j  to  exit  the  SOLVR  environment.  Find  your  way  back  to  the  TVM 
sub-menu  within  the  SOLVE  sub-menu  to  try  the  other  functions  available. 


Function  TVMROOT 

This  function  requires  as  argument  the  name  of  one  of  the  variables  in  the 
TVM  problem.  The  function  returns  the  solution  for  that  variable,  given  that 
the  other  variables  exist  and  have  values  stored  previously.  For  example, 
having  solved  a  TVM  problem  above,  we  can  solve  for,  say,  'N',  as  follows: 
[  '  ]  {alph$(n]  (enter)  mm.  The  result  is  1  0. 


Function  AMORT 

This  function  takes  a  value  representing  a  period  of  payment  (between  0  and 
n)  and  returns  the  principal,  interest,  and  balance  for  the  values  currently 
stored  in  the  TVM  variables.  For  example,  with  the  data  used  earlier,  if  we 
activate  function  AMORT  for  a  value  of  1  0,  we  get: 


4: 

3:  -9999.99999995 

£:  -210-808648348 

1 :  . 00000004766 


Page  6-32 


Function  BEG 

If  selected,  the  TMV  calculations  use  payments  at  the  beginning  of  each 
period.  If  deselected,  the  TMV  calculations  use  payments  at  the  end  of  each 
period. 


Page  6-33 


Chapter  7 

Solving  multiple  equations 

Many  problems  of  science  and  engineering  require  the  simultaneous  solutions 
of  more  than  one  equation.  The  calculator  provides  several  procedures  for 
solving  multiple  equations  as  presented  below.  Please  notice  that  no 
discussion  of  solving  systems  of  linear  equations  is  presented  in  this  chapter. 
Linear  systems  solutions  will  be  discussed  in  detail  in  subsequent  chapters  on 
matrices  and  linear  algebra. 

Rational  equation  systems 

Equations  that  can  be  re-written  as  polynomials  or  rational  algebraic 
expressions  can  be  solved  directly  by  the  calculator  by  using  the  function 
SOLVE.  You  need  to  provide  the  list  of  equations  as  elements  of  a  vector. 
The  list  of  variables  to  solve  for  must  also  be  provided  as  a  vector.  Make  sure 
that  the  CAS  is  set  to  mode  Exact  before  attempting  a  solution  using  this 
procedure.    Also,  the  more  complicated  the  expressions,  the  longer  the  CAS 
takes  in  solving  a  particular  system  of  equations.    Examples  of  this 
application  follow: 

Example  1  -  Projectile  motion 

Use  function  SOLVE  with  the  following  vector  arguments,  the  first  being  the  list 
of  equations:  ['x  =  xO  +  vO*COS(eO)*t'  'y  =yO+vO*SIN(60)*t  - 
g*tA2/2'](^D,  and  the  second  being  the  variables  to  solve  for,  say  t  and  yO, 
i.e.,  ['t"yO']. 

The  solution  in  this  case  will  be  provided  using  the  RPN  mode.  The  only 
reason  being  that  we  can  build  the  solution  step  by  step.    The  solution  in  the 
ALG  mode  is  very  similar.    First,  we  store  the  first  vector  (equations)  into 
variable  A2,  and  the  vector  of  variables  into  variable  Al .  The  following 
screen  shows  the  RPN  stack  before  saving  the  variables. 


&: 
5: 

3:  'fl2' 
l:  'fli 


Page  7-1 


At  this  point,  we  need  only  press  jsroy J  twice  to  store  these  variables. 
To  solve,  first  change  CAS  mode  to  Exact,  then,  list  the  contents  of  A2  and  Al , 
in  that  order:  IHilil  I 


3: 
2: 


 Ct  $01 


Use  command  SOLVE  at  this  point  (from  the  S.SLV  menu:  C5D^)  After 
about  40  seconds,  maybe  more,  you  get  as  result  a  list: 

{'t  =  (x-x0)/(COS(e0)*v0)' 

'yO  =  (2*COS(e0)A2*v0A2*y+(g*xA2(2*x0*g+2*SIN(e0))*COS(e0)*v0A2)*x+ 
(x0A2*g+2*SIN(e0)*COS(e0)*v0A2*x0)))/(2*COS(e0)A2*v0A2),]} 


Press  [eval)  to  remove  the  vector  from  the  list,  then  use  command  OBJ->,  to  get 
the  equations  listed  separately  in  the  stack  


5: 
4: 
3: 
2: 


y0= 


4: 
3 ' 

-"COSteDJ-uD 

2=  ^a.costao^.ufl^+j-xMa.xfl.j- 


*a.: 


Note:  This  method  worked  fine  in  this  example  because  the  unknowns  t  and 
yO  were  algebraic  terms  in  the  equations.  This  method  would  not  work  for 
solving  for  00,  since  60  belongs  to  a  transcendental  term. 


Example  2  -  Stresses  in  a  thick  wall  cylinder 

Consider  a  thick-wall  cylinder  for  inner  and  outer  radius  a  and  b,  respectively, 
subject  to  an  inner  pressure  P,  and  outer  pressure  PD.  At  any  radial  distance  r 
from  the  cylinder's  axis  the  normal  stresses  in  the  radial  and  transverse 
directions,  arr  and  a0e,  respectively,  are  given  by 

_a2.Pi-b2.P0  a2.b2-(P,-P0) 

ee~  b2-a2 


0\.„  = 


a2-P-b2-P„ 


■a 


r2\b2-a2) 
a2-b2-{P-P0) 
r2-(b2-a2) 


Page  7-2 


Notice  that  the  right-hand  sides  of  the  two  equations  differ  only  in  the  sign 
between  the  two  terms.  Therefore,  to  write  these  equations  in  the  calculator,  I 
suggest  you  type  the  first  term  and  store  in  a  variable  Tl ,  then  the  second  term, 
and  store  it  in  T2.  Writing  the  equations  afterwards  will  be  matter  of  recalling 
the  contents  of  Tl  and  T2  to  the  stack  and  adding  and  subtracting  them. 
Here  is  how  to  do  it  with  the  equation  writer: 
Enter  and  store  term  Tl : 


2  2 
a  -Pi-b  -Po 

■  2  2 
b  -a 


EaaEnaBgaBanMHiHEBini 

Enter  and  store  term  T2: 


4: 

3: 

2: 

2  2 

a  -Pi-b  -Po 

b  -a 

l: 

'Tl ' 

4: 

3: 

2: 

2  2 

a  -b  -CPi-Po) 

r£.(b£-a£) 

l: 

'T2' 

a2-b2<Fi-Fo) 
r%2-*2  ) 

EaaBnaBgiEBMaiBEBini 

Notice  that  we  are  using  the  RPN  mode  in  this  example,  however,  the 
procedure  in  the  ALG  mode  should  be  very  similar.  Create  the  equation  for 

gqq.  (j^T)  1111111 11131!  L+J  [alpha] CrD (S  [dLPHA) CrD (3  Q^D  CD  CnD  = 

Create  the  equation  for  crrr:  C^5D !!!!u!::i!!!!!l!l  C^D  S^CSdl  ^™)(JxJ(3  I^nter] 
CD  CB^ 


Put  together  a  vector  with  the  two  equations,  using  function  ->ARRY  (find  it 


using  the  command  catalog  L  r>  j  cat  )  a 


3: 
2: 
l: 


3B.  k2fti  2k2, 


ba-aa 


lb  -a 

_  .]".Fi-t.".F-M  j*-.ba.^Fi-Foj 


ba-.a 


ra4a-.a) 


ter  typing  a  LU ' 


5: 
4: 
3: 
2: 


1:  r 


,  ja.Fi-ba-F0  .  ja.ba.tFi-F0^ 


baV 


ra-(ba-.a) 


Now,  suppose  that  we  want  to  solve  for  P,  and  PG,  given  a,  b,  r,  crrr,  and  cr^. 
We  enter  a  vector  with  the  unknowns: 


2: 
l: 


[ ga.Fi-ba-Fo  ja-ba-tFi 
— to; 


■FoJ 


CFi  Fo: 


Page  7-3 


To  solve  for  P,  and  Pol  use  the  command  SOLVE  from  the  S.SLV  menu 
([  <i  )s.5Lv  );  jt  may  take  the  calculator  a  minute  to  produce  the  result: 

{['Pi=.(((ae-ar)*rA2-(CT9+ar)*aA2)/(2*aA2))' 
'Po=-(((aO-ar)*rA2-(ae+ar)*bA2)/(2*bA2))'  ]  },  i.e., 

 1 


1!  {[- 


i:-TTH-.Tr;i.r3-i;.TTH+.Tr;i..]3 
2^ 


Notice  that  the  result  includes  a  vector  [  ]  contained  within  a  list  { }.  To  remove 
the  list  symbol,  use  [eval).  Finally,  to  decompose  the  vector,  use  function  OBJ->. 
The  result  is: 


3: 
2: 
l: 


I!  .TTH  -  .Tf  j  ■  |"  *"  ~  l|  iTH  +  .Tf  j  ■  ■]  ^ 


a-a 

■;.TiH-.T,-;i.,-5-i;.TiH+.T,-;i.t.5 


2-b 


*a.J 


These  two  examples  constitute  systems  of  linear  equations  that  can  be  handled 
equally  well  with  function  LINSOLVE  (see  Chapter  1  1).  The  following 
example  shows  function  SOLVE  applied  to  a  system  of  polynomial  equations. 


Example  3  -  System  of  polynomial  equations 

The  following  screen  shot  shows  the  solution  of  the  system  X2+XY=1 0,  X2-Y2=-5, 
using  function  SOLVE: 


[x2+X-Y=10  X2-Y2=-!d] 


2  2  2 

X  +Y-X=10  X  -Y  =-5 
:  SOLVECFlNSm ,TX  YT) 

{[X=2  Y=3]  tx=-2  Y=-3]> 


Solution  to  simultaneous  equations  with  MSLV 


Function  MSLV  is  availab 


e  as  the  last  option  in  the  jj^m^  menu: 


£: 
5: 
4: 
3: 
2: 
l: 


i. "J -2  equation.. 

2. Salve  diFF  eq.. 

3. Solve  poly.. 

H. Solve  Lin  syf.. 

5. Solus  Finance.. 


The  help-facility  entry  for  function  MSLV  is  shown  next: 


Page  7-4 


mur:         ;  I 

■^on-po 1 ynom i  a  1  multi- 
variate so Iyer 
1SLV< ' CSIH(X>+Y,X+SIH( 
y>=l] ' i ' CXiV] ' , [0,0]} 
[1.32384112611  -.9681... 
See:  SOLVE 

bwm«:Dim«mtjm«i;r;ti^ 


Example  1  -  Example  from  the  help  facility 

As  with  all  function  entries  in  the  help  facility,  there  is  an  example  attached  to 
the  MSLV  entry  as  shown  above.  Notice  that  function  MSLV  requires  three 
arguments: 

1 .  A  vector  containing  the  equations,  i.e.,  '[SIN(X)+Y,X+SIN(Y)=1  ]' 

2.  A  vector  containing  the  variables  to  solve  for,  i.e.,  '[X,Y]' 

3.  A  vector  containing  initial  values  for  the  solution,  i.e.,  the  initial  values 
of  both  X  and  Y  are  zero  for  this  example. 

In  ALG  mode,  press  BSE  to  copy  the  example  to  the  stack,  press  [enter)  to  run 
the  example.  To  see  all  the  elements  in  the  solution  you  need  to  activate  the 
line  editor  by  pressing  the  down  arrow  key  ); 


HELP 

MSLVCCSIHCX3+Y  X+SINCY3* 
{[SINCXHY  X+SIN(Y)=1.]  D+ 
«SINOO+Ystt+SIN<Y>  =  l„, 
CX,Y], 

[1.323341 1261 1 ,-,9631 

blitiaHitigCOWI.HCll.HBIHlttj 


In  RPN  mode,  the  solution  for  this  example  is  produced  by  using: 

4:  ] 

3:  CSIHCX3+Y  X+SIHCY)=1  -  ] 

£:  CXY] 

l:  [0.  0.] 


Activating  function  MSLV  results  in  the  following  screen. 
4: 

3:  CSIHCX3+Y  X+SIHCY)=1  -  ] 
2:  CXY] 

You  may  have  noticed  that,  while  producing  the  solution,  the  screen  shows 
intermediate  information  on  the  upper  left  corner.  Since  the  solution  provided 
by  MSLV  is  numerical,  the  information  in  the  upper  left  corner  shows  the 
results  of  the  iterative  process  used  to  obtain  a  solution.   The  final  solution  is  X 
=  1.8238,  Y  = -0.9681. 


Page  7-5 


Example  2  -  Entrance  from  a  lake  into  an  open  channel 

This  particular  problem  in  open  channel  flow  requires  the  simultaneous 

v2 

solution  of  two  equations,  the  equation  of  energy:  H0  =  y  H  ,  and 

2g 

Cu  A5/3 

Manning's  equation:  Q  =  —  •  ^S~^  .  In  these  equations,  HQ 

n  P 

represents  the  energy  head  (m,  or  ft)  available  for  a  flow  at  the  entrance  to  a 
channel,  y  is  the  flow  depth  (m  or  ft),  V  =  Q/A  is  the  flow  velocity  (m/s  or 
ft/s),  Q  is  the  volumetric  discharge  (m3/s  or  ft3/ s),  A  is  the  cross-sectional 
area  (m2  or  ft2),  Q  is  a  coefficient  that  depends  on  the  system  of  units  (Q  = 
1 .0  for  the  SI,  Cu  =  1 .486  for  the  English  system  of  units),  n  is  the  Manning's 
coefficient,  a  measure  of  the  channel  surface  roughness  (e.g.,  for  concrete,  n 
=  0.01  2),  P  is  the  wetted  perimeter  of  the  cross  section  (m  or  ft),  S0  is  the 
slope  of  the  channel  bed  expressed  as  a  decimal  fraction.    For  a  trapezoidal 
channel,  as  shown  below,  the  area  is  given  by  A  =  (b  +  my)y ,  while  the 

wetted  perimeter  is  given  by  P  =  b  +  2y^l  +  in   ,  where  b  is  the  bottom 
width  (m  or  ft),  and  m  is  the  side  slope  (1  V:mH)  of  the  cross  section. 

Typically,  one  has  to  solve  the  equations  of  energy  and  Manning's 
simultaneously  for  y  and  Q.  Once  these  equations  are  written  in  terms  of  the 
primitive  variables  b,  m,  y,  g,  SQ,  n,  Cu,  Q,  and  HG,  we  are  left  with  a  system 
of  equations  of  the  form  fi(y,Q)  =  0,  f2(y,Q)  =  0.   We  can  build  these  two 
equations  as  follows. 

We  assume  that  we  will  be  using  the  ALG  and  Exact  modes  in  the  calculator, 
although  defining  the  equations  and  solving  them  with  MSLV  is  very  similar  in 
the  RPN  mode.    Create  a  sub-directory,  say  CHANL  (for  open  CHANneL), 
and  within  that  sub-directory  define  the  following  variables: 


Ho=y+2^*EQ1 


Ho= 


2-9 


Page  7-6 


Q 

fl 

:  Cb+m-yJ-yH1! 


Q 
fl 

2 

yb+y  to 


:  Cb+m-yJ-yH1! 


:  b+2-y-Jl+m^P 


2 

yb+y 


b+2-y 


To  see  the  original  equations,  EQ1  and  EQ2,  in  terms  of  the  primitive 
variables  listed  above,  we  can  use  function  EVAL  applied  to  each  of  the 
equations,  i.e.,  U^Iil:!!:!!!  \jval)  IBjjEEl.  The  equations  are  listed  in  the  stack 
as  follows  (small  font  option  selected): 


:  E'.'HLiEuij 

Q^.y^j.^+H.y^H.j.b^.y^H3^ 
2      2       3  H3 


"u_        2      2  *3 

H  2 

2.yc.3.bc+H.y^.H.3. 

fr+2-y  -h  .3 

:EYfil_(EG2) 

c_(y.t+ya.H).JSo-Cu 

3 1 — ~2 

.  -Jy.b+y  >h 

3I  1 

n.  -Jb+2.y.-J 

2 

Ha  +  i 

We  can  see  that  these  equations  are  indeed  given  in  terms  of  the  primitive 
variables  b,  m,  y,  g,  SQ,  n,  Cu,  Q,  and  HQ. 


In  order  to  solve  for  y  and  Q  we  need  to  give  values  to  the  other  variables. 
Suppose  we  use  H0  =  5  ft,  b  =  1 .5  ft,  m  =  1 ,  n  =  0.012,  S0  =  0.00001,  g  = 
32.2,  and  Cu  =  1 .486.  Before  being  able  to  use  MSLV  for  the  solution,  we 
need  to  enter  these  values  into  the  corresponding  variable  names.  This  can 
be  accomplished  as  follows:  


1 

:5HHo 

5 

:  .012^ 

.012 

:  1 .  5^ 

1.5 

: -00001 ►So 

.00001 

:  ltm 

:32.2^ 

1 

32.2 

Page  7-7 


I.  00001^0 

.00001 

:32.2^ 

32-2 

:  1 . 4B6Ku 

1.486 

4 

Now,  we  are  ready  to  solve  the  equation.  First,  we  need  to  put  the  two 

equations  together  into  a  vector.  We  can  do  this  by  actually  storing  the 

vector  into  a  variable  that  we  will  call  EQS  (EQuationS): 

:  l .  4obH_.u  1 

1.486 

j[EQl  EQ2]*EQS 

As  initial  values  for  the  variables  y  and  Q  we  will  use  y  =  5  (equal  to  the 
value  of  HQ/  which  is  the  maximum  value  that  y  can  take)  and  Q  =  1 0  (this  is 
a  guess).  To  obtain  the  solution  we  select  function  MSLV  from  the  NUM.SLV 
menu,  e.g.,  [j^Jnumslv  (^gjBM,  to  place  the  command  in  the  screen: 


1 . 

j  CEQ1  EQ2>EQS 
Ho=vf±2^  Q=HjSo£i£3J 


2-9 


n-3JP 


Next,  well  enter  variable  EQS:  (WKWJBH,  followed  by  vector  [y,Q]: 

and  by  the  initial  guesses  C3  »C5D£  i  CDQD- 

Before  pressing  [enter]  f  the  screen  will  look  like  this: 

■  1 .  4yy 

jCEQl  EQ2>EQS 

Press  (enter}  to  solve  the  system  of  equations.  You  may,  if  your  angular 
measure  is  not  set  to  radians,  get  the  following  request: 


Page  7-8 


:[EQ1  EQ2>EQS 

4tfb 

Rod ion  Hod«  on? 

,3., 

Ho 

no™  T  "  T"*'ij"'L*  ^ 

1SLV(EQS3  CyPQ]P  C5P 104) 

Press  !!!!!:![!:!!  and  allow  the  solution  to  proceed.  An  intermediate  solution  step 
may  look  like  this: 

|C4.  99656516277,15.90111 
358822986286 

Hq=  Vf^c,  Q=R.JS^.CU.3J 
1SLV(EQS3  CyPQ]P  C5P 104) 


The  vector  at  the  top  representing  the  current  value  of  [y,Q]  as  the  solution 
progresses,  and  the  value  .358822986286  representing  the  criteria  for 
convergence  of  the  numerical  method  used  in  the  solution.  If  the  system  is 
well  posed,  this  value  will  diminish  until  reaching  a  value  close  to  zero.  At  that 
point  a  numerical  solution  would  have  been  found.  The  screen,  after  MSLV 
finds  a  solution  will  look  like  this: 


n-3JP 

:MSLV(EQS,[y  Q],[5  10]) 

Ho=yf^  Q=*^4 


The  result  is  a  list  of  three  vectors.  The  first  vector  in  the  list  will  be  the 
equations  solved.  The  second  vector  is  the  list  of  unknowns.  The  third  vector 
represents  the  solution.  To  be  able  to  see  these  vectors,  press  the  down-arrow 
key  ^3?  to  activate  the  line  editor.  The  solution  will  be  shown  as  follows: 


:;HD  KYI  HEM  f;-- 
EHOHE  EK21 


hLG 


2'9  *"  n.3JP? 
4  [  Ho=  <  V^2+2*y*9  V  <  2*9. 
'y?Q], 
4.  99369613276,  20.  661, 

ta[fiiH[fiEEgiiigiqNi«tiiaj 


The  solution  suggested  is  [4.9936..,  20.661 ...].  This  means,  y  =  4.99  ft,  and 
Q  =  20.661 ...  ft3/ s .  You  can  use  the  arrow  keys  ((3D  CD  ^>  ^3?)  to  see  the 
solution  in  detail. 


Page  7-9 


Using  the  Multiple  Equation  Solver  (MES) 

The  multiple  equation  solver  is  an  environment  where  you  can  solve  a  system 
of  multiple  equations  by  solving  for  one  unknown  from  one  equation  at  a  time. 
It  is  not  really  a  solver  to  simultaneous  solutions,  rather,  it  is  a  one-by-one 
solver  of  a  number  of  related  equations.  To  illustrate  the  use  of  the  MES  for 
solving  multiple  equations  we  present  an  application  related  to  trigonometry  in 
the  next  section.  The  examples  shown  here  are  developed  in  the  RPN  mode. 

Application  1  -  Solution  of  triangles 

In  this  section  we  use  one  important  application  of  trigonometric  functions: 
calculating  the  dimensions  of  a  triangle.  The  solution  is  implemented  in  the 
calculator  using  the  Multiple  Equation  Solver,  or  MES. 
Consider  the  triangle  ABC  shown  in  the  figure  below. 


B  c  A 

The  sum  of  the  interior  angles  of  any  triangle  is  always  1  80°,  i.e.,  a  +  p  +  y  = 
1  80°.  The  sine  law  indicates  that: 


sin  a     sin  j5    sin  y 


a         o  c 

The  cosine  law  indicates  that: 

a2  =  b2  +  c2  -  2-b-c -cos  a, 
b2  =  o2  +  c2  -  2-o-c -cos  p, 
c2  =  a2  +  b2  -  2-a-b-cos  y 

In  order  to  solve  any  triangle,  you  need  to  know  at  least  three  of  the  following 
six  variables:  a,  b,  c,  a,  p,  y.    Then,  you  can  use  the  equations  of  the  sine  law, 


Page  7-10 


cosine  law,  and  sum  of  interior  angles  of  a  triangle,  to  solve  for  the  other 
three  variables. 

If  the  three  sides  are  known,  the  area  of  the  triangle  can  be  calculated  with 

Heron's  formula  A  =  ^js  -  (s  —  a)  -  (s  —  b)  -  (s  —  c)  , where  s  is  known  as  the 

, .           .  a+b+c 
semi-perimeter  ot  the  triangle,  i.e.,  s  =  . 

Triangle  solution  using  the  Multiple  Equation  Solver  (MES) 

The  Multiple  Equation  Solver  (MES)  is  a  feature  that  can  be  used  to  solve  two 
or  more  coupled  equations.  It  must  be  pointed  out,  however,  that  the  MES 
does  not  solve  the  equations  simultaneously.  Rather,  it  takes  the  known 
variables,  and  then  searches  in  a  list  of  equations  until  it  finds  one  that  can  be 
solved  for  one  of  the  unknown  variables.  Then,  it  searches  for  another 
equation  that  can  be  solved  for  the  next  unknowns,  and  so  on,  until  all 
unknowns  have  been  solved  for. 


Creating  a  working  directory 

We  will  use  the  MES  to  solve  for  triangles  by  creating  a  list  of  equations 
corresponding  to  the  sine  and  cosine  laws,  the  law  of  the  sum  of  interior 
angles,  and  Heron's  formula  for  the  area.  First,  create  a  sub-directory  within 
HOME  that  we  will  call  TRIANG,  and  move  into  that  directory.  See  Chapter 
2  for  instructions  on  how  to  create  a  new  sub-directory. 

Entering  the  list  of  equations 

Within  TRIANG,  enter  the  following  list  of  equations  either  by  typing  them 
directly  on  the  stack  or  by  using  the  equation  writer.  (Recall  that  [alpha] [  r>  j (a) 
produces  the  character  a,  and  [alpha} (j^J®  produces  the  character  p.  The 
character  y  needs  to  be  lllllIEIed  from  chars): 

'SIN(a)/a  =  SIN(p)/b' 

'SIN(a)/a  =  SIN(y)/c' 

'SIN(p)/b  =  SIN(y)/c' 
'cA2  =  aA2+bA2-2*a*b*COS(y)' 
'bA2  =  aA2+cA2-2*a*c*COS(p)' 


Page  7-1 1 


'aA2  =  bA2+cA2-2*b*c*COS(a)' 
a+p+y  =  1  80' 
's  =  (a+b+c)/2' 
'A  =  V  (s*(s-a)*(s-b)*(s-c))' 

Then,  enter  the  number  L  9  j,  and  create  a  list  of  equations  by  using:  function 
-HIST  (use  the  command  catalog  Lr>J  cat  ).  Store  this  list  in  the  variable  EQ. 

The  variable  EQ  contains  the  list  of  equations  that  will  be  scanned  by  the  MES 
when  trying  to  solve  for  the  unknowns. 

Entering  a  window  title 

Next,  we  will  create  a  string  variable  to  be  called  TITLE  to  contain  the  string 
"Triangle  Solution",  as  follows: 

[alpha]  [alpha]  I  <i  j  (alpha) 

(^(S(E)(D®®©(3(B(jrD 

[enter] 

CZD 

[alpha]  [alpha]  (f\  (Tj  (t](l\(J]  [enter] 

[ST0>] 

Creating  a  list  of  variables 

Next,  create  a  list  of  variable  names  in  the  stack  that  will  look  like  this: 

{   abcaPyAs  } 
and  store  it  in  variable  LVARI  (List  of  VARIables).  The  list  of  variables 
represents  the  order  in  which  the  variables  will  be  listed  when  the  MES  gets 
started.  It  must  include  all  the  variables  in  the  equations,  or  it  will  not  work 
with  function  MITM  (see  below).    Here  is  the  sequence  of  keystrokes  to  use  to 
prepare  and  store  this  list: 

Press  G^D,  if  needed,  to  get  your  variables  menu.  Your  menu  should  show 
the  variables  WMm  liiiliilii!  SIII!  . 


Open  double  quotes  in  stack 
Locks  keyboard  into  lower-case  alpha. 
Enter  text:  Triangle_ 
Enter  text:  Solution 

Enter  string  "Triangle  Solution"  in  stack 
Open  single  quotes  in  stack 
Enter  variable  name  'TITLE' 
Store  string  into  'TITLE' 


Page  7-12 


Preparing  to  run  the  MES 

The  next  step  is  to  activate  the  MES  and  try  one  sample  solution.  Before  we 
do  that,  however,  we  want  to  set  the  angular  units  to  DEGrees,  if  they  are  not 
already  set  to  that,  by  typing  (Aum) (alpha} @  ®  @  [enter)  . 

Next,  we  want  to  keep  in  the  stack  the  contents  of  TITLE  and  LVARI,  by  using: 

MEM  W!3M 

We  will  use  the  following  MES  functions 

•  MINIT:  MES  INITialization:  initializes  the  variables  in  the  equations 
stored  in  EQ. 

•  MITM:  MES'  Menu  Item:  Takes  a  title  from  stack  level  2  and  the  list  of 
variables  from  stack  level  1  and  places  the  title  atop  of  the  MES 
window,  and  the  list  of  variables  as  soft  menu  keys  in  the  order 
indicated  by  the  list.  In  the  present  exercise,  we  already  have  a  title 
("Triangle  Solution")  and  a  list  of  variables  ({abcaPyAs}) 
in  stack  levels  2  and  1,  respectively,  ready  to  activate  MITM. 

•  MSOLVR:  MES  SOLVER;  activates  the  Multiple  Equation  Solver  (MES) 
and  waits  for  input  by  the  user. 

/tunning  the  MES  interactively 

To  get  the  MES  started,  with  the  variables  TITLE  and  LVARI  listed  in  the  stack, 
activate  command  MINIT,  then  MITM,  and  finally,  MSOLVR  (find  these 
functions  in  the  catalog  (j^J_q*t  ). 

The  MES  is  launched  with  the  following  list  of  variables  available  (Press  LnxtJ 
to  see  the  next  list  of  variables): 


1      a      II      b      II      c      II      «     II      *      II      *  1 

1    *    II    *  II 

II 

II 

Inn 

Press  InxtJ  to  see  the  third  list  of  variables.  You  should  see: 


Press  jNxrj  once  more  to  recover  the  first  variable  menu. 

Let's  try  a  simple  solution  of  Case  I,  using  a  =  5,  b  =  3,  c  =  5.  Use  the 
following  entries: 


Page  7-13 


L5J  [  a  ]  a:5  is  listed  in  the  top  left  corner  of  the  display. 
LJJ  [  b  ]  b:3  is  listed  in  the  top  left  corner  of  the  display. 
L5J  [  c  ]  c:5  is  listed  in  the  top  left  corner  of  the  display. 
To  solve  for  the  angles  use: 

CjjD  [  a  ]         Calculator  reports  Solving  for  a,  and  shows  the  result  a: 
72.5423968763. 

Note:  If  you  get  a  value  that  is  larger  than  1  80,  try  the  following: 


CDCE)[  a  ] 
ED  [    a  ] 


Re-initialize  a  to  a  smaller  value. 
Calculator  reports  Solving  for  a 


Next,  we  calculate  the  other  two  values: 
C3D  [    P    ]  The  result  is  P:  34.9152062474 

C3D  [   y    ]  The  result  is  y:  72.5423968763. 

You  should  have  the  values  of  the  three  angles  listed  in  stack  levels  3  through 
1 .  Press  L+J  twice  to  check  that  they  add  indeed  to  1  80°. 

~  432. 542396376  " 

«: 72.5423963762 
p: 34. 9152062475 


4: 
3: 
2: 
l: 


72.54239637621 
SI 


lj  *:432. 

*rM"iMwTnr~: 


54239687^ 
130 

F-ii-r-ir^r 


Press  [nxtJ  to  move  to  the  next  variables  menu.  To  calculate  the  area  use: 
C5D[  A  ].  The  calculator  first  solves  for  all  the  other  variables,  and  then 
finds  the  area  as  A:  7.15454401063. 


4: 

3: 

«: 

432.542396376 

£: 

130. 

l: 

fl: 

7. 15454401063 

A  HI 

f  HI 

II      II  IIH1 

Note:  When  a  solution  is  found,  the  calculator  reports  the  conditions  for  the 
solution  as  either  Zero,  or  Sign  Reversal.    Other  messages  may  occur  if  the 
calculator  has  difficulties  finding  a  solution. 


Pressing  L*i JlilHBI 
intermediate  results. 


will  solve  for  all  the  variables,  temporarily  showing  the 
Press  C5DEH  to  see  the  solutions: 


Page  7-14 


-v:  72- 

p:  34.9152062474 
«:  72.542396876S 
s:  6.5 

fl:  7.1545440106c 

i 

When  done,  press  L0A/ J  to  return  to  the  MES  environment.  Press  Cjjj j  to  exit 
the  MES  environment  and  return  to  the  normal  calculator  display. 

Organizing  the  variables  in  the  sub  directory 

Your  variable  menu  will  now  contain  the  variables  (press  [nxtJ  to  see  the 
second  set  of  variables): 


Variables  corresponding  to  all  the  variables  in  the  equations  in  EQ  have  been 
created.  There  is  also  a  new  variable  called  Mpar  (MES  parameters),  which 
contains  information  regarding  the  setting  up  of  the  MES  for  this  particular  set 
of  equations.  If  you  use  LrLJ EREH  to  see  the  contents  of  the  variable  Mpar. 
You  will  get  the  cryptic  message:  Library  Data.  The  meaning  of  this  is  that  the 
MES  parameters  are  coded  in  a  binary  file,  which  cannot  be  accessed  by  the 
editor. 

Next,  we  want  to  place  them  in  the  menu  labels  in  a  different  order  than  the 
one  listed  above,  by  following  these  steps: 

1.  Create  a  list  containing  {  EQ  Mpar  LVARI  TITLE  },  by  using: 

(    y      \  I  1  ::::::"::v:::::i:  :::  :::  :  :!":"!::":  :::::"::":":--:::"::  ( rhll-rn\ 

LJnjy   !!.:.!  ::!::.!.!!=!  II.::iL:i.:.L-jLi  iyLIL:=Ll.:i::[  [ENTER} 

2.  Place  contents  of  LVARI  in  the  stack,  by  using:  HEH3. 

3.  Join  the  two  lists  by  pressing  L±J . 

Use  function  ORDER  (use  the  command  catalog  L J  cat  )  to  order  the 
variables  as  shown  in  the  list  in  stack  level  1 . 

4.  Press  (jgjj  to  recover  your  variables  list.  It  should  now  look  like  this: 

ti™iMlimiliHfchH«MM^  ^ 

5.  Press  [nxT)  to  recover  the  first  variable  menu. 


Page  7-15 


Programming  the  MES  triangle  solution  using  User  RPL 

To  facilitate  activating  the  MES  for  future  solutions,  we  will  create  a  program 
that  will  load  the  MES  with  a  single  keystroke.  The  program  should  look  like 
this:  «  DEG  MINIT  TITLE  LVARI  MITM  MSOLVR  »,  and  can  be  typed  in 
by  using: 

L  r>  J  <K>>  Opens  the  program  symbol 

{alpha} [alpha}  Locks  alphanumeric  keyboard 

@(J}@ispcJ  Type  in  DEG  (angular  units  set  to  DEGrees) 

®@($}(B(B(J!!D  Type  in  MINIT_ 

{alpha}  Unlocks  alphanumeric  keyboard 

List  the  name  TITLE  in  the  program 
List  the  name  LVARI  in  the  program 
Locks  alphanumeric  keyboard 


&CD(S&  Cz3  Type  in  MITM_ 

®®@®(v}(El  Type  in  MSOLVR 

[enter}  Enter  program  in  stack 

Store  the  program  in  a  variable  called  TRISOL,  for  TRIangle  SOLution,  by 

using:  CZD(^)(^)(3(3(3(I1(§1(3^^)  (£Z2t) 

Press  G^D,  if  needed,  to  recover  your  list  of  variables.   A  soft  key  label 
liilillilE!  should  be  available  in  your  menu. 

Running  the  program  -  solution  examples 

To  run  the  program,  press  the  SIlililH  soft  menu  key.  You  will  now  have  the 
MES  menu  corresponding  to  the  triangle  solution.  Let's  try  examples  of  the 
three  cases  listed  earlier  for  triangle  solution. 

Example  1  -  Right  triangle 

Use  a  =  3,  b  =  4,  c  =  5.  Here  is  the  solution  sequence: 

CB[a]CE[b]GD[c]To  enter  data 

C5D[    a   ]  The  result  is  a:  36.8698976458 

C5D[    P    ]  The  result  is  p:  53.1301023541. 

C5D[   y    ]  The  result  is  y:  90. 

jNxrj  To  move  to  the  next  variables  menu. 

[<n][   A   ]  The  result  is  A:  6. 

jNxrj  Inxt]  To  move  to  the  next  variables  menu. 


Page  7-16 


Example  2  -  Any  type  of  triangle 

Use  a  =  3,  b  =  4,  c  =  6.  The  solution  procedure  used  here  consists  of  solving 
for  all  variables  at  once,  and  then  recalling  the  solutions  to  the  stack: 
L^J  IliiilliliEl  To  clear  up  data  and  re-start  MES 

CE[a]CE[b]CS[c]To  enter  data 

jNxrj  To  move  to  the  next  variables  menu. 

C5D  EH!  Solve  for  all  the  unknowns. 

C3  EH!  Show  the  solution 


The  solution  is: 


Triansl.*  Solution 


117.279612736 


36.3360575147 
26.3S43297495 
6.5 

5.3326S225193 


At  the  bottom  of  the  screen,  you  will  have  the  soft  menu  keys: 
BID?!!-  IHIilll  llliili:!  ill!! 


The  square  dot  in  lllillillL!"  indicates  that  the  values  of  the  variables,  rather  than 
the  equations  from  which  they  were  solved,  are  shown  in  the  display.  To  see 
the  equations  used  in  the  solution  of  each  variable,  press  the  EQ}3  soft  menu 
key.    The  display  will  now  look  like  this: 


Triansl.*  Solution 


b^2=a^2+c^2-2*a*„ 
^2=bA2+cA2-2*b*.. 
'£=(a+b+c)/2' 
'fi=J"<s*<s-a>*<s-b„ 


VALUE  EGnS-  FRinT 


The  soft  menu  key  SlIIill  is  used  to  print  the  screen  in  a  printer,  if  available. 
And  IIIEiil  returns  you  to  the  MES  environment  for  a  new  solution,  if  needed. 
To  return  to  normal  calculator  display,  press  G^D- 

The  following  table  of  triangle  solutions  shows  the  data  input  in  bold  face  and 
the  solution  in  italics.  Try  running  the  program  with  these  inputs  to  verify  the 
solutions.  Please  remember  to  press  jvM  j  101113  at  the  end  of  each  solution  to 
clear  up  variables  and  start  the  MES  solution  again.  Otherwise,  you  may 


Page  7-17 


carry  over  information  from  the  previous  solution  that  may  wreck  havoc  with 
your  current  calculations. 


a 

b 

c 

a(°) 

PC) 

r(°) 

A 

2.5 

6.9837 

7.2 

20.299 

75 

84.771 

8.6933 

7.2 

8.5 

74.26 

22.616 

27 

130.38 

23.309 

21.92 

17.5 

13.2 

90 

52.97 

37.03 

115.5 

41.92 

23 

29.6 

75 

32 

73 

328.81 

10.27 

3.26 

10.5 

77 

18 

85 

16.66 

17 

25 

32 

37.79 

50.78 

97.44 

210.71 

Adding  an  INFO  button  to  your  directory 

An  information  button  can  be  useful  for  your  directory  to  help  you  remember 
the  operation  of  the  functions  in  the  directory.  In  this  directory,  all  we  need  to 
remember  is  to  press  liLiilii  to  get  a  triangle  solution  started.  You  may  want 
to  type  in  the  following  program:  «"Press  [TRISO]  to  start."  MSGBOX  »,  and 
store  it  in  a  variable  called  INFO.  As  a  result,  the  first  variable  in  your 
directory  will  be  the  lililliE!  button. 

Application  2  -  Velocity  and  acceleration  in  polar  coordinates 

Two-dimensional  particle  motion  in  polar  coordinates  often  involves 
determining  the  radial  and  transverse  components  of  the  velocity  and 
acceleration  of  the  particle  given  r,  r'  =  dr/dt,  r"  =  d2r/dt2,  0,  9'  =  d  6  /dt, 
and,  0"  =  d20/dt2.  The  following  equations  are  used: 

vr  =  r       ar  =  r  —  ra 

ve  -  r6     ae  —r0  +  2r6 
Create  a  subdirectory  called  POLC  (POLar  Coordinates),  which  we  will  use  to 
calculate  velocities  and  accelerations  in  polar  coordinates.     Within  that 
subdirectory,  enter  the  following  variables: 


Page  7-18 


Program  or  value 

Store  into  variable: 

«  PEQ  STEQ  MINIT  NAME  LIST  MITM  MSOLVR  » 

SOLVEP 

"vel.  &  acc.  polar  coord." 

NAME 

{  r  rD  rDD  9D  ODD  vr  v9  v  ar  a9  a  } 

LIST 

{'vr=rD'  'v9  =  r*9D'    'v  =  V(vrA2  +  v9A2)' 

'ar  =  rDD  -  r*9DA2'  'a9  =  r*9DD  +  2*rD*9D' 

'a  =  V(arA2  +  a9A2)'  } 

PEQ 

An  explanation  of  the  variables  follows: 

SOLVEP  =  a  program  that  triggers  the  multiple  equation  solver  for  the 
particular  set  of  equations  stored  in  variable  PEQ; 

NAME  =  a  variable  storing  the  name  of  the  multiple  equation  solver,  namely, 
"vel.  &  acc.  polar  coord. "; 

LIST  =  a  list  of  the  variable  used  in  the  calculations,  placed  in  the  order 
we  want  them  to  show  up  in  the  multiple  equation  solver 
environment; 

PEQ  =  list  of  equations  to  be  solved,  corresponding  to  the  radial  and 
transverse  components  of  velocity  (vr,  v9)  and  acceleration  (ar, 
a6  )  in  polar  coordinates,  as  well  as  equations  to  calculate  the 
magnitude  of  the  velocity  (v)  and  the  acceleration  (a)  when  the 
polar  components  are  known. 

r,  rD,  rDD  =  r  (radial  coordinate),  r-dot  (first  derivative  of  r),  r-double  dot 
(second  derivative  of  r). 

0D,  0DD  =  0-dot  (first  derivative  of  0),  0-double  dot  (second  derivative  of  0). 


Suppose  you  are  given  the  following  information:  r  =  2.5,  rD  =  0.5,  rDD  =  - 
1 .5,  0D  =  2.3,  0DD  =  -6.5,  and  you  are  asked  to  find  vr,  v0,  ar,  a0,  v,  and  a. 


Page  7-19 


Start  the  multiple  equation  solver  by  pressing  L m J H3SH.  The  calculator 
produces  a  screen  labeled  ,  "vel.  &  acc.  polar  coord.",  that  looks  as  follows: 


l: 


r  II  re  II  m  II  ?c  II  m  WvTl 


To  enter  the  values  of  the  known  variables,  just  type  the  value  and  press  the 
button  corresponding  to  the  variable  to  be  entered.  Use  the  following 
keystrokes:  2.5  [  r  ]  0.5  [  rD  ]  1 .5  CltD  [  rDD  ]  2.3  [9D  ]6.5H[  0DD  ]. 

Notice  that  after  you  enter  a  particular  value,  the  calculator  displays  the 
variable  and  its  value  in  the  upper  left  corner  of  the  display.  We  have  now 
entered  the  known  variables.  To  calculate  the  unknowns  we  can  proceed  in 
two  ways: 

a).  Solve  for  individual  variables,  for  example,  LjnJ  [  vr  ]  gives  vr:  0.500. 
Press  (jwiJLtlJ  [  v6  ]  to  get  v6  :  5.750  ,  and  so  on.  The  remaining  results 
are  v:  5.77169819031;  or:  -14.725;  a9:  -13.95;  and  a: 
20.2836911089.;  or, 


b).  Solve  for  all  variables  at  once,  by  pressing  CSEHi-  The  calculator  will 
flash  the  solutions  as  it  finds  them.  When  the  calculator  stops,  you  can 
press  C3EH3  to  list  all  results.  For  this  case  we  have: 


y:  5.77169819031 
ar:  -14.725 
a9:  -13.95 
a:  20.2336911039 


Pressing  the  soft-menu  key  lililili  will  let  you  know  the  equations  used  to 
solve  for  each  of  the  values  in  the  screen: 


Sue!..  &  acc.  polar  coord. 


1 u9=r*8D 1 


ar :    1 ar =r DD-r *G  D^2 .  ' 
a6 :    '  ae=r*GDD+2.  #rD*„. 
a:    la=f<.ar'-2.+atiy-2.  >  ' 


VALUE  EMS"  PRim 


Page  7-20 


To  use  a  new  set  of   values  press,  either  E 

HTTUT1 


[nxt j [nxtJ f  or  jVARj 


Let's  try  another  example  using  r  =  2.5,  vr  =  rD  =  -0.5,  rDD  =  1 .5,  v  =  3.0,  a 
=  25.0.  Find,  0D,  6DD,  v9,  ar,  and  a6.  You  should  get  the  following  results: 


0D:   1. 18321595662 

a9:  -24.9198715888 
GDD:  -9.4946622529 


iwi..  &  acc.  polar  coord. 


1 yr=rD 1 


1  v=\  0^2.  +v$*2. 
'ar=rDD-r*eD^2. 1 
'ae=r*eDD+2.*rD.. 


Page  7-21 


Chapter  8 

Operations  with  lists 

Lists  are  a  type  of  calculator's  object  that  can  be  useful  for  data  processing 
and  in  programming.  This  Chapter  presents  examples  of  operations  with  lists. 

Definitions 

A  list,  within  the  context  of  the  calculator,  is  a  series  of  objects  enclosed 
between  braces  and  separated  by  spaces  (Lspc J )/  in  the  RPN  mode,  or 

commas  (Lr*J  » ),  in  both  modes.  Objects  that  can  be  included  in  a  list  are 

numbers,  letters,  character  strings,  variable  names,  and/or  operators.  Lists 
are  useful  for  manipulating  data  sets  and  in  some  programming  applications. 
Some  examples  of  lists  are: 

i  t  I  } ,  i  "BETA"  h2  4>,  i  1  1.5  2B@}, 
i  a  a  a  a  } ,  {  i  1  2  3  >  i  3  2  1  >  i  1  2  3  >  > 

In  the  examples  shown  below  we  will  limit  ourselves  to  numerical  lists. 

Creating  and  storing  lists 

To  create  a  list  in  ALG  mode,  first  enter  the  braces  key  CSDU   (associated 

with  the  L  +  J  key),  then  type  or  enter  the  elements  of  the  list,  separating  them 

with  commas  (LrU  » ).   The  following  keystrokes  will  enter  the  list  {1  2  3  4} 

and  store  it  into  variable  LI . 

C5D{j      CD  C3  >  CTD  C3  >  (JJ  C3  >  CZJ 


The  screen  will  show  the  following: 


C1,2,3,4»L1 


+^kip  skiph  -kjel  del-i-  del  l  ins 


<!■  2.  3.  4.) 

^!^JH!^gE.]jll.]jgll.]jWIHlTJ1 


The  figure  to  the  left  shows  the  screen  before  pressing  [enter]  f  while  the  one  to 
the  right  shows  the  screen  after  storing  the  list  into  LI .  Notice  that  before 
pressing  (enter}  the  list  shows  the  commas  separating  its  elements.  However, 
after  pressing  (enter)  ,  the  commas  are  replaced  with  spaces. 

Entering  the  same  list  in  RPN  mode  requires  the  following  keystrokes: 


Page  8-1 


S™)  O  CD  (flv^D  CD  d^t) 


The  figure  below  shows  the  RPN  stack  before  pressing  the  G^T)  key: 


2: 

{1.  2.  3. 

4-> 

l: 

LI  ' 

T F; I H H |  HESi  | 

STRS  |  F  Rii-J  |  GRflV  | 

PIPE 

Composing  and  decomposing  lists 

Composing  and  decomposing  lists  makes  sense  in  RPN  mode  only.  Under 
such  operating  mode,  decomposing  a  list  is  achieved  by  using  function 
OBJ->.  With  this  function,  a  list  in  the  RPN  stack  is  decomposed  into  its 
elements,  with  stack  level  1 :  showing  the  number  of  elements  in  the  list.  The 
next  two  screen  shots  show  the  stack  with  a  small  list  before  and  after 
application  of  function  OBJ->: 


4: 
3: 
2: 
l: 


{3  -2  0} 


I  Hpar  | L H F; 1 1 T I T L E | 


4: 

3 

3: 

-2 

£: 

0 

l: 

3. 

EC   |  Hpar  | L '.' m F; 1 1 T I T L E |    a    |  I 

Notice  that,  after  applying  OBJ->,  the  elements  of  the  list  occupy  levels  4: 
through  2:,  while  level  1 :  shows  the  number  of  elements  in  the  list. 


To  compose  a  list  in  RPN  mode,  place  the  elements  of  the  list  in  the  stack, 
enter  the  list  size,  and  apply  function  ->LIST  (select  it  from  the  function 
catalog,  as  follows:  LlU_c^  (j^J^^ ,  then  use  the  up  and  down  arrow  keys 
(^^^▼7 )  to  locate  function  -HIST).  The  following  screen  shots  show  the 


elements  of  a  list  of  size  4  before  and  after  application  of  function  ->L 

5: 
4: 
3: 
2: 


I  Hpar  | L '.' h F; 1 1 T I T L E 


ST: 


Note:  Function  OBJ->  applied  to  a  list  in  ALG  mode  simply  reproduces  the 
list,  adding  to  it  the  list  size]  


:0BJ+({3  2  -1» 


{3  2  -1  3-> 


I  Hpar  | L '.' H F; 1 1 T I T L E 


Page  8-2 


Operations  with  lists  of  numbers 

To  demonstrate  operations  with  lists  of  numbers,  we  will  create  a  couple  of 
other  lists,  besides  list  LI  created  above:  L2={-3,2, 1 ,5},  L3={-6,5,3,1 ,0,3,-4}, 
L4={3,-2,1 ,5,3,2,1 }.  In  ALG  mode,  the  screen  will  look  like  this  after  entering 
lists  L2,  L3,  L4: 


:  {-3  2  1  5>H_2 

{-3  2  1  5> 
:  {-6  5  3  1  0  3  -4>H_3 

{-653  1  0  3  -4> 
:  {3  -2  1  5  3  2  1>H_4 
,   {3-2  1  532  1> 


In  RPN  mode,  the  following  screen  shows  the  three  lists  and  their  names  ready 
to  be  stored.  To  store  the  lists  in  this  case  you  need  to  press  jsroy J  three  times. 


Changing  sign 

The  sign-change  key  (L+/- J)  ,  when  applied  to  a  list  of  numbers,  will  change 
the  sign  of  all  elements  in  the  list.  For  example: 


:|_1 
-LI 


{1.  2.  3.  4-> 


{-!■  -2.  z3.  -4.} 
El-MMliBMfTiHiljlQai 


Addition,  subtraction,  multiplication,  division 

Multiplication  and  division  of  a  list  by  a  single  number  is  distributed  across  the 
list,  for  example: 


-5-L2 

{15  -10  -5  -25^ 

-ki 

5 

{.2  .4  .6  .8> 


Subtraction  of  a  single  number  from  a  list  will  subtract  the  same  number  from 
each  element  in  the  list, 


or  example: 

:|_2 

{-3  2  1  5> 

:|_2-10 

{-13. 

-3-  -9.  -5-> 

Page  8-3 


Addition  of  a  single  number  to  a  list  produces  a  list  augmented  by  the 
number,  and  not  an  addition  of  the  single  number  to  each  element  in  the  list. 
For  example: 


LI 

Ll+6 


{12  3  4} 


<12  3  4  6> 


Subtraction,  multiplication,  and  division  of  lists  of  numbers  of  the  same  length 
produce  a  list  of  the  same  length  with  term-by-term  operations.  Examples: 


L1-L2 
L1-L2 


{4.  0-  2.  -1.) 


{-3.  4.  3.  20. > 


{4-  0.  2.  -1.) 
{-3.  4.  3.  20. > 


L1-L.1 

:|_1-L2 

Li 
L2 

{-.333333333333  1.3..* 


The  division  L4/L3  will  produce  an  infinity  entry  because  one  of  the  elements 
in  L3  is  zero: 


If  the  lists  involved  in  the  operation  have  different  lengths,  an  error  message  is 
produced  (Error:  Invalid  Dimension). 


The  plus  sign  (L  +  J ),  when  applied  to  lists,  acts  a  concatenation  operator, 
putting  together  the  two  lists,  rather  than  adding  them  term-by-term.  For 
example: 


L1+L2 

,  {1  234-32  1  5} 

b«M—«Mi7»mnffi«d 


In  order  to  produce  term-by-term  addition  of  two  lists  of  the  same  length,  we 
need  to  use  operator  ADD.  This  operator  can  be  loaded  by  using  the  function 
catalog  (LrU_c^  ).  The  screen  below  shows  an  application  of  ADD  to  add 
lists  LI  and  L2,  term-by-term:  


LI  ADD  L2 

{-2  4  4  9} 


Page  8-4 


Real  number  functions  from  the  keyboard 

Real  number  functions  from  the  keyboard  (ABS,  ex,  LN,  10x,  LOG,  SIN,  x2,  V, 
COS,  TAN,  ASIN,  ACOS,  ATAN,  yx)  can  be  used  on  lists.  Here  are  some 
examples: 


ABS 


L2 
:IL2I 


{-3  2  1  5> 


{3  2  15} 


EXP  and  LN 


=  eL1 

|  1    2   3  41 

:  LHCL1 J 

,          {0  LHC2)  LHC3)  2-LHC2)> 


LOG  and  ANTILOG 


iLOGCLli 

{0  L0GC2)  L0GC3)  L0G(4^ 
:  RL0GCL2) 


SQ  and  square  root 


SQCLli 

{14  9  16^ 

JL2 

{■PT-J3  J2  1  Js) 
■^■■^■■^■■■■ffftwnn*H 


SIN,  ASIN 


SIHCLli 
KSIHCli  SIHC2i  SIMC33  SI  NO 

{-.304692654015  .20135* 


COS,  ACOS 


C0SCL2) 
[{C0SC3)  C0SC2)  coscii  cosc^H 

RC0S(m 

{1.47062390563  1. 36943* 


AN,  ATAN 


TFMLl) 
KTfMl)  TflHC23  TflHC33  TflNO 
:  RTRHCL2) 

-flTflNO)  flTfM2)  ^  flTflNd* 


INVERSE  (1/x) 


IHV(Ll) 

I1  2  3  4J 


Real  number  functions  from  the  MTH  menu 

Functions  of  interest  from  the  MTH  menu  include,  from  the  HYPERBOLIC  menu: 
SINH,  ASINH,  COSH,  ACOSH,  TANH,  ATANH,  and  from  the  REAL  menu:  %, 
%CH,  %T,  MIN,  MAX,  MOD,  SIGN,  MANT,  XPON,  IP,  FP,  RND,  TRNC, 
FLOOR,  CEIL,  D->R,  R->D.  Some  of  the  functions  that  take  a  single  argument 
are  illustrated  below  applied  to  lists  of  real  numbers: 


Page  8-5 


SINK  ASINH 


:SINH(L1) 

{SINH(l)  SINHC2)  SIHHC3)  SH 

RSINH^f] 

{-.295673847563  .  19869* 
ba<n:i™<n:iMm:iriMiH:iBfiin:iriifiin:i 


TANH,  ATANH 


TRNHCL2) 
K-TRHHC3)  TAHHC23  TflHHCl  J 

RTRHHCL1) 
{flTflHH(l)  RTRNHC2)  RTRHHC* 
b*<n:ir:^<n:iM.H;ir:Tro;iBfnn;ir:ifnn;i 


COSH,  ACOSH 


C0SHCL2) 
KC0SHC3)  C0SHC2)  COSH(l)  CH 

RCOSHCLl) 
<0  RC0SHC2)  RC0SHC3)  RCOS* 
ba<n:i™<n:iMm:iriMiH:iBfiin:iriifiin:i 


SIGN,  MANT,  XPON 


:  ol  UNI.  LI  J 

{1111 

:MRHTC100-L23 

<3.  2.  1.  5 

:XPON(LM00) 

{2-  2-  2-  2.) 

b:MBC1lll,nilim']ll»«'»^ 


P,  FP 


IPCU.2  2.3  -l.5» 

U.  2.  -l^ 
FPCU.2  2.3  -1.5H 

{.2  .3  -.5> 


FLOOR,  CEIL 


FL00RCU.2  2.3  -l.5>i 

U.  2.  -2.^ 

CEILCU.2  2.3  -l.5» 

{2.  3.  -I 


D^R,  R^D 


:  D+RK30  60  90» 
(.523593775593  1.04719H 

Hit  fill 

{30.  60.0000000002  90. 


Examples  of  functions  that  use  two  arguments 

The  screen  shots  below  show  applications  of  the  function  %  to  list  arguments. 
Function  %  requires  two  arguments.  The  first  two  examples  show  cases  in 
which  only  one  of  the  two  arguments  is  a  list.  


:*K10  20  30U) 

{.  I  .2.3> 

:*(5,U0  20  30» 

r  10  D  5  D  i0j 

The  results  are  lists  with  the  function  %  distributed  according  to  the  list 
argument.  For  example, 


Page  8-6 


while 


%({10,  20,  30},  1)  =  {%( 1 0, 1  ),%(20, 1  ),%(30, 1 )}, 
%(5,{1 0,20,30})  =  {%(5,10),%(5,20),%(5,30)} 


In  the  following  example,  both  arguments  of  function  %  are  lists  of  the  same 
size.  In  this  case,  a  term-by-term  distribution  of  the  arguments  is  performed, 
i.e., 

%({10,20,30},{1,2,3})  =  {%(10,1),%(20,2),%(30,3)} 


:*(U0  £@  30M1  2  3» 

{10TO201A30-ll0} 
^■MiiiMMiMiMirfl-amifiia 


This  description  of  function  %  for  list  arguments  shows  the  general  pattern  of 
evaluation  of  any  function  with  two  arguments  when  one  or  both  arguments 
are  lists.  Examples  of  applications  of  function  RND  are  shown  next: 


Lists  of  complex  numbers 

The  following  exercise  shows  how  to  create  a  list  of  complex  numbers  given 
two  lists  of  the  same  length,  one  representing  the  real  parts  and  one  the 
imaginary  parts  of  the  complex  numbers.  Use  LI  ADD  i*L2.  The  screen  also 
shows  that  the  resulting  complex-number  list  is  stored  into  variable  L5: 


i  L2  ADD  * 

U  +  i-3  2+i-2  3+i  4+i-5> 
:FlNSmH_5 

{l  +  i-3  2+i-2  3+i  4+i-5> 
■"■"■"■""■'"fa*™ 


Functions  such  as  LN,  EXP,  SQ,  etc.,  can  also  be  applied  to  a  list  of  complex 
numbers,  e.g., 


Page  8-7 


SQCL5) 

SSQd  +  i-3)  SQ(2+i-2)  SQC3-* 
JL5 

[t3+ii-Jg-2-i-J5)-Jl+JT8  h> 
6  u 


eL5 

l  +  i-3   2+i-2   3+i  4+ 
e  e         e  e 

:LHCL5) 

<LH(l  +  i-3)  LH(2+i-23  LNC3-* 


MLUbi. L~i J 
(RLOGCl  +  i-3)  RL0G(2+i-2)  H 

L0GCL5J 
KLOGCl  +  i— 33  L0G(2+i-2)  LOH 

IHVCL5J 

J  1   1        1  1 

,  ll  +  i-3  2+i-2  3+i  4+_i-5  , 


:SIHCL5i 

(SIHCl  +  i-33  SIHC2+i-23  SI* 
:SIHHCL53 

(SIHHCl  +  i-33  SIHHC2+i-23  ► 
:RSIN(L5) 

(nSIHCl  +  i-33  nSIHC2+i-23  ► 


The  following  example  shows  applications  of  the  functions  RE(Real  part), 
IM(imaginary  part),  ABS(magnitude),  and  ARG(argument)  of  complex 
numbers.  The  results  are  lists  of  real  numbers: 


:RECL5) 
IMCL5) 
:IL5I 


{1  2  3  4} 
{-3  2  1  5> 


{■HI  2-J2  HI 


RRGCL5J 


RTRNC3)  t  RTRHl^l  RTRHlH 


■*MT-«TM?MTMiiaiil]] 


[1]  flTRNl. 


Lists  of  algebraic  objects 

The  following  are  examples  of  lists  of  algebraic  objects  with  the  function  SIN 
applied  to  them: 


SINtflHSdH 


SIH(ot-p)  SIH 


The  MTH/LIST  menu 

The  MTH  menu  provides  a  number  of  functions  that  exclusively  to  lists.  With 
flag  1 1 7  set  to  CHOOSE  boxes: 


Page  8-8 


HHTH  HEHU 

1 .  VECTOR.. 

2 .  HflTRIK.. 


H. HYPERBOLIC. 

5 .  REAL.. 

6.  EASE.. 


LIST  HEHU 

a.ELIST 
3.iTLIST 
H.SORT 

5.  REVLIST 

6.  RDD 


Next,  with  system  flag  1  1  7  set  to  SOFT  menus: 


This  menu  contains  the  following  functions: 


ALIST  :  Calculate  increment  among  consecutive  elements  in  list 

HIST  :  Calculate  summation  of  elements  in  the  list 

nLIST  :  Calculate  product  of  elements  in  the  list 

SORT  :  Sorts  elements  in  increasing  order 

REVLIST  :  Reverses  order  of  list 

ADD  Operator  for  term-by-term  addition  of  two  lists  of  the  same  length 
(examples  of  this  operator  were  shown  above) 


Examples  of  application  of  these  functions  in  ALG  mode  are  shown  next: 


{-6  5  3  1  0  3  -4> 
*LIST(L3) 

{11  -2  -2  -1  3  -7> 


L3 

{-6  5  3  1  0  3  -4> 

SORTCL33 

{-6  -40  1  33  5> 


:|_3 


{-653  1  0  3  -4> 


SLISTCL33 


L3 

{-653  1  0  3  -4> 
REVLISTCL33 

{-430  1  3  5  -6> 
bniiMniininmiJiiiiii^iiMiUid 


SORT  and  REVLIST  can  be  combined  to  sort  a  list  in  decreasing  order: 


L3 

{-653  1  0  3  -4H 
:REVLIST£S0RT(L3H 

{53  3  1  0  -4  -0 


*LIST  ELIST  FTLIST  SORT  REVLI  ADD 


Page  8-9 


Manipulating  elements  of  a  list 

The  PRG  (programming)  menu  includes  a  LIST  sub-menu  with  a  number  of 
functions  to  manipulate  elements  of  a  list.   With  system  flag  1  1  7  set  to 
CHOOSE  boxes: 


PROG  HEMJ 

i. STACK..  \ 

Z.HEHORY.. 

3 .  GRflnCH.. 

H .  TEST.. 

5. TYPE.. 

le.LIST..  II 

LIST  HEHU 


i .  ELEHEnTS.. 


Z.  PROCEDURES.. 

3.0EJ+ 

H.-H.IST 

5.  SUE 

C.REPL 


Item  1 .  ELEMENTS.,  contains  the  following  functions  that  can  be  used  for  the 
manipu  < 


ation  of  elements  in  lists: 

ELEHEnT  METlU 

™ett7 

3. PUT  1 

H.PUTI  1 

S . SIZE 

e . pos  y 

ELEHEnT  HEHU 


H.PUTI 
S . SIZE 
G.POS 
? . HEAD 
S.TfllL 


List  size 

Function  SIZE,  from  the  PRG/LIST/ELEMENTS  sub-menu,  can  be  used  to 
obtain  the  size  (also  known  as  length)  of  the  list,  e.g., 


L3 

:  SIZECL33 


{-653  1  0  3  -4> 


Extracting  and  inserting  elements  in  a  list 

To  extract  elements  of  a  list  we  use  function  GET,  available  in  the 
PRG/LIST/ELEMENTS  sub-menu.  The  arguments  of  function  GET  are  the  list 
and  the  number  of  the  element  you  want  to  extract.  To  insert  an  element  into 
a  list  use  function  PUT  (also  available  in  the  PRG/LST/ELEMENTS  sub-menu). 
The  arguments  of  function  PUT  are  the  list,  the  position  that  one  wants  to 
replace,  and  the  value  that  will  be  replaced.  Examples  of  applications  of 
functions  GET  and  PUT  are  shown  in  the  following  screen: 


:  GET(L3,5) 

:PUT(L3,5.10) 

;-6  5  3 


0 


1  10  3 -4> 
ElMMtMliBMflillil]] 


Page  8-10 


Functions  GETI  and  PUTI,  also  available  in  sub-menu  PRG/  ELEMENTS/,  can 
also  be  used  to  extract  and  place  elements  in  a  list.  These  two  functions, 
however,  are  useful  mainly  in  programming.  Function  GETI  uses  the  same 
arguments  as  GET  and  returns  the  list,  the  element  location  plus  one,  and  the 
element  at  the  location  requested.  Function  PUTI  uses  the  same  arguments  as 
GET  and  returns  the  list  and  the  list  size. 

Element  position  in  the  list 

To  determine  the  position  of  an  element  in  a  list  use  function  POS  having  the 

list  and  the  element  of  interest  as  arguments.  For  example, 
■~L3  I 
{-653  1  0  3  -4> 

:  P0S(L3,5i 

 2- 

■aMaMaMsmmfHiiEi 

HEAD  and  TAIL  functions 

The  HEAD  function  extracts  the  first  element  in  the  list.  The  TAIL  function 
removes  the  first  element  of  a  list,  returning  the  remaining  list.  Some  examples 
are  shown  next: 


:|_3 

{-653103 

-4; 

:  HEADCL3) 

-6 

:TRILCL3) 

{53103 

-4; 

The  SEQ  function 

Item  2.  PROCEDURES.,  in  the  PRG/LIST  menu  contains  the  following  functions 
that  can  be  used  to  operate  on  lists. 

|FF;ml  HEHU 
3.nSUE= 

H.EnDSUB  I 

5 .  l"TF;EHM  1; 

6 .  REYLIST  U 


FRmC  HEHU 
H.EnDSUB 
S .  :"TF;EHM 
G.REYLIST 
7. SORT 


Functions  REVLIST  and  SORT  were  introduced  earlier  as  part  of  the  MTH/LIST 
menu.  Functions  DOLIST,  DOSUBS,  NSUB,  ENDSUB,  and  STREAM,  are 
designed  as  programming  functions  for  operating  lists  in  RPN  mode.  Function 


Page  8-1 1 


SEQ  is  useful  to  produce  a  list  of  values  given  a  particular  expression  and  is 
described  in  more  detail  here. 


The  SEQ  function  takes  as  arguments  an  expression  in  terms  of  an  index,  the 
name  of  the  index,  and  starting,  ending,  and  increment  values  for  the  index, 
and  returns  a  list  consisting  of  the  evaluation  of  the  expression  for  all  possible 
values  of  the  index.  The  general  form  of  the  function  is  SEQ(express/on/  index, 
start,  end,  increment). 


In  the  following  example,  in  ALG  mode,  we  identify  expression  =  n2,  index 
n,  start  =  1 ,  end  =4,  and  increment  =  1 : 


SEQ(n2\n,l.,4.  ,1 J 


{V-  V.  9.  16-> 


The  list  produced  corresponds  to  the  values  {I2,  22,  32,  42}.  In  RPN  mode, 


arguments  of  the 

function  as 

5: 

2. 

n 

4: 

'n' 

3: 

1. 

2: 

4. 

l: 

1. 

ollows: 


before  applying  function  SEQ. 


The  MAP  function 

The  MAP  function,  available  through  the  command  catalog  (Lr>J  cat  )f  takes 
as  arguments  a  list  of  numbers  and  a  function  f(X)  or  a  program  of  the  form 
«  ->  a  ...  »,  and  produces  a  list  consisting  of  the  application  of  function  f 
or  the  program  to  the  list  of  numbers.    For  example,  the  following  call  to 
function  MAP  applies  the  function  SIN(X)  to  the  list  {1,2,3}: 


MRP({1  2  3>,SIHCXi) 

{SINClJ  SIHC2)  SIHC3)> 


The  following  call  to  function  MAP  uses  a  program  instead  of  a  function  as 
second  argument: 


MflP<£8,l,2},*  ■*  x  'x 
{-1  0  3> 


Page  8-12 


Defining  functions  that  use  lists 

In  Chapter  3  we  introduced  the  use  of  the  DEFINE  function  ( (jT3?^—  )  to 
create  functions  of  real  numbers  with  one  or  more  arguments.  A  function 
defined  with  DEF  can  also  be  used  with  list  arguments,  except  that,  any 
function  incorporating  an  addition  must  use  the  ADD  operator  rather  than  the 
plus  sign  (L±J).  For  example,  if  we  define  the  function  F(X,Y)  =  (X-5)*(Y-2), 
shown  here  in  ALG  mode: 


DEFIHE('F(X,Y)=EX-5.  MY-* 
HOVRL 


we  can  use  lists  (e.g.,  variables  LI  and  L2,  defined  earlier  in  this  Chapter)  to 
evaluate  the  function,  resulting  in: 


:  DEFIHE('F(X,Y)=EX-5.  MY-* 
HOVRL 

F(L1,L2) 

{20-  0.  2.  -3.} 


Since  the  function  statement  includes  no  additions,  the  application  of  the 
function  to  list  arguments  is  straightforward.  However,  if  we  define  the 
function  G(X,Y)  =  (X+3)*Y,  an  attempt  to  evaluate  this  function  with  list 
arguments  (LI,  L2)  will  fail:  


:DEFIHEE'G(X,Y)=(X+3.:i-Y') 
HOVRL 

G<L1PL2> 

:DE 

A  *  Error: 
Inual id 
Dimension 

n 

VRL 

:G(Li,i_£j 

" I nu a  lid  D  i  mens  i  on " 

To  fix  this  problem  we  can  edit  the  contents  of  variable  I 
list  in  the  stack  by  using  03 !! 


! ,  which  we  can 


:DEFIHE('G(X,Y)=EX+3.:i-Y') 
HOVRL 

:GEL1,L2) 

11 1  nu a  lid  D  i  mens  i  on 11 
*  ■*  X  Y  'CX+3.W  * 


to  replace  the  plus  sign  (+)  with  ADD: 


Page  8-13 


GCL1.L2) 

" I nya lid  D  i  mens  i  on " 
*  ■+  X  Y  '  <!X+3.  )*Y'  * 
:  *  ■*  X  Y  '  <X  ADD  3.}* 

«  ■*  X  Y  1  £X  ADD  3.W 

ft 


Next,  we  store  the  edited  expression  into  variable 


X  Y  1  <X  RDD 

3.>* 

y  * 

«  ■*  X 

Y  '  <!X  RDD  3. 

>*Y' 

ft 

:RNS(1 

«  ■*  X 

Y  '  <!X  RDD  3. 

>*Y' 

ft 

Evaluating  G(L1,L2)  now  produces  the  following  result: 


G(L1,L2) 


{-12.  10.  6.  35.> 


As  an  alternative,  you  can  define  the  function  with  ADD  rather  than  the  plus 
sign  (+),  from  the  start,  i.e.,  use  DEFINED  5  GCX,  Y>  =  CX  HDD  3)*Y  5  > 


:DEFIHE('G(X,Y)=EX  RDD  3  J 
NOVRL 

:G(L1,L2) 

{-12.  10.  6.  35.> 


You  can  also  define  the  function  as  G(X,Y)  =  (X-3)*Y. 


Applications  of  lists 

This  section  shows  a  couple  of  applications  of  lists  to  the  calculation  of 
statistics  of  a  sample.  By  a  sample  we  understand  a  list  of  values,  say,  {s^ 
s2/      sn}.    Suppose  that  the  sample  of  interest  is  the  list 

{    I  «      5  »      3  1  2.  1  3         4  »      2  »       1  } 


Page  8-14 


and  that  we  store  it  into  a  variable  called  S  (The  screen  shot  below  shows  this 
action  in  ALG  mode,  however,  the  procedure  in  RPN  mode  is  very  similar. 
Just  keep  in  mind  that  in  RPN  mode  you  place  the  arguments  of  functions  in 
the  stack  before  activating  the  function):  


£i.  s.  3.  i.  a.  i.  3.  H .  a.  i.J 
a.  s.  3.  i.  a.  i.  3.  h.  a.  l.i 


Harmonic  mean  of  a  list 

This  is  a  small  enough  sample  that  we  can  count  on  the  screen  the  number  of 
elements  (n=l  0).  For  a  larger  list,  we  can  use  function  SIZE  to  obtain  that 
number,  e.g., 


{1.  5-  3.  1.  2.  1.  3.  4-  ► 
Kl.  5.  3.  1-  2.  1.  3.  4.  2* 

SIZECSi 

10. 


Suppose  that  we  want  to  calculate  the  harmonic  mean  of  the  sample,  defined 
as 

1  1 


1     n  1 

n  k=\  sn 


1  1 

—  +  +  • 


•  + 


1 


Vs  1  ^2 


To  calculate  this  value  we  can  follow  this  procedure: 
1 .   Apply  function  INV  ()  to  list  S:  


<1.  5.  3.  1.  2-  1.  3.  4-  ► 
Kl-  5-  3.  1.  2.  1.  3.  4. 

SIZECS) 

10 

INV(S) 

Kl.  -2  .333333333333  1.  H 


2.   Apply  function  SLIST()  to  the  resulting  list  inl . 

Kl.  5.  3.  1.  2.  1.  3.  4.  £H 
SIZECS3 

10. 

IHVCSi 

Kl-  -2  .333333333333  1.  H 
ELISTCFlNSCl.H 

6. 1166666666^ 


aLIST  ELIST  FTLIST  SORT  REYLI  ADD 


Page  8-15 


3.    Divide  the  result  above  by  n  =  10: 
1=  invi.:i..i  


SI.  .2  .333333333333  1.  H 
ELISTtflNSd.H 

6. 1166666666^ 

RHSdJ 


10. 


,611666666666 


4.   Apply  the  INV()  function  to  the  latest  result: 

1=  i,l_lb  I  l.HNbl.  1 .  II  


6. 1166666666^ 

■  RHSdJ 
"  10. 

.61166666666^ 
:  INVtflNSd.H 

1 . 6348773843 


Thus,  the  harmonic  mean  of  list  S  is  sh  =  1 .6348... 


Geometric  mean  of  a  list 

The  geometric  mean  of  a  sample  is  defined  as 


XS  =  \\flXk   =  a/X1  'X2"'Xn 


k=\ 


To  find  the  geometric  mean  of  the  list  stored  in  S,  we  can  use  the 
following  procedure: 


1 .   Apply  function  nl_ 


STQ  to  list  S: 


.  MMoLl  .  J 

A. 

"  10. 

.611666666666 

:  INVCRNSd 

■  H 

1 . 6343773342 

:  TTLISTCSJ 

720. 

2.   Apply  function  XROOT(x,y),  i.e.,  keystrokes  tj^J^Z  ,  to  the  result  in 
1: 


.61166666666ej 
INVCRNSd.  H 

1 . 6343773343 

:  TTLISTCS1 

720 

KR00T<fiHS<l>, 10)4 


. bl Ibbbbbbbb^ 
INVCRNSd  . )) 

1 . 634S773S42I 

TTLISTCS) 

720. 

RHSd.iJTo7 

1.003203154021 


Thus,  the  geometric  mean  of  list  S  is  sg  =  1 .003203... 


Page  8-16 


Weighted  average 

Suppose  that  the  data  in  list  S,  defined  above,  namely: 

S    :::=     J    1  =S  5  ».  3  :s    1  H  2  =i    1       3  *.  4  *.  2  «    1  } 

is  affected  by  the  weights, 

■  :   -    ■■  ""I  .-:  :~V         .■ 1=         i-":         :™:  ■■  s~.s  S 

!■'■!     ■■-  \  .!.  Z-  ;■    "f  ;■    ..J  ;■    *D  ;■        ;■    O  ;■     \s  ;i     .1.  kJ  X 

If  we  define  the  weight  list  as  W  =  {w1,w2,...,wn},  we  notice  that  the  /c-th 
element  in  list  W,  above,  can  be  defined  by  wk  =  k.  Thus  we  can  use  function 
SEQ  to  generate  this  list,  and  then  store  it  into  variable  !!!!![;!!!!!!!!  as  follows: 

 .■■liL.i. 

i  RNsa .  ).ji07 

1.003203154021 
:SEQ(k,k,l-,10.,l.:i 


Given  the  data  list  {sl7  s2/  sn },  and  the  weight  list  {w^  w2,  wn },  the 
weighted  average  of  the  data  in  S  is  defined  as 


sw  = 


k=\ 


To  calculate  the  weighted  average  of  the  data  in  list  S  with  the  weights  in  list 
W,  we  can  use  th  e  following  steps: 


1 .   Multiply  lists  S  and  W: 

I  1.00320315402) 

:SEQ(k,k,l-,10.,l.:i 

U.  2.  3.  4.  5.  6.  7.  S.  ^ 

:nHSCl.^N 

U.  2.  3.  4.  5.  6.  7.  S.  ^ 

:S-W 

a.  10.  9.  4.  10.  6.  21.  > 


2.    Use  function  ZLIST  in  this  result  to  calculate  the  numerator  of 


Kl.  2.  3.  4.  5.  6.  7.  S.  M 

iflNSd.iHJ 

U.  2.  3.  4.  5.  6.  7.  3.  ^ 

:S-W 

U.  10.  9.  4.  10.  6.  21.  ► 

iSLISTCflNSa.H 

 121. 


Page  8-17 


3.    Use  function  ZLIST,  once  more,  to  calculate  the  denominator  of  sw: 


{1.  2-  3.  4.  5.  6. 

7.  S.  ^ 

(1.  10.  9.  4.  10. 

6.  21.  ► 

:SLIST(fiHS(l.H 

121. 

:SLIST(W) 

55. 

Use  the  expression  ANS(2)/ANS(1 )  to  calculate  the  weighted 
average: 


121. 

:SLIST(W) 

55. 

.  RHSC2J 

"  flHSCU 

2.2 

Thus,  the  weighted  average  of  list  S  with  weights  in  list  W  is  sw=  2.2. 


Note:  ANS(1 )  refers  to  the  most  recent  result  (55),  while  ANS(2)  refers  to 
the  previous  to  last  result  (121). 


Statistics  of  grouped  data 

Grouped  data  is  typically  given  by  a  table  showing  the  frequency  (w)  of  data 
in  data  classes  or  bins.  Each  class  or  bin  is  represented  by  a  class  mark  (s), 
typically  the  midpoint  of  the  class.  An  example  of  grouped  data  is  shown 
next: 


Class 

Frequency 

Class 

mark 

count 

boundaries 

wk 

0-2 

1 

5 

2-4 

3 

12 

4-6 

5 

18 

6-8 

7 

1 

8  -10 

9 

3 

The  class  mark  data  can  be  stored  in  variable  S,  while  the  frequency  count 
can  be  stored  in  variable  W,  as  follows: 


Page  8-18 


:SEQ(2-k-l,k,l,5,U 

{1  3579} 

:flNS(l)*S 

{1  3  5  7  9} 

:{5  12  IS  1  3>Ud 

{5  12  IS  1  3} 


Given  the  list  of  class  marks  S  =  {sl7  s2,      sn },  and  the  list  of  frequency 
counts  W  =  {v/t,  w2,      wn },  the  weighted  average  of  the  data  in  S  with 
weights  W  represents  the  mean  value  of  the  grouped  data,  that  we  call  s,  in 
this  context: 


s  = 


k=\ 


k=\ 


N 


k=\ 


where  N  =  ^^wk  represents  the  total  frequency  count. 


k=l 


The  mean  value  for  the  data  in  lists  S  and  W,  therefore,  can  be  calculated 
using  the  procedure  outlined  above  for  the  weighted  average,  i.e., 

*~*  I  T^1    I~E    TO    I  ^71 


■  SLISTCN) 

55 
13 

:+NUM(flNS(lH 

4.230769230771 


We'll  store  this  value  into  a  variable  called  XBAR: 

2Llb  i  in  J  


55 
13 

:+NUM(flNS(lH 

4.23076923077^ 
:RHSC1>XERR 

4.230769230771 


The  variance  of  this  grouped  data  is  defined  as 

n  n 

Yjwk<sk-*)2  Y,wk<sk-s)2 


v  = 


k=\ 


k=\ 


N 


k=\ 


Page  8-19 


To  calculate  this  last  result,  we  can  use  the  following: 


:flNS(l)*XBflR 

:RHSC1>XBRR 

4.23076923077 

4.23076923077 

:ELISt(u-(S-XBFiR)£) 

:2LISt(w-(S-XBFiR)£] 

156.923076923 

156.923076923 

:SLIST(W) 

:SLIST(H) 

39 

39 

The  standard  deviation  of  the  grouped  data  is  the  square  root  of  the  variance 

I"       i  ■_■  i  n*n  1 


39 

.  nHSC2) 

■  ANsm 

4. 

02366363905 

:  JflHStlJ 

2. 

00590343237 

Page  8-20 


Chapter  9 
Vectors 

This  Chapter  provides  examples  of  entering  and  operating  with  vectors,  both 
mathematical  vectors  of  many  elements,  as  well  as  physical  vectors  of  2  and  3 
components. 

Definitions 

From  a  mathematical  point  of  view,  a  vector  is  an  array  of  2  or  more  elements 
arranged  into  a  row  or  a  column.    These  will  be  referred  to  as  row  and 
column  vectors.  Examples  are  shown  below: 

u  =  [1,-3,5,2] 


v  = 


3 
6 


Physical  vectors  have  two  or  three  components  and  can  be  used  to  represent 
physical  quantities  such  as  position,  velocity,  acceleration,  forces,  moments, 
linear  and  angular  momentum,  angular  velocity  and  acceleration,  etc. 
Referring  to  a  Cartesian  coordinate  system  (x,y,z),  there  exists  unit  vectors  i,  j, 
k  associated  with  each  coordinate  direction,  such  that  a  physical  vector  A  can 
be  written  in  terms  of  its  components  Ax,  Ay,  Az,  as  A  =  Axi  +  Ayj  +  Azk. 
Alternative  notation  for  this  vector  are:  A  =  [Ax,  Ay,  Az],  A  =  (Ax,  Ay,  Az),  or  A 
=  <  Ax,  Ay,  Az  >.  A  two  dimensional  version  of  this  vector  will  be  written  as  A 
=  Axi  +  Ayj,  A  =  [Ax,  Ay],  A  =  (Ax,  Ay),  or  A  =  <  Ax,  Ay  >.  Since  in  the 
calculator  vectors  are  written  between  brackets  [  ],  we  will  choose  the  notation 
A  =  [Ax,  Ay,  AJ  or  A  =  [Ax,  Ay,  AJ,  to  refer  to  two-  and  three-dimensional 
vectors  from  now  on.    The  magnitude  of  a  vector  A  is  defined  as  |  A|  = 

^A2  +  Ay  +  A2z  .  A  unit  vector  in  the  direction  of  vector  A,  is  defined  as  eA  = 

A/ 1 A  | .  Vectors  can  be  multiplied  by  a  scalar,  e.g.,  kA  =  [kAx,  kAy,  kAJ. 
Physically,  the  vector  kA  is  parallel  to  vector  A,  if  k>0,  or  anti-parallel  to 
vector  A,  if  k<0.  The  negative  of  a  vector  is  defined  as  -A  =  (-1  )A  =  [-Ax,  - 
Ay/  -AJ.  Division  by  as  scalar  can  be  interpreted  as  a  multiplication,  i.e., 
A/k  =  (l/k)-A.   Addition  and  subtraction  of  vectors  are  defined  as  A+B  =  [Ax 
±  Bx,  Ay±  By,  Az±  By],  w h e re  B  is  the  vector  B  =  [Bx,  By/  BJ. 


Page  9-1 


There  are  two  definitions  of  products  of  physical  vectors,  a  scalar  or  internal 
product  (the  dot  product)  and  a  vector  or  external  product  (the  cross  product). 
The  dot  product  produces  a  scalar  value  defined  as  A*B  =  |  A  |  |  B  |  cos(0), 
where  9  is  the  angle  between  the  two  vectors.  The  cross  product  produces  a 
vector  Ax B  whose  magnitude  is  |AxB|  =  |A|  |B|sin(0),  and  its  direction  is 
given  by  the  so-called  right-hand  rule  (consult  a  textbook  on  Math,  Physics,  or 
Mechanics  to  see  this  operation  illustrated  graphically).    In  terms  of  Cartesian 
components,  A*B  =  AxBx+AyBy+AzBz,  and  AxB  =  [AyBz-AzBy,AzBx-AxBz,AxBy- 
AyBJ.  The  angle  between  two  vectors  can  be  found  from  the  definition  of  the 
dot  product  as  cos(0)  =  A*B/ 1 A  |  |  B  |  =  eA*eB.  Thus,  if  two  vectors  A  and  B  are 
perpendicular  (9  =  90°  =  7i/2rad),  A*B  =  0. 


Entering  vectors 

In  the  calculator,  vectors  are  represented  by  a  sequence  of  numbers  enclosed 
between  brackets,  and  typically  entered  as  row  vectors.  The  brackets  are 

generated  in  the  calculator  by  the  keystroke  combination  L*~i )[l   , 

associated  with  the  L  x  J  key.  The  following  are  examples  of  vectors  in  the 
calculator: 

L  3  ::  5  :■      2  ::  2  "  1  ::  3         5  ::  6         2  ::  3  J  A  gene^l    TOW  VCCtOr 

CI.  5,  -2.  2]  A  2-D  vector 

C  3  =:  ■■■■■  1 2  "I  A  3-D  vector 

C  5 1 5  ,  5 1--2   ,  5  3 1 N  ( t  >  5  ]  A  vector  of  algebraics 


Typing  vectors  in  the  stack 

With  the  calculator  in  ALG  mode,  a  vector  is  typed  into  the  stack  by  opening 

a  set  of  brackets  (CjjD(Z  )  and  typing  the  components  or  elements  of  the 

vector  separated  by  commas  (Lr*  J  j ).  The  screen  shots  below  show  the 

entering  of  a  numerical  vector  followed  by  an  algebraic  vector.  The  figure  to 
the  left  shows  the  algebraic  vector  before  pressing  LjnJ  •  The  figure  to  the  right 
shows  the  calculator's  screen  after  entering  the  algebraic  vector: 


:  [5  3  -1  2  4] 

[5  3  -1  2  4] 
[tA2Ps-2*t,  T<t-3« 


[5  3  -1  2  4] 

[5  3-1  2  41 
[t2  s-2-t  Jt^] 


[t2  s-2-t  JF^ 


Page  9-2 


In  RPN  mode,  you  can  enter  a  vector  in  the  stack  by  opening  a  set  of  brackets 
and  typing  the  vector  components  or  elements  separated  by  either  commas 

(LrU  » )  or  spaces  (jspcj).  Notice  that  after  pressing  (enter)  f  in  either  mode, 

the  calculator  shows  the  vector  elements  separated  by  spaces. 


Storing  vectors  into  variables 

Vectors  can  be  stored  into  variables.  The  screen  shots  below  show  the  vectors 


u2  =  I"  1  ?  2  "I ,  u3  =  C 
stored  into  variables  II! 
mode: 


2],v2  =  C3!-l],v3  =  C: 


2:1 


!,  IIIEI,  and  llJIsI,  respectively.  First,  in  ALG 


:[1  2>u2 

CI  2] 

:  [-3  2  -2»u3 

[-3  2  -21 

[-3  2  -2]*u3 
[3  -l»y2 
:C1  -5  2]*v3 


Then,  in  RPN  mode  (before  pressing  [sjo>),  repeatedly) 


4: 
3: 
2: 
l: 


[1  2] 
'u2' 
[-3  2  -2] 
'u3' 


4: 
3: 
2: 
l: 


[-3  2  -2] 
C3  -1] 
[1  -5  2] 


[3  -1] 
'v2' 
[1  -5  2] 
■v3' 


Using  the  Matrix  Writer  (MTRW)  to  enter  vectors 

Vectors  can  also  be  entered  by  using  the  Matrix  Writer  [j^mtm_  (third  key  in 
the  fourth  row  of  keys  from  the  top  of  the  keyboard).  This  command  generates 
a  species  of  spreadsheet  corresponding  to  rows  and  columns  of  a  matrix 
(Details  on  using  the  Matrix  Writer  to  enter  matrices  will  be  presented  in  a 
subsequent  chapter).  For  a  vector  we  are  interested  in  filling  only  elements  in 
the  top  row.  By  default,  the  cell  in  the  top  row  and  first  column  is  selected. 
At  the  bottom  of  the  spreadsheet  you  will  find  the  following  soft  menu  keys: 


•:\:;-;;;  1 


The  IllSli!  key  is  used  to  edit  the  contents  of  a  selected  cell  in  the 
Matrix  Writer. 

The  iillJlili  key,  when  selected,  will  produce  a  vector,  as  opposite  to  a 
matrix  of  one  row  and  many  columns. 


Page  9-3 


Vectors  vs.  matrices 

To  see  the  Sill  key  in  action,  try  the  following  exercises: 

(1)  Launch  the  Matrix  Writer  (C5Jmw  ).  With  VSEm  and  HUb  selected, 
enter  CX3f^CS^)CX]^)^) .  This  produces  [3.  5.  2.].  (In  RPN 
mode,  you  can  use  the  following  keystroke  sequence  to  produce  the 
same  result:  CSC^CTDC^CXDt^)^^))- 

(2)  With  lillJlii!  deselected  and  EEUb  selected,,  enter 
CBC^CIDCz3CE(^(^.  This  produces  [[3.  5.  2.]]. 

Although  these  two  results  differ  only  in  the  number  of  brackets  used,  for  the 
calculator  they  represent  different  mathematical  objects.  The  first  one  is  a 
vector  with  three  elements,  and  the  second  one  a  matrix  with  one  row  and 
three  columns.  There  are  differences  in  the  way  that  mathematical  operations 
take  place  on  a  vector  as  opposite  to  a  matrix.  Therefore,  for  the  time  being, 
keep  the  soft  menu  key  mm*  selected  while  using  the  Matrix  Writer. 


The  <-IEI  key  is  used  to  decrease  the  width  of  the  columns  in  the 
spreadsheet.  Press  this  key  a  couple  of  times  to  see  the  column  width 
decrease  in  your  Matrix  Writer. 

The  EH3->  key  is  used  to  increase  the  width  of  the  columns  in  the 
spreadsheet.  Press  this  key  a  couple  of  times  to  see  the  column  width 
increase  in  your  Matrix  Writer. 

The  B3->«  key,  when  selected,  automatically  selects  the  next  cell  to 
the  right  of  the  current  cell  when  you  press  [enter]  .  This  option  is 
selected  by  default. 

The  EiU  key,  when  selected,  automatically  selects  the  next  cell  below 
the  current  cell  when  you  press  (enter}  . 


Moving  to  the  right  vs.  moving  down  in  the  Matrix  Writer 

Activate  the  Matrix  Writer  and  enter  CT](^^CJJ(^^C2l)(^)(^)  with  the 
B3->«  key  selected  (default).  Next,  enter  the  same  sequence  of  numbers  with 
the  iiSlUB  key  selected  to  see  the  difference.  In  the  first  case  you  entered  a 
vector  of  three  elements.  In  the  second  case  you  entered  a  matrix  of  three 
rows  and  one  column. 


Page  9-4 


Activate  the  Matrix  Writer  again  by  using  (<n  Jmtrw  /  and  press  jNxrj  to  check 
out  the  second  soft  key  menu  at  the  bottom  of  the  display.  It  will  show  the 
keys: 

:::!::"!-v:""::  ;::::::»::;::•:;:::  ::       :-r:r"2«:  «"!::»»■■ 

ii:::i.::i:!:i.:.iii     I!:::!.:!!:!:!.:.!     i!:::!:!:!:!:!.::!!!     i!:::!:!:!:!:!.::!!!    .!  —      !!.:!!!!!!  I!::.!:!:!!.!!:!:;!! 

The  lilll;]!!  key  will  add  a  row  full  of  zeros  at  the  location  of  the 
selected  cell  of  the  spreadsheet. 

The  IIIIIIII  key  will  delete  the  row  corresponding  to  the  selected  cell  of 
the  spreadsheet. 

The  DX3  key  will  add  a  column  full  of  zeros  at  the  location  of  the 
selected  cell  of  the  spreadsheet. 

The  illliili  key  will  delete  the  column  corresponding  to  the  selected 
cell  of  the  spreadsheet. 

The  il^iill!  key  will  place  the  contents  of  the  selected  cell  on  the  stack. 
The  EED3  key,  when  pressed,  will  request  that  the  user  indicate  the 
number  of  the  row  and  column  where  he  or  she  wants  to  position  the 
cursor. 

Pressing  [nxt)  once  more  produces  the  last  menu,  which  contains  only  one 
function  ill!!!!!  (delete). 

The  function  EES  will  delete  the  contents  of  the  selected  cell  and 
replace  it  with  a  zero. 

To  see  these  keys  in  action  try  the  following  exercise: 

(1)  Activate  the  Matrix  Writer  by  using  (j^/ww  m   Make  sure  the  EE3a  and 
IZEI->b  keys  are  selected. 

(2)  Enter  the  following: 

CD  (enter)  CT3  [enter}  [JJ  (enter) 

(jvxT)  iiijiiiij;!!!     1131311  r~r~)  12311  iiion 

[  4  j  (enter)  [  5  j  (enter)  [  6  j  [enter) 
[JJ  (enter)  £jj  (enter)  (^9j  (enter) 

(3)  Move  the  cursor  up  two  positions  by  using  .  Then  press  11I3I!. 
The  second  row  will  disappear. 

(4)  Press  lilllil.  A  row  of  three  zeroes  appears  in  the  second  row. 


Page  9-5 


(5)  Press  1II3II.  The  first  column  will  disappear. 

(6)  Press  Mlil!!!!!!!!!!!!!!! .  A  row  of  two  zeroes  appears  in  the  first  row. 

(7)  Press  B9Q  (JJWM  (JJWM  lilll  to  move  to  position  (3,3). 

(8)  Press  ii^ii!:!.    This  will  place  the  contents  of  cell  (3,3)  on  the  stack, 
although  you  will  not  be  able  to  see  it  yet. 

(9)  Press  [enter]  to  return  to  normal  display.   Element  (3,3)  and  the  full  matrix 
will  be  available  in  the  screen. 

Summary  of  Matrix  Writer  use  for  entering  vectors 

In  summary,  to  enter  a  vector  using  the  Matrix  Writer,  simply  activate  the 
writer  (L*i  Jmtrw  ),  and  place  the  elements  of  the  vector,  pressing  (enter}  after 
each  of  them.  Then,  press  (enter) [enter]  .  Make  sure  that  the  VB&m  and  EH->«I 
keys  are  selected. 

Example:  (j^mjrw  (^(^[^(gc^cxJt^ 
produces:  ['xA2'  2  -5  ] 


Building  a  vector  with  ->ARRY 

The  function  ->ARRY,  available  in  the  function  catalog  (Lr>J  cat  ^  r>  j  — ► ; 
use  <^N> ^3?  to  locate  the  function),  can  also  be  used  to  build  a  vector  or 
array  in  the  following  way.  In  ALG  mode,  enter  ->ARRY(vecfor  elements, 
number  of  elements),  e.g.,  


:+FlRRY(  1,2,3,4,43 

[12  3  4] 

:  +FIRRY(  1,-2,-3,33 

[1  -2  -3] 

:-HWiV.p,a,33 

[o:  p  61 


In  RPN  mode: 

(1)  Enter  the  n  elements  of  the  array  in  the  order  you  want  them  to  appear  in 
the  array  (when  read  from  left  to  right)  into  the  RPN  stack. 

(2)  Enter  n  as  the  last  entry. 

(3)  Use  function  ^ARRY. 


Page  9-6 


The  following  screen  shots  show  the  RPN  stack  before  and  after  applying 
function  -^ARRY: 


6: 
5: 
4: 
3: 
2: 
l: 


-5 
-4 
-3 
-2 
1 
5 


£: 
5: 
4: 
3: 
2: 
l: 


[-5  -4  -3-2  1] 


In  RPN  mode,  the  function  [->ARRY]  takes  the  objects  from  stack  levels  n+1,  n, 
n-7,  down  to  stack  levels  3  and  2,  and  converts  them  into  a  vector  of  n 
elements.  The  object  originally  at  stack  level  n+1  becomes  the  first  element, 
the  object  originally  at  level  n  becomes  the  second  element,  and  so  on. 


Note:  Function  -^ARRY  is  also  available  in  the  PRG/TYPE  menu  (C5D^  ) 


Identifying,  extracting,  and  inserting  vector  elements 

If  you  store  a  vector  into  a  variable  name,  say  A,  you  can  identify  elements  of 
the  vector  by  using  A(i),  where  i  is  an  integer  number  less  than  or  equal  to  the 
vector  size.  For  example,  create  the  following  array  and  store  it  in  variable 
A:  [-1,-2,-3,-4,-5]: 


-2  -3  -4  -5^R 

[-1  -2  -3  -4  -53 


To  recall  the  third  element  of  A,  for  example,  you  could  type  in  A(3)  into  the 
calculator.    In  ALG  mode,  simply  type  A(3).    In  RPN  mode,  type  'A(3)' 

[enter]  [EVALj  . 


You  can  operate  with  elements  of  the  array  by  writing  and  evaluating 
algebraic  expressions  such  as: 


:  nC23+nC53 
:nC13-nC43 
:  nC33-nC23 


-7 

3 
6 


.  nC33 
"  RC5) 


:  LNCFK5H 


3 
5 

LHC53+i-TTl 


Page  9-7 


no) 


More  complicated  expressions  involving  elements  of  A  can  also  be  written. 
For  example,  using  the  Equation  Writer  (L  r>  J  eqw  ),  we  can  write  the  following 
summation  of  the  elements  of  A: 


EaaEnaBgaBEiMHiHEBini 

Highlighting  the  entire  expression  and  using  the  IIIISIII  soft  menu  key,  we  get 

the  result:  "-3.5.  

Note:  The  vector  A  can  also  be  referred  to  as  an  indexed  variable  because 
the  name  A  represents  not  one,  but  many  values  identified  by  a  sub-index. 
To  replace  an  element  in  an  array  use  function  PUT  (you  can  find  it  in  the 
function  catalog  C3_c^  /  or  in  the  PRG/LIST/ELEMENTS  sub-menu  -  the  later 
was  introduced  in  Chapter  8).  In  ALG  mode,  you  need  to  use  function  PUT 
with  the  following  arguments:  ?U\[array,  location  to  be  replaced,  new  value). 
For  example,  to  change  the  contents  of  A(3)  to  4.5,  use: 


PUT(Fl,3,4.5) 

C-l  -2  4.5-4  -53 


In  RPN  mode,  you  can  change  the  value  of  an  element  of  A,  by  storing  a  new 
value  in  that  particular  element.  For  example,  if  we  want  to  change  the 
contents  of  A(3)  to  read  4.5  instead  of  its  current  value  of  -3.,  use: 


To  verify  that  the  change  took  place  use:  C3 
[-1  -2  4.5  -4  -5]. 


The  result  now  shown  is: 


Page  9-8 


Note:  This  approach  for  changing  the  value  of  an  array  element  is  not 
allowed  in  ALG  mode,  if  you  try  to  store  4.5  into  A(3)  in  this  mode  you  get 
the  following  error  message:  Invalid  Syntax. 


To  find  the  length  of  a  vector  you  can  use  the  function  SIZE,  available  through 
the  command  catalog  (N)  or  through  the  PRG/LIST/ELEMENTS  sub-menu. 
Some  examples,  based  on  the  arrays  or  vectors  stored  previously,  are  shown 
below: 


:SIZE(u3) 

{3.} 

:SIZE(u2) 

{2.} 

:SIZECm 

 {5.} 

■iiifiBiiwiriM 


Simple  operations  with  vectors 

To  illustrate  operations  with  vectors  we  will  use  the  vectors  A,  u2,  u3,  v2,  and 
v3,  stored  in  an  earlier  exercise. 


Changing  sign 

To  change  the  sign  of  a  vector  use  the  key  L^jJ,  e.g., 


-[2  3  5] 

-,'3 

-H 


[-2  -3  -5] 
[-1  5  -2] 
[1  2  3  4  5] 


Addition,  subtraction 

Addition  and  subtraction  of  vectors  require  that  the  two  vector  operands  have 
the  same  length: 


: u2+u2 
: u3+u3 


[4  1] 
[-2  -3  0] 
[-2  -4  -6  -3  -10] 


Page  9-9 


Attempting  to  add  or  subtract  vectors  of  different  length  produces  an  error 
message  (Invalid  Dimension),  e.g.,  v2+v3,  u2+u3,  A+v3,  etc. 


Multiplication  by  a  scalar,  and  division  by  a  scalar 

Multiplication  by  a  scalar  or  division  by  a  scalar  is  straightforward: 


3-u2 
-5-u3 


C9  -3] 
[15  -10  10] 
[-16  10] 


.u3 

2 


Absolute  value  function 

The  absolute  value  function  (ABS),  when  applied  to  a  vector,  produces  the 
magnitude  of  the  vector.  For  a  vector  A  =  [A1,A2,...,An],  the  magnitude  is 

defined  as  |  A  \=  ^A2  +  A2y  H  \-  A2z  .  In  the  ALG  mode,  enter  the  function 

name  followed  by  the  vector  argument.  For  example:  MBS  J  > , 

RBS  <!■■■!>,  RB3<u3>,  will  show  in  the  screen  as  follows: 


|[1  -2  6]l 

IHI 

Iu3l 


J4l 
■JT7 


The  MTH/VECTOR  menu 

The  MTH  menu  (L*i  Jmth  )  contains  a  menu  of  functions  that  specifically  to 
vector  objects: 


HATH  HEnU 

2 .  MhTRI!!.. 

3.  LIST..  J 
H. HYPERBOLIC. 

5 .  REAL.. 

6.  EASE..  1 

The  VECTOR  menu  contains  the  following  functions  (system  flag  1  1  7  set  to 
CHOOSE  boxes): 


Page  9-10 


VECTOR  HERU 

3.  CROSS  II 

VECTOR  HERU 

C.-*V3 

S.CVLIR  x 
5. SPHERE  x 

Magnitude 

The  magnitude  of  a  vector,  as  discussed  earlier,  can  be  found  with  function 
ABS.  This  function  is  also  available  from  the  keyboard  (LjnJ^—  )•  Examples 
of  application  of  function  ABS  were  shown  above. 


Dot  product 

Function  DOT  is  used  to  calculate  the  dot  product  of  two  vectors  of  the  same 
length.  Some  examples  of  application  of  function  DOT,  using  the  vectors  A, 
u2,  u3,  v2,  and  v3,  stored  earlier,  are  shown  next  in  ALG  mode.  Attempts  to 
calculate  the  dot  product  of  two  vectors  of  different  length  produce  an  error 
message]  


:DOT(fl,fl) 

:D0T(u2,u2) 

:D0T(u3,u3) 


55 
1 

-17 


:  D0T(u2,u3) 

11 1  nua  lid  D  i  mens  i  on 11 
D0T(fl,v3) 

11 1  nua  lid  D  i  mens  i  on 11 
:  D0T(u2,u3) 

11 1  nua  lid  D  i  mens  i  on 11 


Cross  product 

Function  CROSS  is  used  to  calculate  the  cross  product  of  two  2-D  vectors,  of 
two  3-D  vectors,  or  of  one  2-D  and  one  3-D  vector.  For  the  purpose  of 
calculating  a  cross  product,  a  2-D  vector  of  the  form  [Ax,  Ay],  is  treated  as  the 
3-D  vector  [Ax,  Ay,0].  Examples  in  ALG  mode  are  shown  next  for  two  2-D  and 
two  3-D  vectors.  Notice  that  the  cross  product  of  two  2-D  vectors  will  produce 
a  vector  in  the  z-direction  only,  i.e.,  a  vector  of  the  form  [0,  0,  CJ: 


:  CR0S3(u2,u2) 

[0  9  -7] 

:  CR0SS(u2,C2  -3]) 

[0  0  -7] 
:CR0SS([1.5  -2H,u2) 

[0  0  4.5] 


CR0SS(u3,u3) 
:  CR0SS(u3,u3) 


[-6  4  13] 


C0  0  0] 
CR0SSCC1  3  -5],[1  2  3]) 

[19  -3  -1] 


Page  9-1 1 


Examples  of  cross  products  of  one  3-D  vector  with  one  2-D  vector,  or  vice 
versa,  are  presented  next]  


:  CR0SS(u3,u2) 

[-2  -6  -3] 

:  CR0SS(u2,u3) 

[-2  -6-14] 
iCROSSCl  2  3],[5  -6]) 

 [IS  15  -16] 

■iiifiBiiwiriM 


Attempts  to  calculate  a  cross  product  of  vectors  of  length  other  than  2  or  3, 
produce  an  error  message  (Invalid  Dimension),  e.g.,  CROSS(v3,A),  etc. 


Decomposing  a  vector 

Function  V->  is  used  to  decompose  a  vector  into  its  elements  or  components. 
If  used  in  the  ALG  mode,  V->  will  provide  the  elements  of  the  vector  in  a  list, 
e.g., 


{-1.  -2.  -3-  -4.  -5 
{1.  -5. 

a. 


In  the  RPN  mode,  application  of  function  V->  will  list  the  components  of  a 
vector  in  the  stack,  e.g.,  V->(A)  will  produce  the  following  output  in  the  RPN 
stack  (vector  A  is  listed  in  stack  level  6:). 


&: 
5: 
4: 
3: 
2: 
l: 


C-l  -2  -3  -4  -5] 
-1. 
-2. 
-3. 
-4. 
-5. 


Building  a  two-dimensional  vector 

Function  ->V2  is  used  in  the  RPN  mode  to  build  a  vector  with  the  values  in 
stack  levels  1 :  and  2:.  The  following  screen  shots  show  the  stack  before  and 


Page  9-12 


Building  a  three-dimensional  vector 

Function  ->V3  is  used  in  the  RPN  mode  to  build  a  vector  with  the  values  in 
stack  levels  1 :  ,  2:,  and  3:.  The  following  screen  shots  show  the  stack  before 
and  after  applying  function  ->V2: 

3:  8  3: 

2:  6  2: 

l:  2     l:  C8.  6.  2. 


Changing  coordinate  system 

Functions  RECT,  CYLIN,  and  SPHERE  are  used  to  change  the  current 
coordinate  system  to  rectangular  (Cartesian),  cylindrical  (polar),  or  spherical 
coordinates.  The  current  system  is  shown  highlighted  in  the  corresponding 
CHOOSE  box  (system  flag  1  1  7  unset),  or  selected  in  the  corresponding 
SOFT  menu  label  (system  flag  1  1  7  set).  In  the  following  figure  the 
RECTangu' 


£: 
5: 
4: 
3: 
2: 
l: 


VECTOR  HEMJ 

S.CYLin  k 

5. SPHERE  * 

RECT"  CVLin  SPHER 


When  the  rectangular,  or  Cartesian,  coordinate  system  is  selected,  the  top 
line  of  the  display  will  show  an  XYZ  field,  and  any  2-D  or  3-D  vector  entered 
in  the  calculator  is  reproduced  as  the  (x,y,z)  components  of  the  vector.  Thus, 
to  enter  the  vector  A  =  3i+2j-5k,  we  use  [3,2,-5],  and  the  vector  is  shown  as: 


If  instead  of  entering  Cartesian  components  of  a  vector  we  enter  cylindrical 
(polar)  components,  we  need  to  provide  the  magnitude,  r,  of  the  projection  of 
the  vector  on  the  x-y  plane,  an  angle  0  (in  the  current  angular  measure) 
representing  the  inclination  of  r  with  respect  to  the  positive  x-axis  ,  and  a  z- 
component  of  the  vector.  The  angle  0  must  be  entered  preceded  by  the  angle 
character  (Z),  generated  by  using  ^^JLlULU  •  For  example,  suppose  that 
we  have  a  vector  with  r  =  5,  0  =  25°  (DEG  should  be  selected  as  the  angular 
measure),  and  z  =  2.3,  we  can  enter  this  vector  in  the  following  way: 


Page  9-13 


ED  {LCD  CB  >  (^C3(3J  CX3CD  CB  ?  CSraCX} 


Before  pressing  t^j,  the  screen  will  look  as  in  the  left-hand  side  of  the 
following  figure.  After  pressing  {enter),  the  screen  will  look  as  in  the  right-hand 
side  of  the  figure  (For  this  example,  the  numerical  format  was  changed  to  Fix, 
with  three  decimals). 


,[5,^25,2.3] 


C4.5S 

32  2-113  2.300:| 

Notice  that  the  vector  is  displayed  in  Cartesian  coordinates  ,  with  components 
x  =  r  cos(9),  y  =  r  sin(0),  z  =  z,  even  though  we  entered  it  in  polar  coordinates. 
This  is  because  the  vector  display  will  default  to  the  current  coordinate  system. 
For  this  case,  we  have  x  =  4.532,  y  =  2. 1  1  2,  and  z  =  2.300. 


Suppose  that  we  now  enter  a  vector  in  spherical  coordinates  (i.e.,  in  the  form 
(p,9,(|)),  where  p  is  the  length  of  the  vector,  0  is  the  angle  that  the  xy  projection 
of  the  vector  forms  with  the  positive  side  of  the  x-axis,  and  §  is  the  angle  that  p 
forms  with  the  positive  side  of  the  z  axis),  with  p  =  5,  9  =  25°,  and  §  =  45°. 
We  will  use:CfD£  CD  CS  >  (^C3CX3  CX3CD   I 


The  figure  below  shows  the  transformation  of  the  vector  from  spherical  to 
Cartesian  coordinates,  with  x  =  p  sin(4>)  cos(0),  y  =  p  sin  (§)  cos  (0),  z  =  p 
cos((|)).  For  this  case,  x  =  3.204,  y  =  1 .494,  and  z  =  3.536.  

|l:  [3.204  1.494  3.536l| 
III  hi  II W  MM  Id  1 1  I^M^MIH  ■ 


,[5,^25,^454  _ 


If  the  CYLINdrical  system  is  selected,  the  top  line  of  the  display  will  show  an 
RZZ  field,  and  a  vector  entered  in  cylindrical  coordinates  will  be  shown  in  its 
cylindrical  (or  polar)  coordinate  form  (r,0,z).    To  see  this  in  action,  change 
the  coordinate  system  to  CYLINdrical  and  watch  how  the  vector  displayed  in 
the  last  screen  changes  to  its  cylindrical  (polar)  coordinate  form.  The  second 
component  is  shown  with  the  angle  character  in  front  to  emphasize  its  angular 
nature. 

l:  [3.536  ^25.000  3.  536> 


Page  9-14 


The  conversion  from  Cartesian  to  cylindrical  coordinates  is  such  that  r  = 
(x2+y2)1/2,  9  =  tan'^y/x),  and  z  =  z.  For  the  case  shown  above  the 
transformation  was  such  that  (x,y,z)  =  (3.204,  2.1  12,  2.300),  produced 
(r,6,z)  =  (3.536,25°,3.536). 


At  this  point,  change  the  angular  measure  to  Radians.  If  we  now  enter  a 
vector  of  integers  in  Cartesian  form,  even  if  the  CYLINdrical  coordinate  system 
is  active,  it  will  be  shown  in  Cartesian  coordinates,  e.g., 

4:  " 
3: 

2:  [3.536  ^25.000  3.= 
l:  [2 


This  is  because  the  integer  numbers  are  intended  for  use  with  the  CAS  and, 
therefore,  the  components  of  this  vector  are  kept  in  Cartesian  form.  To  force 
the  conversion  to  polar  coordinates  enter  the  vector  components  as  real 
numbers  (i.e.,  add  a  decimal  point),  e.g.,  [2.,  3.,  5.]. 


l:    [3.606  ,£0.933  5.000! 


With  the  cylindrical  coordinate  system  selected,  if  we  enter  a  vector  in 
spherical  coordinates  it  will  be  automatically  transformed  to  its  cylindrical 
(polar)  equivalent  (r,9,z)  with  r  =  p  sin     0  =  9,  z  =  p  cos  <|>.  For  example,  the 
following  figure  shows  the  vector  entered  in  spherical  coordinates,  and 
transformed  to  polar  coordinates.  For  this  case,  p  =  5,  0  =  25°,  and  §  =  45°, 
while  the  transformation  shows  that  r  =  3.563,  and  z  =  3.536.  (Change  to 
DEG): 


3:  4: 

2:  [3.536  ^25.000  3.  536^  3:  [3.  536  ^25.  000  3.  536> 

l:                           [2  3  5]  2:  [2  3  5] 

[5,  ^25,  ^454  1 :  [3.  536  ^25.  000  3.  536> 

Next,  let's  change  the  coordinate  system  to  spherical  coordinates  by  using 
function  SPHERE  from  the  VECTOR  sub-menu  in  the  MTH  menu.  When  this 
coordinate  system  is  selected,  the  display  will  show  the  RZZ  format  in  the  top 
line.  The  last  screen  will  change  to  show  the  following: 


4: 

3:  [5.000  ^25. 000  <£45. 0* 
2  s  [2  3  5] 

,1:  [5.000  ^25. 000  ^45. 0> 
li^lH'^iilHiilfl^W^Hia 


Page  9-15 


Notice  that  the  vectors  that  were  written  in  cylindrical  polar  coordinates  have 
now  been  changed  to  the  spherical  coordinate  system.  The  transformation  is 
such  that  p  =  (r2+z2)1/2,  9  =  0,  and    =  tan"1(r/z).  However,  the  vector  that 
originally  was  set  to  Cartesian  coordinates  remains  in  that  form. 

Application  of  vector  operations 

This  section  contains  some  examples  of  vector  operations  that  you  may 
encounter  in  Physics  or  Mechanics  applications. 

Resultant  of  forces 

Suppose  that  a  particle  is  subject  to  the  following  forces  (in  N):  ?}  =  3i+5j+2k, 
F2  =  -2i+3j-5k,  and  F3  =  2i-3k.  To  determine  the  resultant,  i.e.,  the  sum,  of  all 
these  forces,  you  can  use  the  following  approach  in  ALG  mode: 

:  [3  5  2H+C-2  3  -5H+C2  0  -3> 
C3  3  -6] 

■Man  ■■■■  "^'i  m^m 

Thus,  the  resultant  is  R  =       F2  +  F3  =  (3i+8j-6k)N.    RPN  mode  use: 

C3,5,2]  (enter)  [-2,  3, -5]  (enter)  [2,0,3]  (enter)  CB 


Angle  between  vectors 

The  angle  between  two  vectors  A,  B,  can  be  found  as  0  =cos"](A*B/ 1 A  |  |  B  | ) 
Suppose  that  you  want  to  find  the  angle  between  vectors  A  =  3i-5j+6k,  B  = 
2i+j-3k,  you  could  try  the  following  operation  (angular  measure  set  to  degrees) 
in  ALG  mode: 

1  -  Enter  vectors  [3,-5,6],  press  (enter),  [2,1,-3],  press  (enter). 

2  -  DOT(ANS(l),ANS(2))  calculates  the  dot  product 

3  -  ABS(ANS(3))*ABS((ANS(2))  calculates  product  of  magnitudes 

4  -  ANS(2)/ANS(1 )  calculates  cos(6) 

5  -  ACOS(ANS(l)),  followed  by  ,^NUM(ANS(1 )),  calculates  6 
The  steps  are  shown  in  the  following  screens  (ALG  mode,  of  course): 


[3  -5  61 
12  1  -3] 

D0T(flNS(l),flNS(2H 


[3  -5  6] 
[2  1  -3] 
-17 


:  [2  1  -3] 

:D0TCRHSC1)JRHSC2)3 
:  lflHS(3)MflHS(2)l 


Ld  --■  bJ 
C2  1  -3] 
-17 
J70--JT4 


Page  9-16 


:  IRNSC3)I-IRHS(S)I 

■J70--JT4 

.  RHSC23 

"  HHS(l) 

-17 

J70--JT4 

A. 

J70--JT4 

:flCOS(HHS(lH 

RCOS 

-17  1 

J70-JT4J 

:+NUM(flNS(lH 

122.891 

Thus,  the  result  is  6  =  122.891°.     In  RPN  mode  use  the  following: 

C  3  -i  -5  =■  6  J  (enter)  i2~  1     3  J  [enter)  DOT 
[3,-5,6]  (enter)  RBS  [2,  1,-3]  (py^D  MBS  QD 
C±D     H  C 0  S       H  U  M 


Moment  of  a  force 

The  moment  exerted  by  a  force  F  about  a  point  O  is  defined  as  the  cross- 
product  M  =  rxF,  where  r,  also  known  as  the  arm  of  the  force,  is  the  position 
vector  based  at  O  and  pointing  towards  the  point  of  application  of  the  force. 
Suppose  that  a  force  F  =  (2i+5j-6k)  N  has  an  arm  r  =  (3i-5j+4k)m.  To 
determine  the  moment  exerted  by  the  force  with  that  arm,  we  use  function 
CROSS  as  shown  next: 


:C3  -5  4] 
: [2  5  -61 


C3  -5  4] 


C2  5  -6] 
:CR0SSCRHSC2),RHSCD) 

[10  26  25] 
lilMl'MiMil-H-M'aE'MKM 


Thus,  M  =  (1 0i+26j+25k)  m-N.  We  know  that  the  magnitude  of  M  is  such 
that  |  M  |  =  |  r  |  |  F  |  sin(0),  where  6  is  the  angle  between  r  and  F.  We  can  find 
this  angle  as,  9  =  sin"1  ( |  M  |  / 1  r  |  |  F  | )  by  the  following  operations: 

1  -  ABS(ANS(1))/(ABS(ANS(2))*ABS(ANS(3))  calculates  sin(6) 

2  -  ASIN(ANS(1)),  followed  by  ^NUM(ANS(1))  calculates  6 
These  operations  are  shown,  in  ALG  mode,  in  the  following  screens: 


C  R  0  S  S  i  R  H  S  f  2  i ,  R  H  S  L I  i  i 

[10  26  25] 

IFlHSmi 


IRHSC23I-IRHSC33I 


■JT40T 


J65-5-J2 


:flSIH(flHS(lH 

RSIH 
+NUM(RNSm) 


■J65-5--J2 
JI40T  ' 


J65-5-J2 


41.038 


Page  9-17 


Thus  the  angle  between  vectors  r  and  F  is  6  =  41 .038°.    RPN  mode,  we  can 
use:  L3«  ■■■■5«  4"!  [enter)  12«.S«-&1  (enter)    CROSS     FIBS  L3«  -5i  4"!  [enter) 
MBS  C2,5,-6]  (enter)  RBS  CZD  S  MS  IN  ->NUM 


Equation  of  a  plane  in  space 

Given  a  point  in  space  Po(*oyo/Zo)  and  a  vector  N  =  Nxi+Nyj+Nzk  normal  to  a 
plane  containing  point  P0/  the  problem  is  to  find  the  equation  of  the  plane. 
We  can  form  a  vector  starting  at  point  P0  and  ending  at  point  P(x,y,z),  a 
generic  point  in  the  plane.  Thus,  this  vector  r  =  P0P  =  (x-x0)i+  (y-y0)i  +  (z~zo)k, 
is  perpendicular  to  the  normal  vector  N,  since  r  is  contained  entirely  in  the 
plane.  We  learned  that  for  two  normal  vectors  N  and  r,  N*r  =0.  Thus,  we 
can  use  this  result  to  determine  the  equation  of  the  plane. 

To  illustrate  the  use  of  this  approach,  consider  the  point  P0(2,3,-l)  and  the 
normal  vector  N  =  4i+6j+2k,  we  can  enter  vector  N  and  point  P0  as  two 
vectors,  as  shown  below.  We  also  enter  the  vector  [x,y,z]  last: 


:  [4  6  21 

C4  6  21 

:[2  3  -1] 

12  3  -1] 

:  [x  y  z] 

Cx  y  z] 

lillil'I'HEilH-M'JCMEa 

Next,  we  calculate  vector  P0P  =  r  as  ANS(1 )  -  ANS(2),  i.e., 


[4  6  2] 

:[2  3  -1] 

[2  3  -1] 

:  [x  y  z] 

[x  y  z] 

:  FlNSm-FlNS(2) 

[x-2  y 

-3  z— 1] 

Finally,  we  take  the  dot  product  of  ANS(1 )  and  ANS(4)  and  make  it  equal  to 
zero  to  complete  the  operation  N*r  =0: 


[2  3  -1] 
[x  y  z] 

[x  y  z] 

FlNSm-FlNS(2) 

[x-2  y-3  z— 1] 
DOT(FlNSm..FlNS(4h=0 
Jz—  1 3-2+Cy-33-6+Cx-2M=0 


Page  9-18 


We  can  now  use  function  EXPAND  (in  the  ALG  menu)  to  expand  this 
expression: 


:flNSU)-flNS(2) 


[x  y  z] 


[x-2  y-3  z— 1] 
DOT(fiHS(l),fiHS(4))=0 
Cz— 13-2+Cy-33-6+Cx-2M=^ 

4-x+6-y+2-z-24=6 

^gi^i^iiifiiiii^ngiii^giitijjiiimj 


Thus,  the  equation  of  the  plane  through  point  P0(2, 3,-1 )  and  having  normal 
vector  N  =  4i+6j+2k,  is  4x  +  6y  +  2z  -  24  =  0.  In  RPN  mode,  use: 


1 1  J  [enter]  [ 


J  (enter)  (3=3  [  4  :■  6  :=  2 1  DOT  ExpflND 


Row  vectors,  column  vectors,  and  lists 

The  vectors  presented  in  this  chapter  are  all  row  vectors.  In  some  instances,  it 
is  necessary  to  create  a  column  vector  (e.g.,  to  use  the  pre-defined  statistical 
functions  in  the  calculator).  The  simplest  way  to  enter  a  column  vector  is  by 
enclosing  each  vector  element  within  brackets,  all  contained  within  an 
external  set  of  brackets.  For  example,  enter: 

■"  r  ■:  "5  ™  ill"  ":  : :  ":  :"  .■:  i™  ™  ™  ."  ": 
!..  !..  .!.  ::  si::.  ..!  »    !..  £.  ■■  ..!  «    !..  .I:::  ::  ^  ..!  «    !..  !  i   I  ..!  »    !..        ::  ..! 


This  is  represented  as  the  following  column  vector: 


In  this  section  we  will  showing  you  ways  to  transform:  a  column  vector  into  a 
row  vector,  a  row  vector  into  a  column  vector,  a  list  into  a  vector,  and  a 
vector  (or  matrix)  into  a  list. 

We  first  demonstrate  these  transformations  using  the  RPN  mode.  In  this  mode, 
we  will  use  functions  OBJ^,  ^LIST,  ^ARRY  and  DROP  to  perform  the 
transformation.  To  facilitate  accessing  these  functions  we  will  set  system  flag 
1 1  7  to  SOFT  menus  (see  Chapter  1 ).  With  this  flag  set,  functions  OBJ->, 
->ARRY,  and  -HIST  will  be  accessible  by  using  C3D^—  llilllill.  Functions 


Page  9-19 


OBJ^,  -^ARRY,  and  ^LIST  will  be  available  in  soft  menu  keys  QD  ,  GD  , 
and  (jD  •  Function  DROP  is  available  by  using  (jT]??g_  StHSD^a. 

Following  we  introduce  the  operation  of  functions  OBJ->,  LIST,  ->ARRY,  and 
DROP  with  some  examples. 

Function  OBJ^ 

This  function  decomposes  an  object  into  its  components.  If  the  argument  is  a 
list,  function  OBJ->  will  list  the  list  elements  in  the  stack,  with  the  number  of 
elements  in  stack  level  1,  for  example:  i  1  :■  £?  3>  (enter)  (jT]ffi^_  jjililill  IEEE -> II 
results  in: 


4: 

1 

3: 

2 

2: 

3 

l: 

3. 

OEJ-h  |- 

RRY|-H_IST|  -+5TR  1  -+TRG  |-KJniT 

When  function  OBJ->  is  applied  to  a  vector,  it  will  list  the  elements  of  the 
vector  in  the  stack,  with  the  number  of  elements  in  level  1 :  enclosed  in  braces 
(a  list).  The  following  example  illustrates  this  application:  C         33  [enter] 

CZD??G_fl!aS  ilin^jj   results  in:  


4: 

1 

3: 

2 

2: 

3 

l: 

{3.> 

OEJ-h  |- 

RRY|-H_IST|  -+5TR  1  -+TAG  |-KJniT 

If  we  now  apply  function  OBJ->  once  more,  the  list  in  stack  level  1 :,  {3.},  will 
be  decomposed  as  follows:  


?: 

&: 

5: 

1 

4: 

2 

3: 

3 

2: 

3. 

l: 

1. 

OEJ-h  |- 

RRYKISTI  -+5TR  |  -+TRG  |-KJniT 

Function  -HIST 

This  function  is  used  to  create  a  list  given  the  elements  of  the  list  and  the  list 
length  or  size.  In  RPN  mode,  the  list  size,  say,  n,  should  be  placed  in  stack 
level  1 :.   The  elements  of  the  list  should  be  located  in  stack  levels  2:,  3:, 


Page  9-20 


n+1 :.  For  example,  to  create  the  list  {1,2,  3},  type:  CTD(^D  CS(^™D 

[JJijNTER)  [JJ(enter)  rgHfflG  fflin 

Function  -^ARRY 

This  function  is  used  to  create  a  vector  or  a  matrix.  In  this  section,  we  will  use 
it  to  build  a  vector  or  a  column  vector  (i.e.,  a  matrix  of  n  rows  and  1  column). 
To  build  a  regular  vector  we  enter  the  elements  of  the  vector  in  the  stack,  and 
in  stack  level  1 :  we  enter  the  vector  size  as  a  list,  e.g.,  L  /  Renter)  L  2  j [enter) 

i^JJijNTER)  (JT]U          [JJijNTER)  CfD5TC_fflH  l-»MI 

To  build  a  column  vector  of  n  elements,  enter  the  elements  of  the  vector  in  the 
stack,  and  in  stack  level  1  enter  the  list  {n  1}.  For  example,CD(^D  CB(^D 

[JJ(enter)  CSi^-  CDCS  :CD^  csD?^-ffla  \^mm. 

Function  DROP 

This  function  has  the  same  effect  as  the  delete  key  (L4  J). 

Transforming  a  row  vector  into  a  column  vector 

We  illustrate  the  transformation  with  vector  L         3  "I .    Enter  this  vector  into 
the  RPN  stack  to  follow  the  exercise.  To  transform  a  row  vector  into  a  column 
vector,  we  need  to  carry  on  the  following  operations  in  the  RPN  stack: 
1  -  Decompose  the  vector  with  function  OBJ-> 


4: 

1 

3: 

2 

£: 

3 

l: 

{3.> 

RRYKISTI  -+5TR  |  -+TAG  |-4JniT 

2  -  Press  LUL+J  to  transform  the  list  in  stack  level  1 :  from  {3}  to  {3, 1 } 


1-  \Q.  l.i 

BaHE3MCIHiK3i«niBEIEHi 

3  -  Use  function  ->ARRY  to  build  the  column  vector 


l: 

i 

[i]:?nF;l;l;VJE*tidE.-irf;lEf;' 

G  |-4JniT| 

These  three  steps  can  be  put  together  into  a  UserRPL  program,  entered  as 

follows  (in  RPN  mode,  still):  (S^CSD?^- (BH  illJ^II  CD  GD  1^311:11 

[enter)    ■  )[alpha)[alpha)(r\(x\(c)  [enter)  [sto>] 


Page  9-21 


A  new  variable, 


,  will  be  available  in  the  soft  menu  labels  after  pressing 


Press  CB 


!  to  see  the  program  contained  in  the  variable  RXC: 

jJP.J->   1    ■■!■■    ->j:::jRRY  V> 


This  variable,  llllllllllll,  can  now  be  used  to  directly  transform  a  row  vector  to  a 
column  vector.  In  RPN  mode,  enter  the  row  vector,  and  then  press  1113111! .  Try, 
for  example:  C  1  ?  2*  33  (enter)  WMM. 

After  having  defined  this  variable  ,  we  can  use  it  in  ALG  mode  to  transform  a 
row  vector  into  a  column  vector.  Thus,  change  your  calculator's  mode  to  ALG 

and  try  the  following  procedure:  1 1  :s  2?  31  (enter)  [WJ  IlEiliii  CfD  (l   C3D 

ans_  f  resulting  in: 


[1  2  3] 
:RXCCRHSC1)3 


CI  2  3] 


L_  i 


Transforming  a  column  vector  into  a  row  vector 

To  illustrate  this  transformation,  we'll  enter  the  column  vector 

"I  in  RPN  mode.  Then,  follow  the  next  exercise  to 
transform  a  row  vector  into  a  column  vector: 
1  -  Use  function  OBJ->  to  decompose  the  column  vector 


4: 
3: 
2: 
l: 


1 

2 
3 

<3.  !■> 


-tfRRY M-IST  -kSTR   -HflG  -HJniT 


2  -  Use  function  OBJ->  to  decompose  the  list  in  stack  level  1 : 


&: 

1 

5: 

2 

4: 

3 

3: 

3. 

2: 

1. 

l: 

2. 

OEJ-h  |- 

RRYKISTI  -KSTR  1  -HAG  |-KJniT 

Page  9-22 


3  -  Press  the  delete  key  J  (also  known  as  function  DROP)  to  eliminate  the 
number  in  stack  level  1 : 


5: 

1 

4: 

2 

3: 

3 

2: 

3. 

l: 

1. 

OEJ-h  |- 

RRY|-H_IST|  -+5TR  1  -+TRG  |-KJniT 

4  -  Use  function  -HIST  to  create  a  list 


4: 

1 

3: 

2 

2: 

3 

l: 

{3.> 

RRY|-H_I£T|  -+5TR  |  -+TAG  |-4JniT 

5  -  Use  function  ->ARRY  to  create  the  row  vector 

4: 
3: 
2: 

l:  CI  2  3] 

BaHE3MCIHiK3i«niBEIEHi 

These  five  steps  can  be  put  together  into  a  UserRPL  program,  entered  as 

follows  (in  RPN  mode,  still): 

(      .         „  „  (    ,      1  DDr        ::::ilT:  :!:  ::::     Sv;~!s5_V8     ::=:=":=:::"  \:i 

( J    <K>>  \  JH... )  PRG  m<ZS ~7 II    m$tt ~7 !! 

[  ■  j  ^iw/)  {alpha}  (q}(x}(r}  [enter)  [STQy] 
A  new  variable,  liiillii,  will  be  available  in  the  soft  menu  labels  after  pressing 

[  var)  : 


Press  CSE321  to  see  the  program  contained  in  the  variable  CXR: 
<<  OBJ-»  OBJ-»  DROP  ^HRRV  >> 

This  variable,  I!!!!!!!:!:!!!:!!!!!,  can  now  be  used  to  directly  transform  a  column  vector  to 
a  row  vector.  In  RPN  mode,  enter  the  column  vector,  and  then  press  IIIESI1. 
Try,  for  example:      L  L  1  "I ».  121    I"  3  "I  "I  (enter)  WSm. 

After  having  defined  variable  illlEEill!,  we  can  use  it  in  ALG  mode  to  transform  a 
row  vector  into  a  column  vector.  Thus,  change  your  calculator's  mode  to  ALG 
and  try  the  following  procedure: 

c  c  i  ::i i2  i  ,  lb::!  ::i  ^  cwj  mm  c?D£^_  czd^ 


Page  9-23 


resulting  in: 


111 

[izaj 

:CXR(ANS(D) 

Transforming  a  list  into  a  vector 

To  illustrate  this  transformation,  well  enter  the  list  i  3>  in  RPN  mode. 
Then,  follow  the  next  exercise  to  transform  a  list  into  a  vector: 

1  -  Use  function  OBJ->  to  decompose  the  column  vector 


4: 

1 

3: 

2 

2: 

3 

l: 

3. 

OEJ-h  |- 

RRY|-H_IST|  -+5TR  1  -+TRG  |-KJniT 

2  -  Type  a  1  and  use  function  -HIST  to  create  a  list  in  stack  level  1 : 


4: 

1 

3: 

2 

2: 

3 

l: 

{3.> 

OEJ-h  |- 

RRY|-H_IST|  -+5TR  1  -+TRG  |-KJniT 

3  -  Use  function  ->ARRY  to  create  the  vector  

|f:  [12  3^ 


OEJ^|-*flRRVKlST|  ■+:" T F;  |  -+TRG  |-KJniT 


These  three  steps  can  be  put  together  into  a  UserRPL  program,  entered  as 
follows  (in  RPN  mode): 

[  '  )[ALPHA)[ALPHA)(L\(x\(y]   [ENTER]  [STO>] 

A  new  variable,  IISIIII,  will  be  available  in  the  soft  menu  labels  after  pressing 

Cj^D:   

P  1 

l:  

■HEMUMaMaia 

Press  CB 11501  to  see  the  program  contained  in  the  variable  LXV: 
s.j ::::::...! -7  .!.  ~7\  !.:::::   -7r\r.r.  \ 

This  variable,  can  now  be  used  to  directly  transform  a  list  into  a  vector. 

In  RPN  mode,  enter  the  list,  and  then  press  DS9.  Try,  for  example: 

\    .!.  j:         j:         /    [ENTER  '    jjjj.jjjY&jjjjjj . 


Page  9-24 


After  having  defined  variable  IIIIIII,  we  can  use  it  in  ALG  mode  to  transform  a 
list  into  a  vector.  Thus,  change  your  calculator's  mode  to  ALG  and  try  the 

following  procedure:  i  1  ?  2j  3>  (enter)  {jmJ  WMM  C3D^      {j^Jans_  f  resulting 

in: 


:LXVlflNS(l)) 

CI  2  3] 

■HEMfflMaMaia 

Transforming  a  vector  (or  matrix)  into  a  list 

To  transform  a  vector  into  a  list,  the  calculator  provides  function  AXL.  You  can 
find  this  function  through  the  command  catalog,  as  follows: 

{alpha}  {alpha}  @)(x}@  {alpha}  mm 

As  an  example,  apply  function  AXL  to  the  vector  L         3  J  in  RPN  mode  by 

using:!!!  1  ;s  2?.  31  [enter}  flXI  The  following  screen  shot  shows  the  application 

of  function  AXL  to  the  same  vector  in  ALG  mode. 


:RXLCC1  2  31) 

{1  2  3> 

■HEuaiiaiaia 


Page  9-25 


Chapter  1 0 

Creating  and  manipulating  matrices 

This  chapter  shows  a  number  of  examples  aimed  at  creating  matrices  in  the 
calculator  and  demonstrating  manipulation  of  matrix  elements. 

Definitions 

A  matrix  is  simply  a  rectangular  array  of  objects  (e.g.,  numbers,  algebraics) 
having  a  number  of  rows  and  columns.  A  matrix  A  having  n  rows  and  m 
columns  will  have,  therefore,  nxm  elements.  A  generic  element  of  the  matrix 
is  represented  by  the  indexed  variable      corresponding  to  row  i  and  column 
j.  With  this  notation  we  can  write  matrix  A  as  A  =  [aij]nXm  .  The  full  matrix  is 
shown  next: 


A  -  [^ij]nxm  - 


a, 


a 


12 


a 


a 


21 


a 


22 


a 


2  m 


a 


nl 


a 


nl 


A  matrix  is  square  if  m  =  n.    The  transpose  of  a  matrix  is  constructed  by 
swapping  rows  for  columns  and  vice  versa.  Thus,  the  transpose  of  matrix  A,  is 
AT  =  [(aT)jj]  mXn  =  [ajj]mXn.  The  main  diagonal  of  a  square  matrix  is  the  collection 
of  elements  aM.  An  identity  matrix,  lnXn,  is  a  square  matrix  whose  main 
diagonal  elements  are  all  equal  to  1,  and  all  off-diagonal  elements  are  zero. 
For  example,  a  3x3  identity  matrix  is  written  as 

"1    0  0" 


I 


An  identity  matrix  can  be  written  as  I 
as  Kronecker's  delta,  and  defined  as 


0  1  0 
0    0  1 

lnxn  -  [Sjj]/  where  SS|  is  a  function  known 


*.=r  ifi=i 

'  l0, 


Page  10-1 


Entering  matrices  in  the  stack 

In  this  section  we  present  two  different  methods  to  enter  matrices  in  the 
calculator  stack:  (1)  using  the  Matrix  Writer,  and  (2)  typing  the  matrix  directly 
into  the  stack. 

Using  the  Matrix  Writer 

As  with  the  case  of  vectors,  discussed  in  Chapter  9,  matrices  can  be  entered 
into  the  stack  by  using  the  Matrix  Writer.  For  example,  to  enter  the  matrix: 


-2.5 
0.3 
2 


4.2  2.0 
1.9  2.8 
-0.1  0.5 


first,  start  the  matrix  writer  by  using  [  *i  Jjxraw  .  Make  sure  that  the  option 
!!&!!!->■  is  selected.  Then  use  the  following  keystrokes: 

CSS  CZDCDSS  CZDCD^™) 
At  this  point,  the  Matrix  Writer  screen  may  look  like  this: 


Press  (£N7h)  once  more  to  place  the  matrix  on  the  stack.  The  ALG  mode  stack  is 


[[-2.5,< 
[.3, 1." 
[2,-.l 

4.2,2] 
3,2.8] 
P  .5]] 

-2.54.2  2 
.3  1.92.3 
2    -.1  .5 

-2.5  4.2  2 
.3  1.92.31 
2    -.1  .5 


Page  1 0-2 


If  you  have  selected  the  textbook  display  option  (using  [mope] HI Illjlj  and  checking 
off  ^Textbook),  the  matrix  will  look  like  the  one  shown  above.  Otherwise, 
the  display  will  show: 


The  display  in  RPN  mode  will  look  very  similar  to  these. 


Note:  Details  on  the  use  of  the  matrix  writer  were  presented  in  Chapter  9 


Typing  in  the  matrix  directly  into  the  stack 

The  same  result  as  above  can  be  achieved  by  entering  the  following  directly 
into  the  stack: 

ED(L_   _  


C3  i 

CB  > 


ram  cb  !  ma®  CB  '  CX3CBCJD  CD 


CTD  CB  '  CBCBCB  CB  '  (BCD   

Thus,  to  enter  a  matrix  directly  into  the  stack  open  a  set  of  brackets  (L*i )[l  

and  enclose  each  row  of  the  matrix  with  an  additional  set  of  brackets 

(CB(^  )•  Commas  (CB  \  CB)  should  separate  the  elements  of  each 

row,  as  well  as  the  brackets  between  rows.  (Note:  In  RPN  mode,  you  can 
omit  the  inner  brackets  after  the  first  set  has  been  entered,  thus,  instead  of 
typing,  for  example,  [[1  2  3]  [4  5  6]  [7  8  9]],  type  [[1  2  3]  4  5  6  7  8  9].) 

For  future  exercises,  let's  save  this  matrix  under  the  name  A.  In  ALG  mode 

use  [s^Pj(ALPHA}(A] .  In  RPN  mode,  use  CBC™)(3  (22t)  • 


Creating  matrices  with  calculator  functions 

Some  matrices  can  be  created  by  using  the  calculator  functions  available  in 
either  the  MTH/ MATRIX/MAKE  sub-menu  within  the  MTH  menu  (CB^™-  ), 


Page  1 0-3 


HATH  HEMJ 

1 .  VECTOR..  I 

EzHZHiBL^iiiiiiiiiiiiiiiiiiiiiiiiiiM 

3.  LIST..  1 
H.  HYPERBOLIC.  [ 
S .  REAL.. 
CEASE.. 

HATRIK  HEHU 


a .  RORHALIZE.. 
3. FACTORS.. 
H.COL.. 
5.R0H.. 
G.LSG 


or  in  the  MATRICES/CR 


:ATE  menu  available  through 

HATRICES  HEAU 

It.  CREATE..  [11 

a.  OPERATION. . 
3.FACT0RI^ATI0n.. 
H. QUADRATIC  FORH..  1 
S  .  LIREAR  SYSTEHS..  B 
C.LinEAR  APPL..  |J 

The  MTH/MATRIX/MAKE  sub  menu  (let's  call  it  the  MAKE  menu)  contains  the 


HATRIK  HAHE  MEHU 

HATRIK  HAKE  MEHU 

? . GET  n 

S.GETI 

r.TRn 

5. PUT  I 

H.RDH 

1D.PUTI  1 

S . RAAH 

11. SUB  | 

6. SIZE 

HATRIK  HAKE  MEHU 


la.REPL 

13.-HHAG 

1H.DIAG+ 

15 . VAADERHOADE 

15 . HILBERT 


while  the  MATRICES/CREATE  sub-menu  (let's  call  it  the  CREATE  menu)  has  the 


HATRIK  CREATE  MEHU 

a.  ROM..  | 

3 . AUGHERT 

H.IDn 

5.  con 

fi.-*IAG  i 

HATRIK  CREATE  MEHU 

3.  GET 
5.GETI 

10.  HILBERT 

11.  PUT 
la.PUTI 

Page  1 0-4 


HATRIK  CREATE  HERU 

IH.RDH  ! 
iE.REPL  1 
16. SUB 

1? . VRRDERHORDE  f, 
IE .  MHTRIlEl"..  1 

As  you  can  see  from  exploring  these  menus  (MAKE  and  CREATE),  they  both 
have  the  same  functions  GET,  GETI,  PUT,  PUTI,  SUB,  REPL,  RDM,  RANM, 
HUBERT,  VANDERMONDE,  IDN,  CON,  ->DIAG,  and  DIAG^.  The  CREATE 
menu  includes  the  COLUMN  and  ROW  sub-menus,  that  are  also  available 
under  the  MTH/MATRIX  menu.   The  MAKE  menu  includes  the  functions  SIZE, 
that  the  CREATE  menu  does  not  include.  Basically,  however,  both  menus, 
MAKE  and  CREATE,  provide  the  user  with  the  same  set  of  functions.  In  the 
examples  that  follow,  we  will  show  how  to  access  functions  through  use  of  the 
matrix  MAKE  menu.  At  the  end  of  this  section  we  present  a  table  with  the 
keystrokes  required  to  obtain  the  same  functions  with  the  CREATE  menu  when 
system  flag  1  1  7  is  set  to  SOFT  menus. 

If  you  have  set  that  system  flag  (flag  1  1  7)  to  SOFT  menu,  the  MAKE  menu  will 
be  available  through  the  keystroke  sequence:  {^Jmjh_  \MSM 


The  functions  available  will  be  shown  as  soft-menu  key  labels  as  follows  (press 
jNxrj  to  move  to  the  next  set  of  functions): 


nttj«B4a«BifirHfia;Hiffin;m*iJ 

GETMET^mT 

■Hiuimiaifltti 

 IN^^rtHM 

With  system  flag  1  1  7  set  to  SOFT  menus,  the  functions  of  the  CREATE  menu, 


HilffllMHMUMfliMBaiMCKE 

■  DIHG-H  GET  |  GETI  |HILBE|  PUT  |  PUTI 

In  the  next  sections  we  present  applications  of  the  matrix  functions  in  the 
MAKE  and  CREATE  menu. 


Page  1 0-5 


Functions  GET  and  PUT 

Functions  GET,  GETI,  PUT,  and  PUTI,  operate  with  matrices  in  a  similar 
manner  as  with  lists  or  vectors,  i.e.,  you  need  to  provide  the  location  of  the 
element  that  you  want  to  GET  or  PUT.  However,  while  in  lists  and  vectors  only 
one  index  is  required  to  identify  an  element,  in  matrices  we  need  a  list  of  two 
indices  {row,  column}  to  identify  matrix  elements.    Examples  of  the  use  of 
GET  and  PUT  follow. 

Let's  use  the  matrix  we  stored  above  into  variable  A  to  demonstrate  the  use  of 
the  GET  and  PUT  functions.  For  example,  to  extract  element  a23  from  matrix  A, 
in  ALG  mode,  can  be  performed  as  follows: 


:  GET(Fl,{2  3H 

2.8 

:  Fl(2,3) 

2.  8 

Notice  that  we  achieve  the  same  result  by  simply  typing  FK2p  3)  and 
pressing  [enter)  .  In  RPN  mode,  this  exercise  is  performed  by  entering  [enter 

[enter)  GET,  or  by  using  FK2?  3)  [enter). 

Suppose  that  we  want  to  place  the  value  V  into  element  a31  of  the  matrix. 
We  can  use  function  PUT  for  that  purpose,  e.g., 


PUT(FI,{3  1>,tt) 

T-2.5  4.2  2 
.3  1.92.31 

TT  -.1.5 


GET    GETI    PUT    PUTI    SUE  P.EPL 


In  RPN  mode  you  can  use:  C^D  EB  <  3p  1  >  (^m)  (jT]7r   PUT. 

Alternatively,  in  RPN  mode  you  can  use:   LlJFK2  ?  3>  [enter)  (TroT)  . 

To  see  the  contents  of  variable  A  after  this  operation,  use  Ell. 

Functions  GETI  and  PUTI 

Functions  PUTI  and  GETI  are  used  in  UserRPL  programs  since  they  keep  track 
of  an  index  for  repeated  application  of  the  PUT  and  GET  functions.  The 
index  list  in  matrices  varies  by  columns  first.   To  illustrate  its  use,  we  propose 
the  following  exercise  in  RPN  mode:  IIIIIIII   {2,2}[enter)  GETI.  Screen  shots 
showing  the  RPN  stack  before  and  after  the  application  of  function  GETI  are 
shown  below: 


Page  1 0-6 


Notice  that  the  screen  is  prepared  for  a  subsequent  application  of  GETI  or 
GET,  by  increasing  the  column  index  of  the  original  reference  by  1,  (i.e.,  from 
{2,2}  to  {2,3}),  while  showing  the  extracted  value,  namely  A(2,2)  =  1 .9,  in 
stack  level  1 . 


Now,  suppose  that  you  want  to  insert  the  value  2  in  element  {3  1}  using  PUTI. 
Still  in  RPN  mode,  try  the  following  keystrokes:  C5D  SK  3  I  >  [enter}  [TD 
(enter)  PUTI .  The  screen  shots  below  show  the  RPN  stack  before  and  after  the 
application  of  function  PUTI: 


In  this  case,  the  2  was  replaced  in  position  {3  1},  i.e.,  now  A(3,l)  =  2,  and 
the  index  list  was  increased  by  1  (by  column  first),  i.e.,  from  {3,1}  to  {3,2}. 
The  matrix  is  in  level  2,  and  the  incremented  index  list  is  in  level  1 . 


Function  SIZE 

Function  SIZE  provides  a  list  showing  the  number  of  rows  and  columns  of  the 
matrix  in  stack  level  1 .  The  following  screen  shows  a  couple  of  applications  of 
function  SIZE  in  ALG  mode: 


SIZECm 

<3.  3 
{£.  2.) 


In  RPN  mode,  these  exercises  are  performed  by  using  ! 

EC  1,2],  C3,4]]  (enter)  SIZE  . 


and 


Page  1 0-7 


Function  TRN 

Function  TRN  is  used  to  produce  the  transconjugate  of  a  matrix,  i.e.,  the 
transpose  (TRAN)  followed  by  its  complex  conjugate  (CONJ).  For  example, 
the  following  screen  shot  shows  the  original  matrix  in  variable  A  and  its 
transpose,  shown  in  small  font  display  (see  Chapter  1): 


\-2.5--2.5>i>2  H.a-H.a.i-2  2-2>i>2  - 
jl      2-2-i-a       -.1— .5- .5>i>2. 


:H-i.2.H 

-a.s—a.s.i-a  H.a-H.a-i-a  a-a-i-a 

.3-.3.i-a  i.S-i.S-i-a  n-n-i-a  * 

a-a-i-a  -.i—.i-i-a  .5-.E-i.a 

:TRntfl-i-a.fl) 

-a.5~a.5.-i-a  .s-.s-i-a  a-a-- 
H.a-H.a-i-a  i.s-i.s-i-a  -.1— .> 
a-a-i-a       n-n-i-a     .5-. 5 


If  the  argument  is  a  real  matrix,  TRN  simply  produces  the  transpose  of  the  real 
matrix.  Try,  for  example,  TRN(A),  and  compare  it  with  TRAN(A). 

In  RPN  mode,  the  transconjugate  of  matrix  A  is  calculated  by  using  BMTRH. 

Note:  The  calculator  also  includes  Function  TRAN  in  the 
MATRICES/OPERATIONS  sub-menu: 


HATRICES  HEHU 


1 .  CREATE.. 


3 .  FftCTORiaflTIOn.. 
H .  QUADRATIC  FORH.. 

5 .  LiriEHF;  SVSTEHS.. 

6 .  LIHEhF;  AFFL. 


HATRIK  OPERATION  HEAU 

13.SI.aE  r 

iH.SD.RH 

15. S RAD 

lg.TRAC^^^^^^^ 

i^ha'trTc^ 

For  example,  in  ALG  mode: 


:  TRRHCRi 

[-2.5  .3    2  1 

4-2  1.9  -.  1 

I     2        TT      -  5  J 

Function  CON 

The  function  takes  as  argument  a  list  of  two  elements,  corresponding  to  the 
number  of  row  and  columns  of  the  matrix  to  be  generated,  and  a  constant 


Page  1 0-8 


value.  Function  CON  generates  a  matrix  with  constant  elements.  For 
example,  in  ALG  mode,  the  following  command  creates  a  4x3  matrix  whose 
elements  are  all  equal  to  -1 .5:  


: C0NK4  3 

^-1.5) 

-1.5-1.5-1.51 
-1.5-1.5-1.5 
-1.5-1.5-1.5 
-1.5  -1.5  -l.sJ 

In  RPN  mode  this  is  accomplished  by  using  i  4».  3>  [enter]  CDCZDCS  (JtJ 
(enter)  CON. 

Function  IDN 

Function  IDN  (IDeNtity  matrix)  creates  an  identity  matrix  given  its  size.  Recall 
that  an  identity  matrix  has  to  be  a  square  matrix,  therefore,  only  one  value  is 
required  to  describe  it  completely.  For  example,  to  create  a  4x4  identity 
matrix  in  ALG  mode  use: 


:  IDHC43 

[1  0  0  01 

0  10  0 

0  0  10 

.0  0  0  lJ 

You  can  also  use  an  existing  square  matrix  as  the  argument  of  function  IDN, 


ri  0  0i 

0  1  0 

10  0  lJ 

The  resulting  identity  matrix  will  have  the  same  dimensions  as  the  argument 
matrix.  Be  aware  that  an  attempt  to  use  a  rectangular  (i.e.,  non-square) 
matrix  as  the  argument  of  IDN  will  produce  an  error. 

In  RPN  mode,  the  two  exercises  shown  above  are  created  by  using:  14  J[enter) 
IDN  and  111  IDN. 


Page  1 0-9 


Function  RDM 

Function  RDM  (Re-DiMensioning)  is  used  to  re-write  vectors  and  matrices  as 
matrices  and  vectors.  The  input  to  the  function  consists  of  the  original  vector 
or  matrix  followed  by  a  list  of  a  single  number,  if  converting  to  a  vector,  or 
two  numbers,  if  converting  to  a  matrix.  In  the  former  case  the  number 
represents  the  vector's  dimension,  in  the  latter  the  number  of  rows  and 
columns  of  the  matrix.  The  following  examples  illustrate  the  use  of  function 
RDM: 

Re-dimensioning  a  vector  into  a  matrix 

The  following  example  shows  how  to  re-dimension  a  vector  of  6  elements  into 
a  matrix  of  2  rows  and  3  columns  in  ALG  mode: 


In  RPN  mode,  we  can  use  C  1  ?  2«.  3*.  4*.     6  "I  [enter]  {  ?«.  3>  [enter]  RDM  to 
produce  the  matrix  shown  above. 

Re-dimensioning  a  matrix  into  another  matrix 

In  ALG  mode,  we  now  use  the  matrix  created  above  and  re-dimension  it  into  a 
matrix  of  3  rows  and  2  columns: 


In  RPN  mode,  we  simply  use  i  3«.  £> [enter]  RDM. 
Re-dimensioning  a  matrix  into  a  vector 

To  re-dimension  a  matrix  into  a  vector,  we  use  as  arguments  the  matrix 
followed  by  a  list  containing  the  number  of  elements  in  the  matrix.  For 
example,  to  convert  the  matrix  from  the  previous  example  into  a  vector  of 
length  6,  in  ALG  mode,  use: 


Page  10-10 


:RDM(flNS(l)/£3  2H 


14  -  bJ 


If  using  RPN  mode,  we  assume  that  the  matrix  is  in  the  stack  and  use  i 


Note:  Function  RDM  provides  a  more  direct  and  efficient  way  to  transform  lists 
to  arrays  and  vice  versa,  than  that  provided  at  the  end  of  Chapter  9. 

Function  RANM 

Function  RANM  (RANdom  Matrix)  will  generate  a  matrix  with  random  integer 
elements  given  a  list  with  the  number  of  rows  and  columns  (i.e.,  the 
dimensions  of  the  matrix).  For  example,  in  ALG  mode,  two  different  2x3 
matrices  with  random  elements  are  produced  by  using  the  same  command, 
namely,  RflNM<<  2~. '. 


:  RHNMK2  3>i 

[-5  -7  -91 

12  5  sJ 

:  RHNMK2  3>i 

[-4  9  41 

i-9  -5  SJ 

In  RPN  mode,  use  {2,3}  (enter}  RflHM. 

Obviously,  the  results  you  will  get  in  your  calculator  will  most  certainly  be 
different  than  those  shown  above.  The  random  numbers  generated  are 
integer  numbers  uniformly  distributed  in  the  range  [-10,10],  i.e.,  each  one  of 
those  21  numbers  has  the  same  probability  of  being  selected.  Function 
RANM  is  useful  for  generating  matrices  of  any  size  to  illustrate  matrix 
operations,  or  the  application  of  matrix  functions. 

Function  SUB 

Function  SUB  extracts  a  sub-matrix  from  an  existing  matrix,  provided  you 
indicate  the  initial  and  final  position  of  the  sub-matrix.  For  example,  if  we 


Page  10-11 


want  to  extract  elements  a12/  a13/  a22/  and  a23  from  the  last  result,  as  a  2x2 
sub-matrix,  in  ALG  mode,  use: 


:  RRNMK2  3>i 


[-4  9  41 

SUB(flNS(l)/£l  2W2  3H 

[9  41 
1-5  SJ 


In  RPN  mode,  assuming  that  the  original  2x3  matrix  is  already  in  the  stack, 

use  {  I  :s  2>  {enter}  ■;;  2  j  3>  {enter}  SUB. 


Function  REPL 

Function  REPL  replaces  or  inserts  a  sub-matrix  into  a  larger  one.  The  input  for 
this  function  is  the  matrix  where  the  replacement  will  take  place,  the  location 
where  the  replacement  begins,  and  the  matrix  to  be  inserted.  For  example, 
keeping  the  matrix  that  we  inherited  from  the  previous  example,  enter  the 
matrix:  L  L  1  ;s    :■  3  J    L  4  :s  5  :s  6  J  :■  C  7 8  ?  9  J  J  .  In  ALG  mode,  the  following 
screen  shot  to  the  left  shows  the  new  matrix  before  pressing  [enter}  .    The  screen 
shot  to  the  right  shows  the  application  of  function  RPL  to  replace  the  matrix  in 
HNS<2>,  the  2x2  matrix,  into  the  3x3  matrix  currently  located  in  RNS<  1  >, 
starting  at  position  i      2>  : 


:SUBCRHSC1W1  £>,{£  3» 

[9  4 
1-5  S 

1  2  3 
4  5  6 

7  S  9. 

krl^IHHljl^lillUKHI^jd 


If  working  in  the  RPN  mode,  assuming  that  the  2x2  matrix  was  originally  in 
the  stack,  we  proceed  as  follows: 

C  C  1  ;i  2  :i  3  "I  :=  C  4 5 6  "I ::  C  7  :=  3  :s  9  "I  "I  {enter} (T)  (this  last  key  swaps  the 
contents  of  stack  levels  1  and  2)  -i!      2>  {enter}        (another  swapping  of 
levels  1  and  2)  REP!  


Page  10-12 


Function  -^DIAG 

Function  ->DIAG  takes  the  main  diagonal  of  a  square  matrix  of  dimensions 
nxn,  and  creates  a  vector  of  dimension  n  containing  the  elements  of  the  main 
diagonal.  For  example,  for  the  matrix  remaining  from  the  previous  exercise, 


ir1 

:REPL(flNS(l)/£2  2>,f 

=INS(2H 

2  31 

II. 

9  4 

-5  SJ 

:+DIflG(flNS(lH 

[1 

9  S] 

In  RPN  mode,  with  the  3x3  matrix  in  the  stack,  we  simply  have  to  activate 
function  ->DIflG  to  obtain  the  same  result  as  above. 


Function  DIAG^ 

Function  DIAG->  takes  a  vector  and  a  list  of  matrix  dimensions  {rows, 

columns},  and  creates  a  diagonal  matrix  with  the  main  diagonal  replaced 

with  the  proper  vector  elements.  For  example,  the  command 
I  f\Q     (  ["  ;{  ..  ■■■■■  ;|  ..    ..  3  ]  „  \  3 ..  "»:;  )■  j 

produces  a  diagonal  matrix  with  the  first  3  elements  of  the  vector  argument: 


L'lHLjHHLl  -1  v  :=:_!,■:.:=:  :=:.:■  i 

"1  0  01 
10  -1  0l 
L0  0  2J 


In  RPN  mode,  we  can  use  L  1  ?  - 1  *  2  *  3 1  (enter)  ■■:  3  ;s  3  }  (enter)  D I  AG  to 
obtain  the  same  result  as  above. 


Another  example  of  app 

ication  of  the  DIAG->  function  follows,  in  ALG  mode: 

iDlHU-HLl  ^4  1 

[1  01 

0  2l 

J00J 

In  RPN  mode,  use  L  1 2  :■  3  :■  4  ■■  5  "I  [enter)  ■{  3 2  >  [enter)  D I  AG  ->  . 


In  this  case  a  3x2  matrix  was  to  be  created  using  as  main  diagonal  elements 
as  many  elements  as  possible  form  the  vector  [1,2,3,4,5].  The  main  diagonal, 
for  a  rectangular  matrix,  starts  at  position  (1,1)  and  moves  on  to  position  (2,2), 
(3,3),  etc.  until  either  the  number  of  rows  or  columns  is  exhausted.  In  this 
case,  the  number  of  columns  (2)  was  exhausted  before  the  number  of  rows  (3), 


Page  10-13 


so  the  main  diagonal  included  only  the  elements  in  positions  (1,1)  and  (2,2). 
Thus,  only  the  first  two  elements  of  the  vector  were  required  to  form  the  main 
diagonal. 


Function  VANDERMONDE 

Function  VANDERMONDE  generates  the  Vandermonde  matrix  of  dimension  n 
based  on  a  given  list  of  input  data.  The  dimension  n  is,  of  course,  the  length 
of  the  list.    If  the  input  list  consists  of  objects  {x]f  x2/...  xn},  then,  a 
Vandermonde  matrix  in  the  calculator  is  a  matrix  made  of  the  following 
elements: 


1 

1 

1 

1 

x„ 

4  ■ 

•■  x"~ 

For  example,  the  following  command  in  ALG  mode  for  the  list  {1,2,3,4}: 


:VRHDERM0HDEC{1  2  3  4» 

[1  1 

1 

1  1 

1  2 

4 

1  3 

9 

27 

.1  4 

16 

64J 

In  RPN  mode,  enter  <  3.  ;i  £?  3?  4>  (^D  VANDERMONDE. 
Function  HUBERT 

Function  HUBERT  creates  the  Hilbert  matrix  corresponding  to  a  dimension  n. 
By  definition,  the  nxn  Hilbert  matrix  is  Hn  =  [hjk]nXn,  so  that 


Page  10-14 


The  Hilbert  matrix  has  application  in  numerical  curve  fitting  by  the  method  of 
linear  squares. 

A  program  to  build  a  matrix  out  of  a  number  of  lists 

In  this  section  we  provide  a  couple  of  UserRPL  programs  to  build  a  matrix  out 
of  a  number  of  lists  of  objects.  The  lists  may  represent  columns  of  the  matrix 
(program  H333)  or  rows  of  the  matrix  (program  B333).  The  programs  are 
entered  with  the  calculator  set  to  RPN  mode,  and  the  instructions  for  the 
keystrokes  are  given  for  system  flag  1  17  set  to  SOFT  menus.  This  section  is 
intended  for  you  to  practice  accessing  programming  functions  in  the  calculator. 
The  programs  are  listed  below  showing,  in  the  left-hand  side,  the  keystrokes 
necessary  to  enter  the  program  steps,  and,  in  the  right-hand  side,  the 
characters  entered  in  the  display  as  you  perform  those  keystrokes.  First,  we 
present  the  steps  necessary  to  produce  program  CRMC. 

Lists  represent  columns  of  the  matrix 

The  program  13339  allows  you  to  put  together  a  pxn  matrix  (i.e.,  p  rows,  n 
columns)  out  of  n  lists  of  p  elements  each.  To  create  the  program  enter  the 
following  keystrokes: 

Keystroke  sequence:  Produces: 


Q=T]  _«o> 


THEN 

M  + 
ROLL 
END 


« 


< 


OBJ^ 

^ARRY 

IF 


DUP 

^  n 


1  SWAP 
FOR 


n 


Page  10-15 


05?G_i3aaa  11:111  mm 

NEXT 

C535?c_liaaa  III  lilii 

IF 

S  ED®  C^DCD 

n  1 

rgnfflG  nan  bum 

> 

C5D^_!!l        !!  nan 

THEN 

1 

(alpha)  C^3CDC3 

n  1  - 

L*~l J ^               iJi:!:!.L!!!!  ILiMJJj!! 

FOR 

i 

n  + 

rgn^  siaaa  r^nsm 

ROLL 

J  ,               ::.:::.:.::::•.:.::  ::.:::::::.:.:••• 

NEXT 

(jTJ^G    mm  fTI!  Pi i ill 

END 

[alpha)  (  <n  j  (n\  jspcj 

n 

r<T)MTH  mum  llSEll  bct-» 

COL^ 

(^)  Program  is  displayed  in  level  1 


To  save  the  program:  ^^(a^(a^(Q(^(m\(c](a^ha)  {sfo>) 

Note:  if  you  save  this  program  in  your  HOME  directory  it  will  be  available 
from  any  other  sub-directory  you  use. 


To  see  the  contents  of  the  program  use  L^J  C5DB333.  The  program  listing  is 
the  following: 

■:!::  D!  IP  _»  !■■  1  !.=.! H P  PHP  i  i_>  -^HPPV  TP  i  n  •(  TI-IPN  i  1  ■■!■■ 
ROLL  END  NEXT  IF  n  l'  >""tHEN  1  n'f-  FOR  A  j  1 ' h- "ROLL." 


!■■.!!»"':■:'"!''  !" !■■.!!"■■  \"\"\\ 


To  use  this  program,  in  RPN  mode,  enter  the  n  lists  in  the  order  that  you  want 
them  as  columns  of  the  matrix,  enter  the  value  of  n,  and  press  ESS.  As  an 
example,  try  the  following  exercise: 

■;;  1  ,  v,  3,  4>  (enter)  i  :!. ::  4,  9,  16>  (enter)  ■:;  1  :s  8:5  27,  64 }  (enter)  3  (enter)  m$\m 


The  following  screen  shots  show  the  RPN  stack  before  and  after  running 
program  illlH: 


J- 

4: 

{1  2  3  4} 

3: 

{14  9  16) 

£: 

{1  S  27  64) 

l: 

3 

l: 

[1  1    1  " 

£43 

3  9  27 

.4  16  64. 

Page  10-16 


To  use  the  program  in  ALG  mode,  press  ill!!!  followed  by  a  set  of  parentheses 

(1*1  YJ.  ).    Within  the  parentheses  type  the  lists  of  data  representing  the 

columns  of  the  matrix,  separated  by  commas,  and  finally,  a  comma,  and  the 
number  of  columns.     The  command  should  look  like  this: 


■     ::-:!::     :    ■:  : 


■    ..  ::.:.  ..         ..     :  :-. 


The  ALG  screen  showing  the  execution  of  program  CRMC  is  shown  below: 


:CRMC(U  2  3  4M1 

4  9  16>,> 

[111 

2  4  S 

3  9  27 
.4  16  64. 

HYP  | H 'I ij L" 2 1 H L" I Tl □  | H L" I Tl 2 1 H T M Tl 2 1 H H L F T 

Lists  represent  rows  of  the  matrix 

The  previous  program  can  be  easily  modified  to  create  a  matrix  when  the 
input  lists  will  become  the  rows  of  the  resulting  matrix.  The  only  change  to  be 
performed  is  to  change  COL^  for  ROW^  in  the  program  listing.  To  perform 
this  change  use: 


OS 

3D  3D  3D  

[alpha)  [alpha]  (r\  (q\  (wj  [alpha]  (enter) 


List  program  CRMC  in  stack 
Move  to  end  of  program 
Delete  COL 

Type  in  ROW,  enter  program 


To  store  the  program  use:  ^^(^h^(^ha)(c)(r\(m\(r\(^ha)  C^D 


■;;  1  ,  v,  3,  4>  (enter)  •;  1  |S  4:;  9S  16>  (enter)  •;  1  i5  S,  27  j  64  >  [enter]  3  (enter]  MEM 
The  following  screen  shots  show  the  RPN  stack  before  and  after  running 
program  lajiLLu: 


4: 
3: 
2: 
l: 


U.  2.  3.  4-> 
{1.  2-  9.  16. > 
{1.  3.  27.  64. > 
3. 


l: 

2. 

3. 

4-  i 

0: 

2. 

9. 

16.  1 

3. 

27. 

64.  J 

These  programs  can  be  useful  for  statistical  applications,  specifically  to  create 
the  statistical  matrix  ZDAT.  Examples  of  the  use  of  these  program  are  shown 
in  a  latter  chapters. 


Page  10-17 


Manipulating  matrices  by  columns 

The  calculator  provides  a  menu  with  functions  for  manipulating  matrices  by 
operating  in  their  columns.  This  menu  is  available  through  the 
MTH/MATRIX/COL.  sequence:  (C5D^-  )  shown  in  the  figure  below  with 
system  flag  1  1  7  set  to  CHOOSE  boxes: 


HATH  HEnU 

i .  VECTOR..  i 

3.  LIST..  i 
H. HYPERBOLIC. 

5 .  REAL.. 

6.  BASE.. 

HATRIK  HEnU 

i.HAHE..  ij. 
a.nORHALIZE.. 
3.FACT0RS..  I 

5.  R0H.. 

6.  LSG 

or  through  the  MATRICES/CREATE/COLUMN  sub-menu: 


HATRICES  HEnU 


a.OPERATIOnS.. 
3. FACTORISATION. 
H .  QUADRATIC  FORH.. 

5 .  LIREAR  SYSTEHS.. 

6 .  LIREAR  APFL.. 


HATRIK  CREATE  HEMJ 

a.ROH..  1 

3 . AUGHEnT 

H.IDn 

5.  con 
c.-hhag 

Droaches  will  show  the  same 

unctions: 

CREATE  COL  HEMJ 

CREATE  COL  HEHU  [ 

3.C0L+ 
H.COL- 
5.CSHP 
fi. HATRIK.. 

3.C0L+ 
H.COL- 
5.CSHP 
fi.  CREATE.. 

When  system  flag  1  1  7  is  set  to  SOFT  menus,  the  COL  menu  is  accessible 
through  [jhJmth  WmMMIISM  W32M  ,  or  through  rfT\ matrices  flMIi  lllllSulll  Both 


approaches  will  show  the  same  set  o 


c:<nHi]aHi]aH]gKiiii!Tiit 


functions: 


The  operation  of  these  functions  is  presented  below. 


Function  -^COL 

Function  -^COL  takes  as  argument  a  matrix  and  decomposes  it  into  vectors 
corresponding  to  its  columns.  An  application  of  function  ->COL  in  ALG  mode 
is  shown  below.  The  matrix  used  has  been  stored  earlier  in  variable  A.  The 
matrix  is  shown  in  the  figure  to  the  left.  The  figure  to  the  right  shows  the  matrix 


Page  10-18 


decomposed  in  columns.  To  see  the  full  result,  use  the  line  editor  (triggered 
by  pressing  ^7 ).  


:fl 

-2.5  4.2  2. 

.3  1.92.81 
_  2.    -.1  .5 

:  +COLCFI) 

<[-2.5  .3  2.]  [4.2  1. 


T 


.5 

.9  - 


To  I 
2.  -.1 

-^coLcni 

tt-2.5  .3  2.]  [4.2  1 
£[-2.5? .3,2. ]? 
[4.2, 1.9,-. 1], 
[ 2 . ,2.3, .511, 3.  y  , 
ffaitiiHitiieEgiiigiqi'JiMtua 


In  RPN  mode,  you  need  to  list  the  matrix  in  the  stack,  and  the  activate  function 
->COL,  i.e.,  IIIIIIII  ->COL.  The  figure  below  shows  the  RPN  stack  before  and 


after  the  application  of  function  ->COL. 


4: 
3: 
2: 
l: 


-2.5  4.2  2. 
.3  1.92.31 
2.    -.1  .5 


P1 
£: 
5: 

4:  [-2.5.32.] 
3:  [4.21.9-.1] 
2:  [2.  2.3  .5] 

1 '  3 


In  this  result,  the  first  column  occupies  the  highest  stack  level  after 
decomposition,  and  stack  level  1  is  occupied  by  the  number  of  columns  of  the 
original  matrix.  The  matrix  does  not  survive  decomposition,  i.e.,  it  is  no 
longer  available  in  the  stack. 


Function  COL^ 

Function  COL^  has  the  opposite  effect  of  Function  -^COL,  i.e.,  given  n 
vectors  of  the  same  length,  and  the  number  n,  function  COL->  builds  a  matrix 
by  placing  the  input  vectors  as  columns  of  the  resulting  matrix.  Here  is  an 
example  in  ALG  mode.  The  command  used  was: 

COL  ->  (  L  I ■■■ !■::!  ■■■  3 1 »  L  4  ■■■  3  ■■■  6  J  »  L  7  S  9  J  3  ) 


:C0L-K[1.  2.  3.]I4. 

s! 

6. 

In  RPN  mode,  place  the  n  vectors  in  stack  levels  n+1,  n,  n-l,...,2,  and  the 
number  n  in  stack  level  1 .  With  this  set  up,  function  COL->  places  the  vectors 


Page  10-19 


as  columns  in  the  resulting  matrix.  The  following  figure  shows  the  RPN  stack 
before  and  after  using  function  COL->. 


4: 

[1. 

2. 

3.] 

3: 

[4. 

5. 

6.] 

2: 

[7. 

3. 

9.] 

l: 

3. 

2: 

l: 

4. 

5. 

3.  1 

L3. 

6. 

9j 

Function  COL+ 

Function  COL+  takes  as  argument  a  matrix,  a  vector  with  the  same  length  as 
the  number  of  rows  in  the  matrix,  and  an  integer  number  n  representing  the 
location  of  a  column.  Function  COL+  inserts  the  vector  in  column  n  of  the 
matrix.  For  example,  in  ALG  mode,  we'll  insert  the  second  column  in  matrix  A 
with  the  vector  [-1,-2,-3],  i.e., 


In  RPN  mode,  enter  the  matrix  first,  then  the  vector,  and  the  column  number, 
before  applying  function  COL+.  The  figure  below  shows  the  RPN  stack  before 


4: 
2: 

l: 

[-2.5  4-2  2.  1 
■  3  1-92.3 
.  2.    -.1  .5J 
C-l.  -2.  -3.] 
2. 

4: 

3: 

2: 

l: 

[-2.5 

-1. 

4.2  2.  1 

.3 

-2. 

1.9  2.  Si 

.  2. 

-3. 

-.1  .5j 

Function  COL- 

Function  COL-  takes  as  argument  a  matrix  and  an  integer  number  representing 
the  position  of  a  column  in  the  matrix.  Function  returns  the  original  matrix 
minus  a  column,  as  well  as  the  extracted  column  shown  as  a  vector.  Here  is 
an  example  in  the  ALG  mode  using  the  matrix  stored  in  A: 


:C0L-CR,3.) 
1-2.5  4.2 


[[-2.5  4.2]  I 
||  ^3    l.|j  [2.  2.3  .5]j 


Page  1 0-20 


In  RPN  mode,  place  the  matrix  in  the  stack  first,  then  enter  the  number 
representing  a  column  location  before  applying  function  COL-.  The  following 
figure  shows  the  RPN  stack  before  and  after  applying  function  COL- 


Function  CSWP 

Function  CSWP  (Column  SWaP)  takes  as  arguments  two  indices,  say,  i  and  j, 
(representing  two  distinct  columns  in  a  matrix),  and  a  matrix,  and  produces  a 
new  matrix  with  columns  i  and  j  swapped.    The  following  example,  in  ALG 
mode,  shows  an  application  of  this  function.  We  use  the  matrix  stored  in 


L  Z. 

-.     1           .    !D  J 

[-2.5 

2.  4.21 

1 

2.S  1.9 

.2? 

.5  -.lJ 

In  RPN  mode,  function  CSWP  lets  you  swap  the  columns  of  a  matrix  listed  in 
stack  level  3,  whose  indices  are  listed  in  stack  levels  1  and  2.  For  example, 
the  following  figure  shows  the  RPN  stack  before  and  after  applying  function 
CSWP  to  matrix  A  in  order  to  swap  columns  2  and  3: 


As  you  can  see,  the  columns  that  originally  occupied  positions  2  and  3  have 
been  swapped.  Swapping  of  columns,  and  of  rows  (see  below),  is  commonly 
used  when  solving  systems  of  linear  equations  with  matrices.  Details  of  these 
operations  will  be  given  in  a  subsequent  Chapter. 

Manipulating  matrices  by  rows 

The  calculator  provides  a  menu  with  functions  for  manipulating  matrices  by 
operating  in  their  rows.  This  menu  is  available  through  the 


Page  10-21 


MTH/MATRIX/ROW. .  sequence:  (QD^™-  )  shown  in  the  figure  below  with 


system  f 


^9 


1  1 7  set  to  CHOOSE  boxes: 


HATH  HEMJ 

i.VECTOR..  1 

\2 .  HATRI 

mm 

3.  LIST.. 

H. HYPERBOLIC.  ? 
S .  REAL.. 

G .  EASE..  L 

or  through  the  MATRICES/CREATE/ROW  sub-menu 


HATRIK  HEnU 

i.HAHE..  1 

a.nORHALISE.. 

3. FACTORS..  > 

H.COL.. 

G.LSG 

HATRICES  HEnU 


a.OPERATIOnS.. 
3. FACTORIZATION. 
H.  QUADRATIC  FORM.. 
S .  LIREAR  SYSTEHS.. 
G.LinEAR  AFPL.. 


Both  approaches  will  show  the  same 


HATRIK  CREATE  HEMJ 

3 . AUGHEhT 
H.IDn 

5.  con 

G.-4IAG  I 

CREATE  ROM  HEHU 


a.ROH-n 

3.R0H+ 

H.ROH- 

5.RCI 

G.RCIJ 


unctions: 


CREATE  RON  HEHU 

3.R0H-* 

3.R0H+ 

H.ROH- 

5.RCI 

G.RCIJ 

When  system  flag  1  1  7  is  set  to  SOFT  menus,  the  ROW  menu  is  accessible 


through  [jT}mw 


••H!:!::"""n"  iiFlv""":::::  iillr:5i-"i"i"ii 

!.s j.-.l JL-Jll  II: J.! j.illli  IHII.lIlslsLi-lll 


or  through  [^matrices 


Both 


approaches  will  show  the  same  set  of  functions: 


The  operation  of  these  functions  is  presented  below. 

Function  -^ROW 

Function  ^ROW  takes  as  argument  a  matrix  and  decomposes  it  into  vectors 
corresponding  to  its  rows.  An  application  of  function  ->ROW  in  ALG  mode  is 
shown  below.  The  matrix  used  has  been  stored  earlier  in  variable  A.  The 
matrix  is  shown  in  the  figure  to  the  left.  The  figure  to  the  right  shows  the  matrix 
decomposed  in  rows.  To  see  the  full  result,  use  the  line  editor  (triggered  by 
pressing  ^3? ). 


Page  10-22 


:fl 

f-2.5  4. 

2  2.  1 

■3  1. 

9  2.8 

l  2.  -. 

1  .5] 

:  +RCMfl) 

{[-2.5  4.2  2.]  [.3  1 

.9  2.  ► 

I  2.    -.1  .5 

■*ROH(fl) 
{[-2.5  4.2  2.]  [.  3  1.9  2.  H 
4C-2.5i4.2i2.], 
[.3, 1.9,2.8], 
[2. ,-. 1, .5], 3. > 


In  RPN  mode,  you  need  to  list  the  matrix  in  the  stack,  and  the  activate  function 
^ROW,  i.e.,  ill  ^ROW.  The  figure  below  shows  the  RPN  stack  before 


and  after  the  application  of  function  ->ROW. 


[-2.5  4.2  2.] 
C.3  1.9  2.3] 
[2.  -.1  .5] 

3 


In  this  result,  the  first  row  occupies  the  highest  stack  level  after  decomposition, 
and  stack  level  1  is  occupied  by  the  number  of  rows  of  the  original  matrix. 
The  matrix  does  not  survive  decomposition,  i.e.,  it  is  no  longer  available  in  the 
stack. 


Function  ROW-> 

Function  ROW^  has  the  opposite  effect  of  the  function  -^ROW,  i.e.,  given  n 
vectors  of  the  same  length,  and  the  number  n,  function  ROW->  builds  a 
matrix  by  placing  the  input  vectors  as  rows  of  the  resulting  matrix.  Here  is  an 
example  in  ALG  mode.  The  command  used  was: 

:  :  _V       :"  ■■  "":  ":        :"  .■:       ■::"      .■"  ":        : ?      sS      s":  ":       "":  "■. 

!":. !■'■!  ~7  s-.  !..  .!.  ;■      ;■      ..!  ;■  !..  ~v  ;■  ..J  ;■  D  ..!     !..     ;■  O  ji  ::;!  ..I     Z-  .-! 


:R0N+(C1.  2.  3.] 

14.  5.  6* 
1.  2.  3.1 
4.  5.  6. 
.7.  3.  9. J 

In  RPN  mode,  place  the  n  vectors  in  stack  levels  n+1,  n,  n-l,...,2,  and  the 
number  n  in  stack  level  1 .  With  this  set  up,  function  ROW->  places  the 
vectors  as  rows  in  the  resulting  matrix.  The  following  figure  shows  the  RPN 
stack  before  and  after  using  function  ROW->. 


Page  10-23 


4: 
3: 
2: 
l: 


CI.  2.  3.] 
C4.  5.  6.] 
[7.  3.  9.] 
3. 


1- 

§! 

II; 

3. 

Function  ROW+ 

Function  ROW+  takes  as  argument  a  matrix,  a  vector  with  the  same  length  as 
the  number  of  rows  in  the  matrix,  and  an  integer  number  n  representing  the 
location  of  a  row.  Function  ROW+  inserts  the  vector  in  row  n  of  the  matrix. 
For  example,  in  ALG  mode,  we'll  insert  the  second  row  in  matrix  A  with  the 
vector  [-1  ,-2,-3],  i.e., 


:R0N+(FI,[-1 

.  -2.  -3.],2.:i 
-2.5  4.2  2. 
-1.  -2.  -3. 
.3  1.92.3 
.  2.    -.1  .5. 

In  RPN  mode,  enter  the  matrix  first,  then  the  vector,  and  the  row  number, 
before  applying  function  ROW+.  The  figure  below  shows  the  RPN  stack 
before  and  after  applying  function  ROW+. 


4-- 

3: 
2: 
l: 

[-2.5-1.  4.2  2.  1 
.3   -2.  1.92.81 
.  2.    -3.  -.  1 

Function  ROW- 

Function  ROW-  takes  as  argument  a  matrix  and  an  integer  number 
representing  the  position  of  a  row  in  the  matrix.  The  function  returns  the 
original  matrix,  minus  a  row,  as  well  as  the  extracted  row  shown  as  a  vector. 
Here  is  an  example  in  the  ALG  mode  using  the  matrix  stored  in  A: 


R0N-CR.3.) 

-2.5  ^.2  2.  1[2   _  t  > 
■  3    1.9  2.  SJ         "  1  "  - 


In  RPN  mode,  place  the  matrix  in  the  stack  first,  then  enter  the  number 
representing  a  row  location  before  applying  function  ROW-.  The  following 
figure  shows  the  RPN  stack  before  and  after  applying  function  ROW-. 


Page  1 0-24 


Function  RSWP 

Function  RSWP  (Row  SWaP)  takes  as  arguments  two  indices,  say,  i  and  j, 
(representing  two  distinct  rows  in  a  matrix),  and  a  matrix,  and  produces  a 
new  matrix  with  rows  i  and  j  swapped.    The  following  example,  in  ALG 
mode,  shows  an  application  of  this  function.  We  use  the  matrix  stored  in 
variable  A  for  the  example   This  matrix  is  listed  first. 


.  J 

[-£.54.2  2. 
2.  -.1  -5 
.3  1.92. 


In  RPN  mode,  function  CSWP  lets  you  swap  the  rows  of  a  matrix  listed  in 
stack  level  3,  whose  indices  are  listed  in  stack  levels  1  and  2.  For  example, 
the  following  figure  shows  the  RPN  stack  before  and  after  applying  function 
CSWP  to  matrix  A  in  order  to  swap  rows  2  and  3: 


As  you  can  see,  the  columns  that  originally  occupied  positions  2  and  3  have 
been  swapped. 


Function  RCI 

Function  RCI  stands  for  multiplying  Row  I  by  a  Constant  value  and  replace  the 
resulting  row  at  the  same  location.    The  following  example,  written  in  ALG 
mode,  takes  the  matrix  stored  in  A,  and  multiplies  the  constant  value  5  into 
row  number  3,  replacing  the  row  with  this  product. 


.3  1.92. S| 
2.    -.1  .5. 

:RCI(fl,5.,3.:i 

"-2.54.2  2. 
.3    1.92. S| 
10.  -.52. 


Page  1 0-25 


This  same  exercise  done  in  RPN  mode  is  shown  in  the  next  figure.  The  left- 
hand  side  figure  shows  the  setting  up  of  the  matrix,  the  factor  and  the  row 
number,  in  stack  levels  3,  2,  and  1 .  The  right-hand  side  figure  shows  the 
resu Iti ng  matrix  after  function  RCI  is  activated. 


Function  RCIJ 

Function  RCIJ  stands  for  "take  Row  I  and  multiplying  it  by  a  constant  C  and 
then  add  that  multiplied  row  to  row  J,  replacing  row  J  with  the  resulting  sum/' 
This  type  of  row  operation  is  very  common  in  the  process  of  Gaussian  or 
Gauss-Jordan  elimination  (more  details  on  this  procedure  are  presented  in  a 
subsequent  Chapter).  The  arguments  of  the  function  are:  (1)  the  matrix,  (2) 
the  constant  value,  (3)  the  row  to  be  multiplied  by  the  constant  in(2),  and  (4) 
the  row  to  be  replaced  by  the  resulting  sum  as  described  above.  For  example, 
taking  the  matrix  stored  in  variable  A,  we  are  going  to  multiply  column  3 
times  1 .5,  and  add  it  to  column  2.  The  following  example  is  performed  in 
ALG  mode: 


:kUlJU-l,l 

■  -Z' i3  ■  i2  ■  J 
[-^.5  4.2    2.  1 
3.3  1.753.55 
.2.     -.1     .5  J 

In  RPN  mode,  enter  the  matrix  first,  followed  by  the  constant  value,  then  by 
the  row  to  be  multiplied  by  the  constant  value,  and  finally  enter  the  row  that 
will  be  replaced.  The  following  figure  shows  the  RPN  stack  before  and  after 
applying  function  RCIJ  under  the  same  conditions  as  in  the  ALG  example 
shown  at 


Dove: 

4: 

[-2.5  4.2  2.  1 

.3  1.92.8 

I  2.    -.1  .5] 

3: 

1.5 

2: 

3. 

l: 

2. 

3  = 

4: 

3: 

2: 

l: 

[-2.5 

4.2    2.  1 

1.75  3.55 

-.1     .5  J 

Page  1 0-26 


Chapter  1 1 

Matrix  Operations  and  Linear  Algebra 

In  Chapter  10  we  introduced  the  concept  of  a  matrix  and  presented  a  number 
of  functions  for  entering,  creating,  or  manipulating  matrices.  In  this  Chapter 
we  present  examples  of  matrix  operations  and  applications  to  problems  of 
linear  algebra. 

Operations  with  matrices 

Matrices,  like  other  mathematical  objects,  can  be  added  and  subtracted. 
They  can  be  multiplied  by  a  scalar,  or  among  themselves.  An  important 
operation  for  linear  algebra  applications  is  the  inverse  of  a  matrix.  Details  of 
these  operations  are  presented  next. 


To  illustrate  the  operations  we  will  create  a  number  of  matrices  that  we  will 
store  in  variables.  The  generic  name  of  the  matrices  will  be  A//  and  B//, 
where  /  represents  the  number  of  rows  and  /  the  number  of  columns  of  the 
matrices.  The  matrices  to  be  used  are  generated  by  using  function  RANM 
(random  matrices).  If  you  try  this  exercise  in  your  calculator  you  will  get 
different  matrices  than  the  ones  listed  herein,  unless  you  store  them  into  your 
calculator  exactly  as  shown  below.  Here  are  the  matrices  A22,  B22,  A23, 
B23,  A32,  B32,  A33  and  B33  created  in  ALG  mode:  


:RAnH(£a  2YMH22 

Lli  a  J 

\?  -si 

L-s  sJ 

In  RPN  mode,  the  steps  to  follow  are: 

■;  2=:  £}  (enter)  RflNM  ;  fl££  E  (TroT) 
{  £  ==  3  y  [enter]  RflHM  ':  fl£3  E  (stq>J 
{  3  :e  2}  [enter]  RflHM  5  H3£  5 
{  3  :e  3}  [enter]  RflNM  5  A3 3  5 


;  .;■  [ENTER] 

; }  [enter) 
! }  [enter] 
\  ';■  [enter] 


Rfli-iH  5 

RflNM 


5  (sToyj 

'*  [STOtJ 

1:3  3  £  5  [sTQtJ 
B33  5  [sjo>) 


Page  11-1 


Addition  and  subtraction 

Consider  a  pair  of  matrices  A  =  [ajj]mXn  and  B  =  [bjj]mXn.  Addition  and 
subtraction  of  these  two  matrices  is  only  possible  if  they  have  the  same 
number  of  rows  and  columns.  The  resulting  matrix,  C  =  A  ±  B  =  [Cjj]mXn  has 
elements  c^  =  a^  ±  bjj.  Some  examples  in  ALG  mode  are  shown  below  using 
the  matrices  stored  above  (e.g.,  EBl(^fJ  lllillil)  


Translating  the  ALG  examples  to  RPN  is  straightforward,  as  illustrated  here. 
The  remaining  examples  of  matrix  operations  will  be  performed  in  ALG  mode 
only. 

Multiplication 

There  are  different  multiplication  operations  that  involve  matrices.  These  are 
described  next. 

Multiplication  by  a  scalar 

Multiplication  of  the  matrix  A  =  [ajj]mXn  by  a  scalar  k  results  in  the  matrix  C  = 
kA  =  [Cjj]mXn  =  [kajj]mXn.    In  particular,  the  negative  of  a  matrix  is  defined  by 
the  operation  -A  =(-1  )A  =  [-a^]  mXn.    Some  examples  of  multiplication  of  a 
matrix  by  a  scalar  are  shown  below. 


:fi23+B23 


In  RPN  mode,  the  steps  to  follow  are: 


fl££  {enter)  B££LflV7^JL  +  J     fl££  G^D  B££G^^G3 

fl£3  [enter)  B2.3[enter)1  +  j      fl£3  [enter)  B2.3[enter)1_  — ) 

R3£  [enter)  B3£Qa^GD     H3£  (enter)  B3£SGED 


Page  11-2 


By  combining  addition  and  subtraction  with  multiplication  by  a  scalar  we  can 
form  linear  combinations  of  matrices  of  the  same  dimensions,  e.g., 


In  a  linear  combination  of  matrices,  we  can  multiply  a  matrix  by  an  imaginary 
number  to  obtain  a  matrix  of  complex  numbers,  e.g., 


S-?.6-i  I 

1  1H--H>6>i  lG--5>G>i 

1  -2-6-6-i 

s-a-e-i  J 

f-tie-an-i)  -(6,6) 

1   (iH.aH)  (16..30) 

ia-na-i 

s-ia-i  J 

Matrix-vector  multiplication 

Matrix-vector  multiplication  is  possible  only  if  the  number  of  columns  of  the 
matrix  is  equal  to  the  length  of  the  vector.  This  operation  follows  the  rules  of 
matrix  multiplication  as  shown  in  the  next  section.   A  couple  of  examples  of 
matrix-vector  multiplication  follow: 


Page  1 1-3 


Vector-matrix  multiplication,  on  the  other  hand,  is  not  defined.  This 
multiplication  can  be  performed,  however,  as  a  special  case  of  matrix 
multiplication  as  defined  next. 


Matrix  multiplication 

Matrix  multiplication  is  defined  by  CmXn  =  AmXp-BpXn,  where  A  =  [ajj]mXp/  B  = 
[bjj]pXn/  and  C  =  [Cjj]mXn.  Notice  that  matrix  multiplication  is  only  possible  if  the 
number  of  columns  in  the  first  operand  is  equal  to  the  number  of  rows  of  the 
second  operand.    The  general  term  in  the  product,      is  defined  as 


p 

cij=Yaaik -bkJ,  for  i  =  1,2,... ,m;  j  =  1,2,... ,n. 

k=\ 


This  is  the  same  as  saying  that  the  element  in  the  i-th  row  and  j-th  column  of 
the  product,  C,  results  from  multiplying  term-by-term  the  i-th  row  of  A  with  the  j- 
th  column  of  B,  and  adding  the  products  together.  Matrix  multiplication  is  not 
commutative,  i.e.,  in  general,  AB  ^  B  A.  Furthermore,  one  of  the 
multiplications  may  not  even  exist.  The  following  screen  shots  show  the  results 
of  multiplications  of  the  matrices  that  we  stored  earlier: 


:fi33-B33 

"  16  31  -63 
-103  3  11? 
-HH  3H  36 

&33.fi33 

?H    13  IS 
33  -35  -13 
  IPS  6? 

Pfliittllttliittllttl 


B32-A23 


The  matrix-vector  multiplication  introduced  in  the  previous  section  can  be 
thought  of  as  the  product  of  a  matrix  mxn  with  a  matrix  nxl  (i.e.,  a  column 
vector)  resulting  in  an  mxl  matrix  (i.e.,  another  vector).  To  verify  this 
assertion  check  the  examples  presented  in  the  previous  section.  Thus,  the 
vectors  defined  in  Chapter  9  are  basically  column  vectors  for  the  purpose  of 
matrix  multiplication. 


Page  1 1  -4 


The  product  of  a  vector  with  a  matrix  is  possible  if  the  vector  is  a  row  vector, 
i.e.,  a  Ixm  matrix,  which  multiplied  with  a  matrix  mxn  produces  a  lxn  matrix 
(another  row  vector).  For  the  calculator  to  identify  a  row  vector,  you  must  use 
double  brackets  to  enter  it,  e.g., 


■  l?-2 

1-5  -331 

:&23 

[0  H  -HI 

U  -6  -SJ 

=  ii  3i-flns(i) 

lis  -in  -asi 

Term-by-term  multiplication 

Term-by-term  multiplication  of  two  matrices  of  the  same  dimensions  is  possible 
through  the  use  of  function  HADAMARD.  The  result  is,  of  course,  another 
matrix  of  the  same  dimensions.  This  function  is  available  through  Function 
catalog  (CB^  ),  or  through  the  MATRICES/OPERATIONS  sub-menu 


:HH[ihMhF;[ii:h??,E:??:i 

-3 

asi 

-HO 

HZ 

1-35 

-6 

3  J 

:Hh[ihHhF;[m:h22,E:22:i 

-56 

0  1 

[ 

0 

id 

:Hh[ihMHR[ii:E:?2,h?2:i 

-isi 

-na 

ho 

o  J 

:Hh[ihMhF;[ii:E:27,h27:i 

-aoi 

1-13  -2H 

-hoJ 

The  identity  matrix 

In  Chapter  9  we  introduce  the  identity  matrix  as  the  matrix  I  =  [5jj]nXn,  where  8^ 
is  the  Kronecker's  delta  function.  Identity  matrices  can  be  obtained  by  using 
function  IDN  described  in  Chapter  9.  The  identity  matrix  has  the  property  that 
A  I  =  I- A  =  A.  To  verify  this  property  we  present  the  following  examples 
using  the  matrices  stored  earlier  on: 


Page  1 1-5 


The  inverse  matrix 

The  inverse  of  a  square  matrix  A  is  the  matrix  A"1  such  that  A  A1  =  A1 -A  =  I, 
where  I  is  the  identity  matrix  of  the  same  dimensions  as  A.  The  inverse  of  a 
matrix  is  obtained  in  the  calculator  by  using  the  inverse  function,  INV  (i.e.,  the 
L  '/*  J  key).  An  example  of  the  inverse  of  one  of  the  matrices  stored  earlier  is 
presented  next: 


-15  -H  35 
3H5  3H5  3H5 
-1  _26_  zll 
2H3  3H3  3H3 
H?     23  H3 


To  verify  the  properties  of  the  inverse  matrix,  we  present  the  following 
multiplications: 


IlI  1  Ll| 

Lli  o  iJ 

ri  oi 
Lli  iJ 

:flaa-inv(flaa) 

ri  oi 
Lli  iJ 

Characterizing  a  matrix  (The  matrix  NORM  menu) 

The  matrix  NORM  (NORMALIZE)  menu  is  accessed  through  the  keystroke 
sequence  Ij^Jmth_  (system  flag  1  1  7  set  to  CHOOSE  boxes):  


HHTH  HERU 


i .  VECTOR.. 


H. HYPERBOLIC. 

5 .  REAL.. 

6.  EASE.. 


This  menu  contains  the  following  functions: 


HATRIK  RORH  HETll 

a.snRH 

3 .  F;  Ti  F;  M 

H.CnRH 

5.SRAD 

e.conD 

HATRIK  HERU 


i.HAKE.. 


a .  AORHALIZE.. 


3. FACTORS. . 
H.C0L.. 

5 .  RON.. 

6.  LSG 


HATRIK  nORH  HERU 

c.cord  [ 

? . RARK 
E .  DET 
5 . TRACE 

iO.TRAA  I 

™  ^^^^1 

Page  1 1-6 


These  functions  are  described  next.  Because  many  of  these  functions  use 
concepts  of  matrix  theory,  such  as  singular  values,  rank,  etc.,  we  will  include 
short  descriptions  of  these  concepts  intermingled  with  the  description  of 
functions. 

Function  ABS 

Function  ABS  calculates  what  is  known  as  the  Frobenius  norm  of  a  matrix.  For 
a  matrix  A  =  [a^]  mXn,  the  Frobenius  norm  of  the  matrix  is  defined  as 


If  the  matrix  under  consideration  in  a  row  vector  or  a  column  vector,  then  the 
Frobenius  norm  ,  |  |A|  |F ,  is  simply  the  vector's  magnitude.  Function  ABS  is 
accessible  directly  in  the  keyboard  as  UjJ^bs_  . 

Try  the  following  exercises  in  ALG  mode  (using  the  matrices  stored  earlier  for 
matrix  operations): 


:  IA22I 

2--JT7 

:  IA23I 

■J170 

:  IB32I 

J95 

:  IE33I 

:  IA33I 

2-J70 

:  IA32I 

■J227 

Function  SNRM 

Function  SNRM  calculates  the  Spectral  NoRM  of  a  matrix,  which  is  defined  as 
the  matrix's  largest  singular  value,  also  known  as  the  Euclidean  norm  of  the 
matrix.    For  example, 


SNRMCFI22) 
SNRMCFI32) 
SNRMCFI33) 


8. 

14.7146399549 
14. 1367419471 


Page  11-7 


Singular  value  decomposition 

To  understand  the  operation  of  Function  SNRM,  we  need  to  introduce  the 
concept  of  matrix  decomposition.  Basically,  matrix  decomposition  involves 
the  determination  of  two  or  more  matrices  that,  when  multiplied  in  a  certain 
order  (and,  perhaps,  with  some  matrix  inversion  or  transposition  thrown  in), 
produce  the  original  matrix.  The  Singular  Value  Decomposition  (SVD)  is  such 
that  a  rectangular  matrix  AmXn  is  written  as  AmXn  =  UmXm  -SmXn  -V  TnXn, 


Where  U  and  V  are  orthogonal  matrices,  and  S  is  a  diagonal  matrix.  The 
diagonal  elements  of  S  are  called  the  singular  values  of  A  and  are  usually 
ordered  so  that  S;  >  si+1,  for  i  =  1 ,  2,       n-1 .  The  columns  [Uj]  of  U  and  [Vj]  of 
V  are  the  corresponding  singular  vectors.  (Orthogonal  matrices  are  such  that 
U-  UT  =  I.  A  diagonal  matrix  has  non-zero  elements  only  along  its  main 
diagonal). 

The  rank  of  a  matrix  can  be  determined  from  its  SVD  by  counting  the  number 
of  non-singular  values.  Examples  of  SVD  will  be  presented  in  a  subsequent 
section. 


Functions  RNRM  and  CNRM 

Function  RNRM  returns  the  Row  NoRM  of  a  matrix,  while  function  CNRM 
returns  the  Column  NoRM  of  a  matrix.  Examples, 


:  RNRMCR22) 
:  CHRMCA22J 
:  RHRMCR33J 


8 
8 


21 


CHRMCR33) 
RHRMCA23) 
:  CHRMCR23) 


~2I 
26 
19 
16 


bnaMMna;iiana;iMna;iMffnaMand  bn=«iana;iiana;iMna;iiaffnaMand 


Row  norm  and  column  norm  of  a  matrix 

The  row  norm  of  a  matrix  is  calculated  by  taking  the  sums  of  the  absolute 
values  of  all  elements  in  each  row,  and  then,  selecting  the  maximum  of  these 
sums.  The  column  norm  of  a  matrix  is  calculated  by  taking  the  sums  of  the 
absolute  values  of  all  elements  in  each  column,  and  then,  selecting  the 
maximum  of  these  sums. 


Page  1 1-8 


Function  SRAD 

Function  SRAD  determines  the  Spectral  RADius  of  a  matrix,  defined  as  the 
largest  of  the  absolute  values  of  its  eigenvalues.    For  example, 


SRFlD(Fl22) 
SRFlD(Fl33) 

E 

SRFlD(B22) 


8. 

8.8339 1257969 
15.5156097709 


Definition  of  eigenvalues  and  eigenvectors  of  a  matrix 

The  eigenvalues  of  a  square  matrix  result  from  the  matrix  equation  Ax  =  Xx. 
The  values  of  X  that  satisfy  the  equation  are  known  as  the  eigenvalues  of  the 
matrix  A.  The  values  of  x  that  result  from  the  equation  for  each  value  of  I  are 
known  as  the  eigenvectors  of  the  matrix.    Further  details  on  calculating 
eigenvalues  and  eigenvectors  are  presented  later  in  the  chapter. 

Function  COND 

Function  COND  determines  the  condition  number  of  a  matrix.  Examples, 


:  C0HDCR22) 
:  C0HDCB33) 

c 

:CQND(Fl33f 


9.83617386179 
6.73714359438 


Condition  number  of  a  matrix 

The  condition  number  of  a  square  non-singular  matrix  is  defined  as  the 
product  of  the  matrix  norm  times  the  norm  of  its  inverse,  i.e., 
cond(A)  =  |  |  A  |  |  x  |  |  A1  |  | .   We  will  choose  as  the  matrix  norm,  |  |  A  |  | ,  the 
maximum  of  its  row  norm  (RNRM)  and  column  norm  (CNRM),  while  the  norm 
of  the  inverse,  |  |  A1  |  |,  will  be  selected  as  the  minimum  of  its  row  norm  and 
column  norm.  Thus,   |  | A|  |  =  max(RNRM(A),CNRM(A)),  and   |  | A1  |  |  = 
minlRNRMlA1),  CNRMJA1)). 


Page  11-9 


The  condition  number  of  a  singular  matrix  is  infinity.  The  condition  number  of 
a  non-singular  matrix  is  a  measure  of  how  close  the  matrix  is  to  being 
singular.  The  larger  the  value  of  the  condition  number,  the  closer  it  is  to 
singularity.    (A  singular  matrix  is  one  for  which  the  inverse  does  not  exist). 

Try  the  following  exercise  for  matrix  condition  number  on  matrix  A33.  The 
condition  number  is  COND(A33)  ,  row  norm,  and  column  norm  for  A33  are 
shown  to  the  left.     The  corresponding  numbers  for  the  inverse  matrix, 
INV(A33),  are  shown  to  the  right: 


C0NDCFI33) 

6.78714859433^ 

:RHRMCn333 

21. 

CNRMCFI33) 

 20. 

i  i  ni  ■  in  ■  i  ■     ■  ■  ni  ■  iTii  ii  ;-■  ■  ii  ajmi£i^ 


2^71 

C0ND(INV(fl33H 

6.73714359437 
:RNRM(INV(fl33H 

.261044176707 
CNRM(INV(fl33H 

. 339357429719 

iirii'Jii.ii^i.rii'aiiTiiiiiiiiiiiiiiiMiajj 


Since  RNRM(A33)  >  CNRM(A33),  then  we  take  |  |  A33  |  |  =  RNRM(A33)  = 
21.  Also,  since  CNRM(INV(A33))  <  RNRM(INV(A33)),  then  we  take 
||INV(A33)||  =  CNRM(INV(A33))  =  0.261044...  Thus,  the  condition 
number  is  also  calculated  as  CNRM(A33)*CNRM(INV(A33))  =  COND(A33) 
=  6.7871485... 


Function  RANK 

Function  RANK  determines  the  rank  of  a  square  matrix.  Try  the  following 
examples: 


RflHKCn223 
RAHKCE223 


The  rank  of  a  matrix 

The  rank  of  a  square  matrix  is  the  maximum  number  of  linearly  independent 
rows  or  columns  that  the  matrix  contains.  Suppose  that  you  write  a  square 
matrix  AnXn  as  A  =  [c}  c2  ...  cn],  where  c;  (i  =  1 ,  2,       n)  are  vectors 
representing  the  columns  of  the  matrix  A,  then,  if  any  of  those  columns,  say  ck, 
can  be  written  as  Ck  =  *c7  ? 

j*k,jG{l,2,...,n} 


Page  11-10 


where  the  values  dj  are  constant,  we  say  that  ck  is  linearly  dependent  on  the 
columns  included  in  the  summation.  (Notice  that  the  values  of  j  include  any 
value  in  the  set  {1 ,  2,       n},  in  any  combination,  as  long  as  j^k.)  If  the 
expression  shown  above  cannot  be  written  for  any  of  the  column  vectors  then 
we  say  that  all  the  columns  are  linearly  independent.   A  similar  definition  for 
the  linear  independence  of  rows  can  be  developed  by  writing  the  matrix  as  a 
column  of  row  vectors.     Thus,  if  we  find  that  rank(A)  =  n,  then  the  matrix  has 
an  inverse  and  it  is  a  non-singular  matrix.  If,  on  the  other  hand,  rank(A)  <  n, 
then  the  matrix  is  singular  and  no  inverse  exist. 


For  example,  try  finding  the  rank  for  the  matrix: 


You  will  find  that  the  rank  is  2.  That  is  because  the  second  row  [2,4,6]  is 
equal  to  the  first  row  [1,2,3]  multiplied  by  2,  thus,  row  two  is  linearly 
dependent  of  row  1  and  the  maximum  number  of  linearly  independent  rows  is 
2.  You  can  check  that  the  maximum  number  of  linearly  independent  columns 
is  3.  The  rank  being  the  maximum  number  of  linearly  independent  rows  or 
columns  becomes  2  for  this  case. 

Function  DET 

Function  DET  calculates  the  determinant  of  a  square  matrix.  For  example, 


DETCB33) 
DETCR33) 


-246 
-498 


DETCE22) 
:  DETCR22) 


-8 
-16 


Page  11-11 


The  determinant  of  a  matrix 

The  determinant  of  a  2x2  and  or  a  3x3  matrix  are  represented  by  the  same 
arrangement  of  elements  of  the  matrices,  but  enclosed  between  vertical  lines, 
i.e., 


a, 


*21 


*12 


*22 


a\3 

a  21 

a3l 

a32 

A  2x2  determinant  is  calculated  by  multiplying  the  elements  in  its  diagonal 
and  adding  those  products  accompanied  by  the  positive  or  negative  sign  as 
indicated  in  the  diagram  shown  below. 


/ 

all  ai2 


a21  a22 


0 


© 


The  2x2  determinant  is,  therefore, 


^^21  ^"22 


Q^i  *  ^^22      ^"12  *  ^^21 


A  3x3  determinant  is  calculated  by  augmenting  the  determinant,  an  operation 
that  consists  on  copying  the  first  two  columns  of  the  determinant,  and  placing 
them  to  the  right  of  column  3,  as  shown  in  the  diagram  below.  The  diagram 
also  shows  the  elements  to  be  multiplied  with  the  corresponding  sign  to  attach 
to  their  product,  in  a  similar  fashion  as  done  earlier  for  a  2x2  determinant. 
After  multiplication  the  results  are  added  together  to  obtain  the  determinant. 


Page  11-12 


ail   ai2  ai3 


a21   a22  a23 


A    4  / 
a 


ail  ai2 


21  22 

X  X  \ 


.  a31  a32  a33  . 

t/     \/    t/  \\ 

©  ©  © 


©  ©  © 


For  square  matrices  of  higher  order  determinants  can  be  calculated  by  using 
smaller  order  determinant  called  cofactors.  The  general  idea  is  to  " expand"  a 
determinant  of  a  nxn  matrix  (also  referred  to  as  a  nxn  determinant)  into  a  sum 
of  the  cofactors,  which  are  (n-l)x(n-l)  determinants,  multiplied  by  the  elements 
of  a  single  row  or  column,  with  alternating  positive  and  negative  signs.  This 
"expansion"  is  then  carried  to  the  next  (lower)  level,  with  cofactors  of  order  (n- 
2)x(n-2),  and  so  on,  until  we  are  left  only  with  a  long  sum  of  2x2 
determinants.  The  2x2  determinants  are  then  calculated  through  the  method 
shown  above. 

The  method  of  calculating  a  determinant  by  cofactor  expansion  is  very 
inefficient  in  the  sense  that  it  involves  a  number  of  operations  that  grows  very 
fast  as  the  size  of  the  determinant  increases.   A  more  efficient  method,  and 
the  one  preferred  in  numerical  applications,  is  to  use  a  result  from  Gaussian 
elimination.  The  method  of  Gaussian  elimination  is  used  to  solve  systems  of 
linear  equations.  Details  of  this  method  are  presented  in  a  later  part  of  this 
chapter. 

To  refer  to  the  determinant  of  a  matrix  A,  we  write  det(A).  A  singular  matrix 
has  a  determinant  equal  to  zero. 

Function  TRACE 

Function  TRACE  calculates  the  trace  of  square  matrix,  defined  as  the  sum  of 
the  elements  in  its  main  diagonal,  or 


tr(X)  =  % 


i=\ 


Examples: 


Page  11-13 


TRnCECfl223 
TRflCECB223 


-6 
15 


|:  TRFlCE(Fl33) 
TRFlCE(B33) 


4 
-7 


Function  TRAN 

Function  TRAN  returns  the  transpose  of  a  real  or  the  conjugate  transpose  of  a 
complex  matrix.  TRAN  is  equivalent  to  TRN.  The  operation  of  function  TRN 
was  presented  in  Chapter  10. 

Additional  matrix  operations  (The  matrix  OPER  menu) 

The  matrix  OPER  (OPERATIONS)  is  available  through  the  keystroke  sequence 
CfTl matrices  (system  flag  1  1  7  set  to  CHOOSE  boxes): 


The  OPERATIONS  menu  includes  the 


HATRICES  HEnU 

1 .  CREATE..  | 

r.FACTORIZATIOn.. 

H .  QUADRATIC  FORM.. 

S .  LiriEHF;  SVSTEHS..  B 

fi.LinEAR  AFFL..  | 

HATRIX  OFERATIOnJ  MEHU 


2.AKL 
?■ .  AKH 
H.CnRH 

F.LiiriD 

fi.DET 


ollowing  functions: 


HATRIK  OFERATIOnS  HEMJ 

? . HADAHARD  [ 
S.LS4 

5.  HAD  j! 

ID . RARK 

ii.RnRH 

HATRIK  OFERATIOnS  HEHU 


13.si:e 

iH.SnRH 
IS. S RAD 
1G . TRACE 
1? . TRAR 


Functions  ABS,  CNRM,  COND,  DET,  RANK,  RNRM,  SNRM,  TRACE,  and 
TRAN  are  also  found  in  the  MTH/MATRIX/NORM  menu  (the  subject  of  the 
previous  section).  Function  SIZE  was  presented  in  Chapter  10.  Function 
HADAMARD  was  presented  earlier  in  the  context  of  matrix  multiplication. 
Functions  LSQ  ,  MAD  and  RSD  are  related  to  the  solution  of  systems  of  linear 
equations  and  will  be  presented  in  a  subsequent  section  in  this  Chapter.  In 
this  section  well  discuss  only  functions  AXL  and  AXM 


Page  11-14 


Function  AXL 

Function  AXL  converts  an  array  (matrix)  into  a  list,  and  vice  versa.  For 
examples,  


E32 


nXLCE323 


0  3 
5  -61 
-4  -3\ 


E33 


nXLCE333 


-4  1  71 

-4  -5  7 
-7  6  2] 


{{0  3>  <5  -6>  {-4  -3»     H-4  1  7>  {-4  -5  7>  {-7  6  2> 


Note:  the  latter  operation  is  similar  to  that  of  the  program  CRMR  presented  in 
Chapter  1  0. 


Function  AXM 

Function  AXM  converts  an  array  containing  integer  or  fraction  elements  into  its 

corresponding  decimal,  or  approximate,  form.  For  example, 

I  I  —  1  Q — ^rr — v=n    I  1  -A-  1  -■  >-  1  -■  >-  1  -■ 


249  249  249 
-1  26  -33 
249  249  249 
47  23  43 
.493  493  493^ 
■■■■IMIMKHlita 


flXMtflNSUH 
-7. 6305220SS35E-2  -1 .  ^ 
-4.01606425703E-3  .  > 
9.43775100402E-2  4.6 


Function  LCXM 

Function  LCXM  can  be  used  to  generate  matrices  such  that  the  element  aij  is  a 
function  of  i  and  j.  The  input  to  this  function  consists  of  two  integers,  n  and  m, 
representing  the  number  of  rows  and  columns  of  the  matrix  to  be  generated, 
and  a  program  that  takes  i  and  j  as  input.    The  numbers  n,  m,  and  the 
program  occupy  stack  levels  3,  2,  and  1,  respectively.  Function  LCXM  is 
accessible  through  the  command  catalog  LeD_^z;  . 

For  example,  to  generate  a  2 '3  matrix  whose  elements  are  given  by  = 
(i+j)2,  first,  store  the  following  program  into  variable  PI  in  RPN  mode.   This  is 
the  way  that  the  RPN  stack  looks  before  pressing  [sto>)  . 


3  • 

2:  *  +  i  j  *  ' (i+j>^2 

'  EVflL  *  * 
l:  'PI 


Page  11-15 


The  implementation  of  function  LCXM  for  this  case  requires  you  to  enter: 


czd(^ctd(^codi  Lcm  (enter) 


The  following  figure  shows  the  RPN  stack  before  and  after  applying  function 
LCXM: 


3:  £. 

1:  *  ■*  i  j  ■£  1  <i+j)Aa! 

'  EVRL  *  *   , 


3: 

1" 

[4-  9.  16.1 

19-  16.  £5.  J 

n  ALG  mode,  this  example  can  be  obtained  by  using: 


:LCXM(2./3.,RCL('Pn)  I 
[4.  9.  16.1 
19.  16.  25 .j 


The  program  PI  must  still  have  been  created  and  stored  in  RPN  mode. 

Solution  of  linear  systems 

A  system  of  n  linear  equations  in  m  variables  can  be  written  as 

dirxi  +a12-x2  +  a13-x3  +  ...+  a1/m.rx  m_i  +  a1/fn-x  m  =bu 
a2rxi  +  a22-x2  +  a23-x3  +  ...+  a2/m.rx  m.}  +  a2m-x  m  =  b2/ 
a3rxi    +  a32-x2   +  a33-x3   +  ...+  a3/m.rx  m.}    +  a3Vx  m   =  b3/ 


an-l,l'Xl  +  Qn-l,2*X2  +  an-l,3*X3  +  •••+  an-l,m-rX  m-1  +  an-l,m'X  m  ~  ^n.i, 

an1-X]+  an2-x2  +  an3-x3  +  ...+   an/m-rx  m-i  +  an,m*x  m  -  bn. 

This  system  of  linear  equations  can  be  written  as  a  matrix  equation,  AnXm-xmXl 
=  bnxi,  if  we  define  the  following  matrix  and  vectors: 


an 

an  ■ 

xx 

V 

A  = 

a2l 

i 

X  = 

i 

b  = 

an2  ■ 

^nm  _ 

nxm 

_Xm_ 

mxl 

nxl 


Page  11-16 


Using  the  numerical  solver  for  linear  systems 

There  are  many  ways  to  solve  a  system  of  linear  equations  with  the  calculator. 
One  possibility  is  through  the  numerical  solver  [^Jnumslv  .  From  the  numerical 
solver  screen,  shown  below  (left),  select  the  option  4.  Solve  I'm  sys.v  and 


press  iiiiiii 


The  following  input  form  wi  I  be  provide  (right): 


i.SoLu*  equation.. 
3.  So  Luc  diFF 
3.£olug  pol.^.. 


5.SoLu«  Finance.. 
6.HSLV 


To  solve  the  linear  system  Ax  =  b,  enter  the  matrix  A,  in  the  format  [[  all7 
a12/ ...  ],  ...  [••••]]  in  the  A:  field.  Also,  enter  the  vector  b  in  the  B:  field. 
When  the  X:  field  is  highlighted,  press  [SOLVE].  If  a  solution  is  available,  the 
solution  vector  x  will  be  shown  in  the  X:  field.  The  solution  is  also  copied  to 
stack  level  1 .  Some  examples  follow. 

A  square  system 

The  system  of  linear  equations 

2x]  +  3x2  -5x3  =  1  3, 
X]  -  3x2  +  8x3  =  -1  3, 
2x}  -  2x2  +  4x3  =  -6, 


can  be  written  as  the  matrix  equation  A-x  =  b,  if 


2  3-5 
1    -3  8 
2-2  4 


x  = 


and    b  = 


13 
-13 
-6 


This  system  has  the  same  number  of  equations  as  of  unknowns,  and  will  be 
referred  to  as  a  square  system.  In  general,  there  should  be  a  unique  solution 
to  the  system.  The  solution  will  be  the  point  of  intersection  of  the  three  planes 
in  the  coordinate  system  (x^  x2,  x3)  represented  by  the  three  equations. 


Page  11-17 


To  enter  matrix  A  you  can  activate  the  Matrix  Writer  while  the  A:  field  is 
selected.  The  following  screen  shows  the  Matrix  Writer  used  for  entering 
matrix  A,  as  well  as  the  input  form  for  the  numerical  solver  after  entering 
matrix  A  (press  (enter}  in  the  Matrix  Writer): 


1.  -3. 
2-  -2. 


B. 
4. 


H'HII'JWOiWHgK'HlMfln 


SOLVE  SYSTEH  ftK=E= 


.,-5-]  CI.,-., 


Enter  coefficients  Hatrix  ft 


EDIT  CHOOS 


Press  <y^?  to  select  the  B:  field.  The  vector  b  can  be  entered  as  a  row  vector 
with  a  single  set  of  brackets,  i.e.,  L  13? 13? 61  111!!!:!!!  . 
After  entering  matrix  A  and  vector  b,  and  with  the  X:  field  highlighted,  we  can 
press  B3E9  to  attempt  a  solution  to  this  system  of  equations: 


SOLVE  SVSTEH  H ■ K=E 
[[2. ,3. ,-5. ]  CI.,-, 
C13.i-13.i6-] 


Enter  solutions  or  press  SOLVE 


A  solution  was  found  as  shown  next. 


SOLVE  SVSTEH  HK=E 

fl=  C  C2. j  3. j -5. ]  C 1 . j -„ 
*:  C13.  i-13.  i-6.  ] 


CI. ,2. i-l. ] 


Enter  solutions  or  press  SOLVE 


To  see  the  solution  in  the  stack  press  (enter)  .    The  solution  is  x  =  [1 ,2,-1  ]. 


Solutions:Cl.  2.  -1.] 


To  check  that  the  solution  is  correct,  enter  the  matrix  A  and  multiply  times  this 
solution  vector  (example  in  algebraic  mode): 


Page  11-18 


Solut ions:[l.  2.  -1.] 


Under-determined  system 

The  system  of  linear  equations 


2xt  +  3x2  -5x3  =  -10, 
x}  -  3x2  +  8x3  =  85, 


can  be  written  as  the  matrix  equation  Ax  =  b,  if 


2  3-5 


-10 


A  = 


x  = 


and    b  = 


1    -3  8 


85 


x 


'3 


This  system  has  more  unknowns  than  equations,  therefore,  it  is  not  uniquely 
determined.  We  can  visualize  the  meaning  of  this  statement  by  realizing  that 
each  of  the  linear  equations  represents  a  plane  in  the  three-dimensional 
Cartesian  coordinate  system  (xl7  x2,  x3).  The  solution  to  the  system  of 
equations  shown  above  will  be  the  intersection  of  two  planes  in  space.  We 
know,  however,  that  the  intersection  of  two  (non-parallel)  planes  is  a  straight 
line,  and  not  a  single  point.  Therefore,  there  is  more  than  one  point  that 
satisfy  the  system.  In  that  sense,  the  system  is  not  uniquely  determined. 

Let's  use  the  numerical  solver  to  attempt  a  solution  to  this  system  of  equations: 
E»  EH  .    Enter  matrix  A  and  vector  b  as  illustrated  in  the 

previous  example,  and  press  l!ii!!I!I;ll  when  the  X:  field  is  highlighted: 


Page  11-19 


■X"X"X"X"X"X"X  "7  f  1 1  IIC    C  '■'  C  T  r  H    M  ■  V  ~  C"  X'X'X'X'X'X'X1 

x-x-x-x-x-x-x-  -•■■■l  1 1     1 1  Ln  n  n-c-  -x-x-x-x-x-x-x 

fl=  CC2-  ,3.  ,-5.  ]  [1. 

Enter  folutionf  or  prgff  SOLVE 


To  see  the  details  of  the  solution  vector,  if  needed,  press  the  Ollili  button. 
This  will  activate  the  Matrix  Writer.  Within  this  environment,  use  the  right- 
and  left-arrow  keys  to  move  about  the  vector,  e.g., 


Thus,  the  solution  is  x  =  [15.373,  2.4626,  9.6268]. 

To  return  to  the  numerical  solver  environment,  press  (enter}  . 

The  procedure  that  we  describe  next  can  be  used  to  copy  the  matrix  A  and 
the  solution  vector  X  into  the  stack.  To  check  that  the  solution  is  correct,  try 
the  following: 

•  Press  ^ ^ ,  to  highlight  the  A:  field. 

•  Press  [nxtJ  MSM(ENm)f  to  copy  matrix  A  onto  the  stack. 

•  Press  li!!!!!:!!!!:!!!!!!!  to  return  to  the  numerical  solver  environment. 

•  Press  ^?  ^3? [enter) ,  to  copy  solution  vector  X  onto  the  stack. 

•  Press  ECU  to  return  to  the  numerical  solver  environment. 

•  Press  (enter)  to  return  to  the  stack. 

In  ALG  mode,  the  stack  will  now  look  like  this: 


Page  11-20 


Bo  1  ut  i  ons :  [  1 5^  373 1 3^3^ 

Let's  store  the  latest  result  in  a  variable  X,  and  the  matrix  into  variable  A,  as 
follows: 

Press  [^)[alpha}(x) [enter]  to  store  the  solution  vector  into  variable  X 
Press         CS  (3D  to  clear  three  levels  of  the  stack 
Press  [s^{Ai^(^[BfER)  to  store  the  matrix  into  variable  A 

Now,  let's  verify  the  solution  by  using:  IIIIIIII CX3  Kill  [enter),  which  results  in 
(press  ^3?  to  see  the  vector  elements):  [-9.99999999999  85.  ],  close  enough 
to  the  original  vector  b  =  [-1 0  85]. 

Try  also  this JIIECZD  [15*  10^3?  10]  (enter)  {J+J^num [enter) f  i.e., 


:fl(l5^10] 

-3S.  255. " 

3  3 
:+NUM(flNS(lH      "        "  ' 
C-10.  85.] 

OQEaEaEaEa 

This  result  indicates  that  x  =  [15,10/3,10]  is  also  a  solution  to  the  system, 
confirming  our  observation  that  a  system  with  more  unknowns  than  equations 
is  not  uniquely  determined  (under-determined). 

How  does  the  calculator  came  up  with  the  solution  x  =  [1  5.37...  2.46... 
9.62...]  shown  earlier?  Actually,  the  calculator  minimizes  the  distance  from  a 
point,  which  will  constitute  the  solution,  to  each  of  the  planes  represented  by 
the  equations  in  the  linear  system.  The  calculator  uses  a  leost-squore  method, 
i.e.,  minimizes  the  sum  of  the  squares  of  those  distances  or  errors. 

Over-determined  system 

The  system  of  linear  equations 

X]  +  3x2  =  1  5, 
2x}  -  5x2  =  5, 
-Xt  +  x2  =  22, 


Page  11-21 


can  be  written  as  the  matrix  equation  Ax  =  b,  if 


A  = 


1  3 

2  -5 
-1  1 


X  = 


,    and    b  = 


15 
5 

22 


This  system  has  more  equations  than  unknowns  (an  over-determined  system). 
The  system  does  not  have  a  single  solution.      Each  of  the  linear  equations  in 
the  system  presented  above  represents  a  straight  line  in  a  two-dimensional 
Cartesian  coordinate  system  (xl7  x2).  Unless  two  of  the  three  equations  in  the 
system  represent  the  same  equation,  the  three  lines  will  have  more  than  one 
intersection  points.  For  that  reason,  the  solution  is  not  unique.  Some 
numerical  algorithms  can  be  used  to  force  a  solution  to  the  system  by 
minimizing  the  distance  from  the  presumptive  solution  point  to  each  of  the 
lines  in  the  system.    Such  is  the  approach  followed  by  the  HP  49  G  numerical 
solver. 


Let's  use  the  numerical  solver  to  attempt  a  solution  to  this  system  of  equations: 
[TnNUM.sLv  MM  .    Enter  matrix  A  and  vector  b  as  illustrated  in  the 

previous  example,  and  press  when  the  X:  field  is  highlighted: 


SOLVE  SVSTEH  HK=E 


fl=  CC1- ,3. ]  C2. ?-5. ] 
E=  [15.  ,5.  ,22.  ] 


.  02054794 


Enter  solutions  or  preff  SOLVE 


To  see  the  details  of  the  solution  vector,  if  needed,  press  the  ilillil!  button. 
This  will  activate  the  Matrix  Writer.  Within  this  environment,  use  the  right- 
and  left-arrow  keys  to  move  about  the  vector,  e.c 


Page  11-22 


Press  [enter]  to  return  to  the  numerical  solver  environment.  To  check  that  the 
solution  is  correct,  try  the  following: 


•  Press  ^ ^ ,  to  highlight  the  A:  field. 

•  Press  [nxt)  mW[§NTER)/  to  copy  matrix  A  onto  the  stack. 

•  Press  IB  to  return  to  the  numerical  solver  environment. 

•  Press  ^z?  ^3?  EEQ3  [enter)  f  to  copy  solution  vector  X  onto  the  stack. 

•  Press  llllllliLiilllllll  to  return  to  the  numerical  solver  environment. 

•  Press  [enter)  to  return  to  the  stack. 

In  ALG  mode,  the  stack  will  now  look  like  this: 


Let's  store  the  latest  result  in  a  variable  X,  and  the  matrix  into  variable  A,  as 
follows: 

Press  \jj^[Ai^0[EtfrER)  to  store  the  solution  vector  into  variable  X 
Press  (3D  L*J  L*J  to  clear  three  levels  of  the  stack 
Press  [^(alp^(a][enter}  to  store  the  matrix  into  variable  A 

Now,  let's  verify  the  solution  by  using:        ixj  ililllll  [enter) f  which  results  in 
the  vector  [8.6917...  -3.4109...  -1.1301...],  which  is  not  equal  to  [15  5  22], 
the  original  vector  b.  The  ''solution''  is  simply  the  point  that  is  closest  to  the 
three  lines  represented  by  the  three  equations  in  the  system,  and  not  an  exact 


Least-square  solution  (function  LSQ) 

The  LSQ  function  returns  the  minimum-norm  least-square  solution  of  a  linear 
system  Ax  =  b,  according  to  the  following  criteria: 


Solut  ions: [3. 02054794^ 


solution. 


Page  11-23 


•  If  A  is  a  square  matrix  and  A  is  non-singular  (i.e.,  it's  inverse  matrix 
exist,  or  its  determinant  is  non-zero),  LSQ  returns  the  exact  solution  to 
the  linear  system. 

•  If  A  has  less  than  full  row  rank  (underdetermined  system  of  equations), 
LSQ  returns  the  solution  with  the  minimum  Euclidean  length  out  of  an 
infinity  number  of  solutions. 

•  If  A  has  less  than  full  column  rank  (over-determined  system  of 
equations),  LSQ  returns  the  "solution"  with  the  minimum  residual  value 
e  =  Ax  -  b.  The  system  of  equations  may  not  have  a  solution, 
therefore,  the  value  returned  is  not  a  real  solution  to  the  system,  just 
the  one  with  the  smallest  residual. 

Function  LSQ  takes  as  input  vector  b  and  matrix  A,  in  that  order.  Function 
LSQ  can  be  found  in  Function  catalog  (Lr>J  cat  ).  Next,  we  use  function  LSQ 
to  repeat  the  solutions  found  earlier  with  the  numerical  solver: 


Square  system 

Consider  the  system 


wi 


ith 


2x}  +  3x2  -5x3  =  1  3, 
X]  -  3x2  +  8x3  =  -1  3, 
2x}  -  2x2  +  4x3  =  -6, 


A  = 


2 

3 

-5" 

"  13 

1 

-3 

8 

,    x  = 

x2 

,    and    b  = 

-13 

2 

-2 

4 

x3 

-6 

The  solution  using  LSQ  is  shown  next: 
12  -2 


-5  o  I 
-2  4  1 


[13-13-6] 


2  3  -51 

1  -3  3 
2-2  4 


[13  -13  -61 


11  -y  y 

L2-2  4 


[13  -13  -61 

[13  -13  -63 
:  LSQ(flNSm,flNS(2H 

CI.  2.  -1 


Page  11-24 


Under-determined  system 

Consider  the  system 


2xt  +  3x2  -5x3  =  -10, 
x}  -  3x2  +  8x3  =  85, 


wi 


ith 


A  = 


2  3-5 
1    -3  8 


x  = 


and    b  = 


The  solution  using  LSQ  is  shown  next: 


-10 
85 


11  -a  y  j 


[2  3  -sj 
li  -3  s  J 


:[-10  35] 

C-10  35] 
LSQ(FlNSm,FlNS(£H 
[-10  353    1 1 5 .  373 1 343234  2 .  46263  > 


+£KIF  5KIF-H +4EL   DEL*  DEL  LI  I  Til"  ■ 


Over-determined  system 

Consider  the  system 


wi 


ith 


X]  +  3x2  =  1 5, 
2x}  -  5x2  =  5, 
-Xt  +  x2  =  22, 


3 

.  -5 
1  1 


x  = 


and    b  = 


15 
5 

22 


The  solution 


LSQ  is  shown  next: 


:[15  5  22] 

[15  5  22] 
LSQ(FlNSm,FlNS(2H 
13.02054794521  1 .39041* 


Page  11-25 


Compare  these  three  solutions  with  the  ones  calculated  with  the  numerical 
solver. 

Solution  with  the  inverse  matrix 

The  solution  to  the  system  Ax  =  b,  where  A  is  a  square  matrix  is  x  =  A1-  b. 
This  results  from  multiplying  the  first  equation  by  A1,  i.e.,  A1 -A-x  =  A]-b.  By 
definition,  A1 -A  =  I,  thus  we  write  l-x  =  A1  b.  Also,  l-x  =  x,  thus,  we  have, 
x  =  A]  b 

For  the  example  used  earlier,  namely, 

2x}  +  3x2  -5x3  =  1  3, 

X]  -  3x2  +  8x3  =  -1  3, 

2xt  -  2x2  +  4x3  =  -6, 
we  can  find  the  solution  in  the  calculator  as  follows: 


[2  3  -5] 
:  1 1  -3  8 
L2-2  4  J 

m\ 

>|1    "5     O    I        A  1 

12-2  4  J 

[2  3  -51 
1-3  3 
12-2  4  J 

:C13-13-6]  f 
C13  -13  -6] 

^» 

1-3  8 
12-2  4  J 

:[13  -13  -6] 

[13  -13  -61 
:IHVCnHSC2)™SCl) 

CI  2  -1] 

which  is  the  same  result  found  earlier. 

Solution  by  "division"  of  matrices 

While  the  operation  of  division  is  not  defined  for  matrices,  we  can  use  the 
calculator's  L±J  key  to  "divide"  vector  b  by  matrix  A  to  solve  for  x  in  the 
matrix  equation  A-x  =  b.  This  is  an  arbitrary  extension  of  the  algebraic 
division  operation  to  matrices,  i.e.,  from  A-x  =  b,  we  dare  to  write  x  =  b/A 
(Mathematicians  would  cringe  if  they  see  this!)  This,  of  course  is  interpreted  as 
(l/A)-b  =  A^-b,  which  is  the  same  as  using  the  inverse  of  A  as  in  the 


Page  11-26 


previous  section.  The  procedure  for  the  case  of  "dividing"  b  by  A  is 
illustrated  below  for  the  case 


2x]  +  3x2  -5x3  =  1  3, 
X]  -  3x2  +  8x3  =  -1  3, 
2x]  -  2x2  +  4x3  =  -6, 


The  procedure  is  shown  in  the  following  screen  shots: 

[i-pc — 5  o  1 — i    

12-2  4  1 


[13  -13  -61 


\2  3  -51 

1  -3  S 
12-2  4, 

[13  -13  -61 


[13  -13  -6] 

flHS(l) 
AHSC2) 


\2~4%\ 

[13  -13  -61 

[1  2  -1] 


The  same  solution  as  found  above  with  the  inverse  matrix. 


Solving  multiple  set  of  equations  with  the  same  coefficient  matrix 

Suppose  that  you  want  to  solve  the  following  three  sets  of  equations: 
X  +2Y+3Z  =14,     2X  +4Y+6Z  =  9,      2X  +4Y+6Z  =  -2, 
3X  -2Y+  Z  =  2,    3X  -2Y+  Z  =  -5,      3X  -2Y+  Z  =  2, 
4X  +2Y  -Z  =  5,  4X  +2Y  -Z  =  1 9,      4X  +2Y  -Z  =  1 2. 

We  can  write  the  three  systems  of  equations  as  a  single  matrix  equation:  AX 

=  B,  where 


"1 

2 

3 

X(D 

X(2) 

X(3) 

A  = 

3 

-2 

1 

X  = 

*0) 

*(2) 

7 

J(3)  ' 

4 

2 

-1 

7 

_  (i) 

7 

^(2) 

7 

^(3)  _ 

"14 

9 

-2" 

B  = 

2 

-5 

2 

5 

19 

12 

Page  11-27 


The  sub-indices  in  the  variable  names  X,  Y,  and  Z,  determine  to  which 
equation  system  they  refer  to.  To  solve  this  expanded  system  we  use  the 
following  procedure,  in  RPN  mode, 


I"  I"    I  4  ::  4  ::  —  >'    j  ::    j  !■ 

CCli  2",  3  iTubV 


,  ■■"D «  i:::!..!  ,  ID,  iy,  I'd  A  ..!  {enter} 

~2,  1] ,  [4,2,-1]]  (^GED 


The  result  of  this  operation  is: 


X  = 


1  2  2 

2  5  1 

3  -1  -2 


Gaussian  and  Gauss-Jordan  elimination 

Gaussian  elimination  is  a  procedure  by  which  the  square  matrix  of  coefficients 
belonging  to  a  system  of  n  linear  equations  in  n  unknowns  is  reduced  to  an 
upper-triangular  matrix  (echelon  form)  through  a  series  of  row  operations. 
This  procedure  is  known  as  forward  elimination.  The  reduction  of  the 
coefficient  matrix  to  an  upper-triangular  form  allows  for  the  solution  of  all  n 
unknowns,  utilizing  only  one  equation  at  a  time,  in  a  procedure  known  as 
backward  substitution. 

Example  of  Gaussian  elimination  using  equations 

To  illustrate  the  Gaussian  elimination  procedure  we  will  use  the  following 
system  of  3  equations  in  3  unknowns: 

2X+4Y+6Z=14, 

3X  -2Y+  Z  =  -3, 

4X+2Y  -Z  =  -4. 

We  can  store  these  equations  in  the  calculator  in  variables  El,  E2,  and  E3, 
respectively,  as  shown  below.  For  backup  purposes,  a  list  containing  the 
three  equations  was  also  created  and  stored  into  variable  EQS.  This  way,  if  a 
mistake  is  made,  the  equations  will  still  be  available  to  the  user. 


Page  11-28 


:2-X+4-Y+6-Z=14^1 

2-  X+4-Y+6-Z=14 
:  3-X-2-Y+Z=-3^2 

3-  X-(2-Y-Z)=-3 
:  4-X+2-Y-Z=-4^3 

4-X+2-Y-Z=-4 


El 
:E2 
E3 


2-  X+4-Y+6-Z=14 

3-  x-i:2-y-z:i=-3 

4-X+2-Y-Z=-4 


To  start  the  process  of  forward  elimination,  we  divide  the  first  equation  (El )  by 
2,  and  store  it  in  El,  and  show  the  three  equations  again  to  produce: 


:^E1 

:E2 
:E3 


X+2-Y+3-Z-7 
3-X-(2-Y-Z)=-3 
4-X+2-Y-Z=-4 


Next,  we  replace  the  second  equation  E2  by  (equation  2  -  3xequation  1,  i.e., 
E1-3xE2),  and  the  third  by  (equation  3  -  4xequation  1 ),  to  get 


E2-3-E^E2 
E3-4'E1*E3 


-CS-Y+S-Z-24) 
-E6-Y+13-Z-32) 


:E1 

E2 
E3 


X+2-Y+3-Z-7 
-CS-Y+S-Z-24) 
-C6-Y+13-Z-32) 


Next,  divide  the  second  equation  by  -8,  to  get 


=  ^E2 


Y+Z-3 


:E1 
:E2 
:E3 


X+2-Y+3-Z-7 
Y+Z-3 
-C6-Y+13-Z-32) 


Next,  replace  the  third  equation,  E3,  with  (equation  3  +  6xequation  2,  i.e., 
E2+6xE3),  to  get  


E3+6■E2^E3 


-C7-Z-14) 


El 
:E2 
E3 


X+2-Y+3-Z-7 
Y+Z-3 
-C7-Z-14) 


Notice  that  when  we  perform  a  linear  combination  of  equations  the  calculator 
modifies  the  result  to  an  expression  on  the  left-hand  side  of  the  equal  sign,  i.e., 
an  expression  =  0.  Thus,  the  last  set  of  equations  is  interpreted  to  be  the 
following  equivalent  set  of  equations: 

X+2Y+3Z  =  7, 


Page  11-29 


Y+  1  =  2, 
-71  =  A  A. 

The  process  of  backward  substitution  in  Gaussian  elimination  consists  in 
finding  the  values  of  the  unknowns,  starting  from  the  last  equation  and 


working  upwards.  Thus,  we  solve  for  Z 


:E2 
:E3 

:  S0LVE(E3,'Z') 


X+2-Y+3-Z-7 
Y+Z-3 
-C7-Z-14) 
Z=2 


irst: 


E3 

:  S0LVE(E3,'Z'} 


SUBST(E2,FlNSm^E2 


t+Z=3 
-C7-Z-14) 
Z=2 


Y+2-3 


Next,  we  substitute  1=2  into  equation  2  (E2),  and  solve  E2  for  Y: 

1  -C7-Z-I4:i 


:  S0LVE(E3,'Z') 
:SUBST(E2,flNS(l)»E2 
:  S0LVE(E2,'Y') 


Z=2 
Y+2-3 
Y=l 


Next,  we  substitute  Z=2  and  Y  =  1  into  El,  and  solve  El  for  X: 


:SUESTCE1,Y=13 

X+2-1+3-Z-7 
:  SUBST(FlNSm..Z=2) 

X+2-1+3-2-7 

:FlNS(l^El 

X+2-1+3-2-7 


X+2-1+3-Z- 
:  SUBST(FlNSm,Z=2) 

X+2- 1+3-2-71 

flNS(l>El 

X+2-1+3-2-7^ 
iSOLVECflHSClVX'J 

X=-l 


The  solution  is,  therefore,  X  =  -1 ,  Y  =  1 ,  Z  =  2. 


Example  of  Gaussian  elimination  using  matrices 

The  system  of  equations  used  in  the  example  above  can  be  written  as  a  matrix 


equation  Ax  =  b,  if 

we 

use: 

(2 

4 

~X' 

"14 

A  = 

3 

-2 

1 

,   x  = 

Y 

,    b  = 

-3 

^4 

2 

Z 

-4 

To  obtain  a  solution  to  the  system  matrix  equation  using  Gaussian  elimination, 
we  first  create  what  is  known  as  the  augmented  matrix  corresponding  to  A, 
i.e., 


Page  1 1-30 


(2 

4 

6 

14  ^ 

A  - 

aug 

3 

-2 

1 

-3 

V4 

2 

-1 

-4J 

The  matrix  Aaug  is  the  same  as  the  original  matrix  A  with  a  new  row, 
corresponding  to  the  elements  of  the  vector  b,  added  (i.e.,  augmented)  to  the 
right  of  the  rightmost  column  of  A. 

Once  the  augmented  matrix  is  put  together,  we  can  proceed  to  perform  row 
operations  on  it  that  will  reduce  the  original  A  matrix  into  an  upper-triangular 
matrix.  For  this  exercise,  we  will  use  the  RPN  mode  {j^^op^l^-jmM),  with 
system  flag  117  set  to  SOFT  menu.  In  your  calculator,  use  the  following 
keystrokes.  First,  enter  the  augmented  matrix,  and  make  an  extra  copy  of  the 
same  in  the  stack  (This  step  is  not  necessary,  except  as  an  insurance  that  you 
have  an  extra  copy  of  the  augmented  matrix  saved  in  case  you  make  a 
mistake  in  the  forward  elimination  procedure  that  we  are  about  to  undertake.): 

C  C  2 ,  4 ,  6 ,  1 4  ] ,  C  3 ,  -2 ,  1 ,  -3  J ,  C  4 ,  2 ,  ■■■■■  1 ,  4  J  "I  (enter)  (enter) 

Save  augmented  matrix  in  variable  AAUG:  CZIDS^S^(3(3(9(S1S™)  I™*) 

With  a  copy  of  the  augmented  matrix  in  the  stack,  press  [j^Jmth_  EEHjjg  033 
to  activate  the  ROW  operation  menu.     Next,  perform  the  following  row 
operations  on  your  augmented  matrix. 
Multiply  row  1  by  Vr.  CD  CD  CD  IS3 

Multiply  row  1  by  -3  add  it  to  row  2,  replacing  it:  CDCD  CD 

Multiply     row     1      by     -4     add     it     to     row     3,     replacing  it: 

GD  CD  Cz3  CD  Cz3  CD  Illiiliiililiililii 

Multiply  row  2  by  -1/8:  □DCDCDCD 

Multiply  row  2  by  6  add  it  to  row  3,  replacing  it:  CDCDCDCDCD  13513! 


Page  11-31 


If  you  were  performing  these  operations  by  hand,  you  would  write  the 
following: 


A 


A  ~ 

Aaug  ~ 


2  4  6 

3  -2  1 

4  2-1 

2  3 


J 


0  -8 
0  -6 


-8 
-13 


A_„  = 


aug 


14 

-3 
-4 

7 
-24 
-32 

y  v 

1  2  3 
0  1  1 
0  0-7 


2 

-2 
2 

2 
1 

-6 


7 
3 

-14 


3 
1 

-1 

3 
1 

-13 


7 

-3 
-4 


J 
1 
3 

-32 


The  symbol  =  ("  is  equivalent  to")  indicates  that  what  follows  is  equivalent  to 
the  previous  matrix  with  some  row  (or  column)  operations  involved. 

The  resulting  matrix  is  upper-triangular,  and  equivalent  to  the  set  of  equations 

X  +2Y+3Z  =  7, 
Y+  Z  =  3, 
-71  =  -14, 

which  can  now  be  solved,  one  equation  at  a  time,  by  backward  substitution, 
as  in  the  previous  example. 

Gauss-Jordan  elimination  using  matrices 

Gauss-Jordan  elimination  consists  in  continuing  the  row  operations  in  the 
upper-triangular  matrix  resulting  from  the  forward  elimination  process  until  an 
identity  matrix  results  in  place  of  the  original  A  matrix.  For  example,  for  the 
case  we  just  presented,  we  can  continue  the  row  operations  as  follows: 

Multiply  row  3  by -1/7:  CZDGtDQD  CD  EH 

Multiply  row  3  by  -1,  add  it  to  row  2,  replacing  it:   C7~)[ +/-  J    lspc)  I  3  J 


Page  11-32 


Multiply  row  3  by  -3,  add  it  to  row  1,  replacing  it: 

CD  CD  CD  CD  CD  CD  IHB 


Multiply  row  2  by  -2,  add  it  to  row  1,  replacing  it:  CDCDCDCDCDCD 
Writing  this  process  by  hand  will  result  in  the  following  steps: 


fl 

2 

3 

7  > 

(\ 

2 

3 

7^ 

fl 

2 

3 

?1 

A  - 

0 

1 

1 

3 

0 

1 

1 

3 

0 

1 

1 

1 

v° 

0 

-7 

~l4J 

v° 

0 

1 

2J 

1° 

0 

1 

2, 

fl 

2 

0 

1] 

(\ 

0 

0 

_1) 

Aug  = 

0 

1 

0 

1 

0 

1 

0 

1 

1° 

0 

1 

V 

0 

1 

2J 

Pivoting 

If  you  look  carefully  at  the  row  operations  in  the  examples  shown  above,  you 
will  notice  that  many  of  those  operations  divide  a  row  by  its  corresponding 
element  in  the  main  diagonal.  This  element  is  called  a  pivot  element,  or 
simply,  a  pivot.  In  many  situations  it  is  possible  that  the  pivot  element 
become  zero,  in  which  case  we  cannot  divide  the  row  by  its  pivot.  Also,  to 
improve  the  numerical  solution  of  a  system  of  equations  using  Gaussian  or 
Gauss-Jordan  elimination,  it  is  recommended  that  the  pivot  be  the  element 
with  the  largest  absolute  value  in  a  given  column.  In  such  cases,  we 
exchange  rows  before  performing  row  operations.  This  exchange  of  rows  is 
called  partial  pivoting.  To  follow  this  recommendation  is  it  often  necessary  to 
exchange  rows  in  the  augmented  matrix  while  performing  a  Gaussian  or 
Gauss-Jordan  elimination. 

While  performing  pivoting  in  a  matrix  elimination  procedure,  you  can 
improve  the  numerical  solution  even  more  by  selecting  as  the  pivot  the  element 
with  the  largest  absolute  value  in  the  column  and  row  of  interest.  This 
operation  may  require  exchanging  not  only  rows,  but  also  columns,  in  some 


Page  1 1-33 


pivoting  operations.  When  row  and  column  exchanges  are  allowed  in 
pivoting,  the  procedure  is  known  as  full  pivoting. 

When  exchanging  rows  and  columns  in  partial  or  full  pivoting,  it  is  necessary 
to  keep  track  of  the  exchanges  because  the  order  of  the  unknowns  in  the 
solution  is  altered  by  those  exchanges.  One  way  to  keep  track  of  column 
exchanges  in  partial  or  full  pivoting  mode,  is  to  create  a  permutation  matrix  P 
=  lnXn,  at  the  beginning  of  the  procedure.  Any  row  or  column  exchange 
required  in  the  augmented  matrix  Aaug  is  also  registered  as  a  row  or  column 
exchange,  respectively,  in  the  permutation  matrix.  When  the  solution  is 
achieved,  then,  we  multiply  the  permutation  matrix  by  the  unknown  vector  x 
to  obtain  the  order  of  the  unknowns  in  the  solution.  In  other  words,  the  final 
solution  is  given  by  P-x  =  b',  where  b'  is  the  last  column  of  the  augmented 
matrix  after  the  solution  has  been  found. 

Example  of  Gauss-Jordan  elimination  with  full  pivoting 

Let's  illustrate  full  pivoting  with  an  example.  Solve  the  following  system  of 
equations  using  full  pivoting  and  the  Gauss-Jordan  elimination  procedure: 

X+2Y  +  3Z  =  2, 

2X+        3Z  =  -1, 

8X+16Y-Z  =  41. 
The  augmented  matrix  and  the  permutation  matrix  are  as  follows: 


"1 

2 

3 

2  " 

"1 

0 

0" 

A  - 

2 

0 

3 

-1 

,    P  = 

0 

1 

0 

8 

16 

-1 

41 

0 

0 

1 

Store  the  augmented  matrix  in  variable  AAUG,  then  press        ll!i!!!i!!I!3  to  get  a 
copy  in  the  stack.   We  want  to  keep  the  CSWP  (Column  Swap)  command 
readily  available,  for  which  we  use:  CS_cat  (al^(al^(c](s}(alp^}  (find  CSWP), 
■39.    You'll  get  an  error  message,  press  jQN J,  and  ignore  the  message. 
Next,  get  the  ROW  menu  available  by  pressing:  [^matrices  EEMi  1393. 

Now  we  are  ready  to  start  the  Gauss-Jordan  elimination  with  full  pivoting. 
We  will  need  to  keep  track  of  the  permutation  matrix  by  hand,  so  take  your 
notebook  and  write  the  P  matrix  shown  above. 


Page  11-34 


First,  we  check  the  pivot  au.  We  notice  that  the  element  with  the  largest 
absolute  value  in  the  first  row  and  first  column  is  the  value  of  a31  =  8.  Since 
we  want  this  number  to  be  the  pivot,  then  we  exchange  rows  1  and  3,  by 
using:  UJL^LiJt^lJ  EKH.  The  augmented  matrix  and  the  permutation 
matrix  now  are: 

8      16      -1       41  0  0  1 

2  0  3-1  010 
12       3        2  0  0  1 


Checking  the  pivot  at  position  (1,1)  we  now  find  that  16  is  a  better  pivot  than 
8,  thus,  we  perform  a  column  swap  as  follows:  LiJC^JCDLr^J—^z;  EDI 


16 

8 

-1 

41 

0 

2 

3 

-1 

2 

1 

3 

2 

0  0  1 

1   0  0 

0  1  0 

Now  we  have  the  largest  possible  value  in  position  (1,1),  i.e.,  we  performed 
full  pivoting  at  (1,1).  Next,  we  proceed  to  divide  by  the  pivot: 
CD CD CD CD CD        .  The  permutation  matrix  does  not  change,  but  the 
augmented  matrix  is  now: 

1  1/2   -1/16  41/16 
0       2  3-1 

2  13  2 


0  0  1 

1  0  0 
0   1  0 


The  next  step  is  to  eliminate  the  2  from  position  (3,2)  by  using: 

CD  CD  CD  CD  CD  CD  IHB 


1      1/2   -1/16  41/16 
0       2  3-1 
0      0     25/8  -25/8 


0  0  1 

1  0  0 
0    1  0 


Having  filled  up  with  zeros  the  elements  of  column  1  below  the  pivot,  now  we 
proceed  to  check  the  pivot  at  position  (2,2).  We  find  that  the  number  3  in 
position  (2,3)  will  be  a  better  pivot,  thus,  we  exchange  columns  2  and  3  by 
using:  CD  CD  CD  GD-^i;  ilSlI 

1     -1/16    1/2  41/16  0    1  0] 


Page  1 1-35 


0      3  2-1 
0    25/8      0  -25/82 


1 

0 


0  0 
0  1 


Checking  the  pivot  at  position  (2,2),  we  now  find  that  the  value  of  25/8,  at 
position  (3,2),  is  larger  than  3.  Thus,  we  exchange  rows  2  and  3  by  using: 


CDGDCD_CD 

1 

0 
0 


-1/16 
25/8 
3 


1/2 
0 
2 


41/16 
-25/8 
-1 


0 
0 
1 


1 

0 
0 


0 
1 

0 


Now,  we  are  ready  to  divide  row  2  by  the  pivot  25/8,  by  using 

CD  CD  CD  CD  CD  0>  CD  CD  (W)I35 

1      -1/16   1/2    41/16  0   1  0 

0  10-1  001 
0        3       2-1  10  0 


Next,     we     eliminate  the 

CD  CD  CD  CD  CD  CD  ESS 
1    -1/16  1/2 
0      1  0 
0      0  2 


3     from     position     (3,2)     by  using: 


41/16 
-1 

2 


0 
0 


0 
1 

0 


Having  filled  with  zeroes  the  position  below  the  pivot,  we  proceed  to  check 
the  pivot  at  position  (3,3).  The  current  value  of  2  is  larger  than  V2  or  0,  thus, 
we  keep  it  unchanged.  We  do  divide  the  whole  third  row  by  2  to  convert  the 
pivot  to  1 ,  by  using:  CDCDCDi 


0   1  0 

0  0  1 

1  0  0 


1    -1/16   1/2  41/16 
0  10-1 
0      0       1  1 


Next,  we  proceed  to  eliminate  the  V2  in  position  (1,3)  by  using: 

CD  CD  CD®  CD  CD  CD  ESS 


1 

0 
0 


-1/16 
1 

0 


0 
0 
1 


33/16 
-1 
1 


0 
0 
1 


1  0 
0  1 

0  0 


Page  1 1-36 


Finally,  we  eliminate  the  -1/16  from  position  (1 ,2)  by  using: 

0  1  0 
0 


1 

0 

0 

2 

0 

1 

0 

-1 

0 

0 

1 

1 

0 
0 


0 


We  now  have  an  identity  matrix  in  the  portion  of  the  augmented  matrix 
corresponding  to  the  original  coefficient  matrix  A,  thus  we  can  proceed  to 
obtain  the  solution  while  accounting  for  the  row  and  column  exchanges  coded 
in  the  permutation  matrix  P.  We  identify  the  unknown  vector  x,  the  modified 
independent  vector  b'  and  the  permutation  matrix  P  as: 


Y 

"  2  " 

"0 

1 

0" 

X 

Y 

,  b'= 

-1 

,    P  = 

0 

0 

1 

Z 

1 

1 

0 

0 

The  solution  is  given  by  P  x=b',  or 

"0    1  0 

0  0  1 

1  0  0 


Which  results  in 


X 

3 

Y 

-1 

Z 

1 

~Y~ 

'  3  " 

Z 

-1 

X 

1 

Step-by-step  calculator  procedure  for  solving  linear  systems 

The  example  we  just  worked  is,  of  course,  the  step-by-step,  user-driven 
procedure  to  use  full  pivoting  for  Gauss-Jordan  elimination  solution  of  linear 
equation  systems.  You  can  see  the  step-by-step  procedure  used  by  the 
calculator  to  solve  a  system  of  equations,  without  user  intervention,  by  setting 
the  step-by-step  option  in  the  calculator's  CAS,  as  follows: 


Page  11-37 


_CflS  MODEL" 
HoduU:  13 

_nuH«ric  _flppfOX  _COHpUX 
_Y«rbos«  ^St«p^St«p  _Incr  Foh 
^ Rigorous ^SiHp  non-Rational 
Enter  injgp gnjgnt  uari abjj  nam 


Then,  for  this  particular  example,  in  RPN  mode,  use: 

C  2 ,    1 1  4 1 3  [enter}  [  [  l  -  2  -  3 "I ,  C  2 ,  8 ,  3  "I ,  C  8 


16,-1]]   (enter)  [^J 


The  calculator  shows  an  augmented  matrix  consisting  of  the  coefficients  matrix 
A  and  the  identity  matrix  I,  while,  at  the  same  time,  showing  the  next 
procedure  to  calculate: 

_2=L2-2-Ll 

1  2   3  1  0  0] 

2  0  3  0  1  0 
S  16  -1  0  0  1 


L2  =  L2-2-L1  stands  for  " replace  row  2  (L2)  with  the  operation  L2  -  2-L1 .  If  we 
had  done  this  operation  by  hand,  it  would  have  corresponded  to: 
[^[^iJ^CD(JK)CD^^-  Press  11,  and  follow  the  operations  in 
your  calculator's  screen.  You  will  see  the  following  operations  performed: 

L3=L3-8-Ll,  LI  =  2-L1--U2,  LI  =25-Ll--3  L3,  L2  =  25-L2-3-L3, 
and  finally  a  message  indicating  "Reduction  result"  showing: 


deduction  result 
50    0      0   -24  25  3" 
0  -100   0    -26  25-3 
0     0    -25  -S   0  1 


When  you  press  EEH,  the  calculator  returns  the  final  result  [1  2-1]. 
Calculating  the  inverse  matrix  step-by-step 

The  calculation  of  an  inverse  matrix  can  be  considered  as  calculating  the 
solution  to  the  augmented  system  [A  |  I  ].  For  example,  for  the  matrix  A  used 
in  the  previous  example,  we  would  write  this  augmented  matrix  as 


aug(I) 


1     2  3 

3  -2  1 

4  2-1 


1  0  0 
0  1  0 
0   0  1 


Page  1 1-38 


To  see  the  intermediate  steps  in  calculating  and  inverse,  just  enter  the  matrix 
A  from  above,  and  press  LAJ,  while  keeping  the  step-by-step  option  active  in 
the  calculator's  CAS.    Use  the  following: 

c  c  i ,  2 , 3  ::i ,  c  3 ,  -2 ,  l  ::i ,  c  4 , 2 ,  ■■■■■  i  :i  ::i  (^cs 


After  going  through  the  different  steps,  the  solution  returned  is: 

-      .  . 


What  the  calculator  showed  was  not  exactly  a  Gauss-Jordan  elimination  with 
full  pivoting,  but  a  way  to  calculate  the  inverse  of  a  matrix  by  performing  a 
Gauss-Jordan  elimination,  without  pivoting.  This  procedure  for  calculating 
the  inverse  is  based  on  the  augmented  matrix  (Aaug)nXn  =  [AnXn  |  lnXn]. 

The  calculator  showed  you  the  steps  up  to  the  point  in  which  the  left-hand  half 
of  the  augmented  matrix  has  been  converted  to  a  diagonal  matrix.  From 
there,  the  final  step  is  to  divide  each  row  by  the  corresponding  main  diagonal 
pivot.  In  other  words,  the  calculator  has  transformed  (Aaug)nXn  =  [AnXn  |  lnXn], 
into  [I  |  A1]. 

Inverse  matrices  and  determinants 

Notice  that  all  the  elements  in  the  inverse  matrix  calculated  above  are  divided 
by  the  value  56  or  one  of  its  factors  (28,  7,  8,  4  or  1).   If  you  calculate  the 
determinant  of  the  matrix  A,  you  get  c/e/(A)  =  56. 
We  could  write,  A1  =  C/c/ef(A),  where  C  is  the  matrix 

0      8  8 


C  = 


7  -13  8 
14     6  -8 


The  result  (A1)^  =  CnXn  /c/e/(AnXn),  is  a  general  result  that  applies  to  any  non- 
singular  matrix  A.  A  general  form  for  the  elements  of  C  can  be  written  based 
on  the  Gauss-Jordan  algorithm. 


Page  11-39 


Based  on  the  equation  A"1  = 
A"1,  is  not  defined  if  c/e/(A) 
singular  matrix. 


C/det(A),  sketched  above,  the  inverse  matrix, 
=  0.  Thus,  the  condition  c/e/(A)  =  0  defines  also  a 


Solution  to  linear  systems  using  calculator  functions 

The  simplest  way  to  solve  a  system  of  linear  equations,  Ax  =  b,  in  the 
calculator  is  to  enter  b,  enter  A,  and  then  use  the  division  function  /.    If  the 
system  of  linear  equations  is  over-determined  or  under-determined,  a 
"solution"  can  be  produced  by  using  Function  LSQ  (Least-SQuares),  as  shown 
earlier.  The  calculator,  however,  offers  other  possibilities  for  solving  linear 
systems  of  equations  by  using  Functions  included  in  the  MATRICES'  LINEAR 
SYSTEMS.,  menu  accessible  through  [^matrices  (Set  system  flag  1  1  7  to 
CHOOSE  boxes):  


MHTRIlEl"  METiU 


i.  CREATE.. 
a.OFERATIOnS.. 
3. FACTORIZATION. 
H. QUADRATIC  FORM.. 


S . LInEAR  SVSTEn: 


G.LinEAR  HFFL.. 


HATRIK  LinEAR  SYS.  HERU 

a.REF 

3.rr«F 

H.RREF 

S . SVST2HAT 

C.HATRICES.. 

The  functions  included  are  LINSOLVE,  REF,  rref,  RREF,  and  SYST2MAT. 


Function  LINSOLVE 

Function  LINSOLVE  takes  as  arguments  an  array  of  equations  and  a  vector 
containing  the  names  of  the  unknowns,  and  produces  the  solution  to  the  linear 
system.  The  following  screens  show  the  help-facility  entry  (see  Chapter  1 )  for 
function  LINSOLVE,  and  the  corresponding  example  listed  in  the  entry.  The 
right-hand  side  screen  shows  the  result  using  the  line  editor  (press  to 
activate): 


-IHSOLVE: 

So lues  a  system  of 
. inear  equations 
.INSOLVEt [X+Y=3?  X-Y=l ] 

[X=2  Y=l] 

See:  SOLVE 


HELP 

LINS0LVE(EX+Y=3  X-Y=l> 
«[X+Y=3  X-Y=l]  [X  Y]>  Spe> 
K[X+Y=3,X-Y=1]? 
[X? Y]>? :Specif ic:  £2? 
-2,2, 1, >,IX=2,Y=1]> 


Here  is  another  example  in  ALG  mode.  Enter  the  following: 
I  T  N i  "! !  v  F  ( \  x  ■■■■■ '?  *  V  ■■!■■  7  ===  ■■■■■  ft , '?  *  x + V  ■■■■■  *  7  ==:  ft  ==  s:::i  *  x  ■■■■■  :":s  *  V + 7  ^  ■■■■■  I  :":s  "I  == 

"CX, V,Z]> 


Page  1 1-40 


to  produce  the  solution:  C  X=—  1  Y=2  ?  Z 


Function  LINSOLVE  works  with  symbolic  expressions.  Functions  REF,  rref,  and 
RREF,  work  with  the  augmented  matrix  in  a  Gaussian  elimination  approach. 

Functions  REF,  rref,  RREF 

The  upper  triangular  form  to  which  the  augmented  matrix  is  reduced  during 
the  forward  elimination  part  of  a  Gaussian  elimination  procedure  is  known  as 
an  "echelon"  form.  Function  REF  (Reduce  to  Echelon  Form)  produces  such  a 
matrix  given  the  augmented  matrix  in  stack  level  1 . 

Consider  the  augmented  matrix, 


1 

-2 

1 

0 

A  - 

2 

1 

-2 

-3 

5 

-2 

1 

12 

Representing  a  linear  system  of  equations,  Ax  =  b,  where 

A_  r  r  -j  „  .....p..  -j  "i  „  r  v..  1  ..  -■■P"i ..  ri:::;..  -  v..  I  "!  "I 

and 

b  =  \ 

Enter  the  augmented  matrix,  and  save  it  into  variable  AAUG,  in  ALG  mode: 

c  c  i ,  -2 ,  i ,  e  ::i ,  c  2 ,  i ,  "2 ,  -3  ::i  c  5 ,  -2 , 1 , 1 2  ::i  ::i  ►  rrug 

Application  of  function  REF  produces: 

 A.    I  >—  1  =  ^"1 

I5-2  1  iz[ 

:  REFCRRUG) 


1-2  1  0 
0  1 

1  .0  0    1    7  ^ 

fiTiTEII^lgi^ll^lIiHM^a 


The  result  is  the  upper  triangular  (echelon  form)  matrix  of  coefficients  resulting 
from  the  forward  elimination  step  in  a  Gaussian  elimination  procedure. 


Page  11-41 


The  diagonal  matrix  that  results  from  a  Gauss-Jordan  elimination  is  called  a 
row-reduced  echelon  form.    Function  RREF  (  Row-Reduced  Echelon  Form)  The 
results  of  this  function  call  is  to  produce  the  row-reduced  echelon  form  so  that 
the  matrix  of  coefficients  is  reduced  to  an  identity  matrix.  The  extra  column  in 
the  augmented  matrix  will  contain  the  solution  to  the  system  of  equations. 

As  an  example,  we  show  the  result  of  applying  function  RREF  to  matrix  AAUG 
in  ALG  mode: 


:  RREFCRRUG) 

0  I 
.0  0 

1 

i    ■_■  ■ 
5  5 
1    7  J 

[1  0  0  31 
0  10  5 
.0  0  1  7] 

The  result  is  final  augmented  matrix  resulting  from  a  Gauss-Jordan  elimination 
without  pivoting. 

A  row-reduced  echelon  form  for  an  augmented  matrix  can  be  obtained  by 
using  function  rref.  This  function  produces  a  list  of  the  pivots  and  an 
equivalent  matrix  in  row-reduced  echelon  form  so  that  the  matrix  of 
coefficients  is  reduced  to  a  diagonal  matrix. 

For  example,  for  matrix  AAUG,  function  rref  produces  the  following  result: 

1 


rref(RRUG) 
Piyots:{3  1.4  1.5  £.>  ^ 


r 


4  1.  ^1  Z.> 
*3, 1. ,4,1., 


t :Piuots 
5.2.  }» 

[[£0,0,0,60] 

[0, 15,0,75] 
[0,0, 12,84]] y 


The  second  screen  above  is  obtained  by  activating  the  line  editor  (press  <s^?). 
The  result  shows  pivots  of  3,  1,4,  1,5,  and  2,  and  a  reduced  diagonal 
matrix. 


Function  SYST2MAT 

This  function  converts  a  system  of  linear  equations  into  its  augmented  matrix 
equivalent.  The  following  example  is  available  in  the  help  facility  of  the 
calculator: 


Page  11-42 


:  HELP 

:  SYST2MRTCCX+Y  X-Y=2]lH 
3VST2MflT<CX+V,X-V=2iP 


HELP 

:  SYST2MRTCCX+Y  X-Y=2],l>> 


The  result  is  the  augmented  matrix  corresponding  to  the  system  of  equations: 


X+Y  =  0 
X-Y  =2 


Residual  errors  in  linear  system  solutions  (Function  RSD) 

Function  RSD  calculates  the  ReSiDuals  or  errors  in  the  solution  of  the  matrix 
equation  Ax=b,  representing  a  system  of  n  linear  equations  in  n  unknowns. 
We  can  think  of  solving  this  system  as  solving  the  matrix  equation:  f(x)  =  b  - 
Ax  =  0.  Suppose  that,  through  a  numerical  method,  we  produce  as  a  first 
approximation  the  solution  x(0).  Evaluating  f(x(0))  =  b  -  A-x(0)  =  e  ^  0. 
Thus,  e  is  a  vector  of  residuals  of  Function  for  the  vector  x  =  x  (0). 

To  use  Function  RSD  you  need  the  terms  b,  A,  and  x(0),  as  arguments.  The 
vector  returned  is  e  =  b  -  A-x(0).  For  example,  using  A  = 
C  C  2 ,  - 1  J  C  0 ,  2  J  J ,  x(0)  =  C  1 .  8 ,  2 .  7  J ,  and  b  =  C  1 ,  6  J ,  we  can  find 
the  vector  of  residuals  as  follows: 


RSD<Cl,6]i [[2,-l][0? 
211,  [1.8,2.7]^ 


RSd[[1  6H,[|  21],H1.S  2.7> 
[-1  .6] 


The  result  is  e  =  b  -  A-x(0)  =  C 


"I , 


Note:  If  we  let  the  vector  Ax  =  x  -  x  (0),  represent  the  correction  in  the 
values  of  x  (0),  we  can  write  a  new  matrix  equation  for  Ax,  namely  A-Ax  = 
e.    Solving  for  Ax  we  can  find  the  actual  solution  of  the  original  system  as  x 
=  x(0)  +  Ax. 


Page  1 1 -43 


Eigenvalues  and  eigenvectors 

Given  a  square  matrix  A,  we  can  write  the  eigenvalue  equation  Ax  =  ^-x, 
where  the  values  of  X  that  satisfy  the  equation  are  known  as  the  eigenvalues 
of  matrix  A.  For  each  value  of  X,  we  can  find,  from  the  same  equation, 
values  of  x  that  satisfy  the  eigenvalue  equation.  These  values  of  x  are  known 
as  the  eigenvectors  of  matrix  A.    The  eigenvalues  equation  can  be  written 
also  as  (A  -  X-I)x  =  0. 

This  equation  will  have  a  non-trivial  solution  only  if  the  matrix  (A  -  ^-1)  is 
singular,  i.e.,  if  det(A  -  ^-1)  =  0. 

The  last  equation  generates  an  algebraic  equation  involving  a  polynomial  of 
order  n  for  a  square  matrix  AnXn.  The  resulting  equation  is  known  as  the 
characteristic  polynomial  of  matrix  A.  Solving  the  characteristic  polynomial 
produces  the  eigenvalues  of  the  matrix. 

The  calculator  provides  a  number  of  functions  that  provide  information 
regarding  the  eigenvalues  and  eigenvectors  of  a  square  matrix.    Some  of 
these  functions  are  located  under  the  menu  MATRICES/EIGEN  activated 
through  [^matrices  . 


HATRICES  HEHU 

3. FACTORIZATION. 
H .  QUADRATIC  FORH.. 
S .  LIREAR  SYSTEHS.. 
e.LinEAR  HFFL.. 

3 .  VECTOR.. 

HATRIK  EIGEJWECT.  HEHU 

2.  EGY  ! 

3.  EGVL  | 
H.JORDAn  ! 
5. PCAR  | 
C.PHini  ! 


Function  PCAR 

Function  PCAR  generates  the  characteristic  polynomial  of  a  square  matrix 
using  the  contents  of  variable  VX  (a  CAS  reserved  variable,  typically  equal  to 
'X')  as  the  unknown  in  the  polynomial.  For  example,  enter  the  following 
matrix  in  ALG  mode  and  find  the  characteristic  equation  using  PCAR: 

■■■  ■-  ■■     ■::: :  "■     r  ■■      .-■  "■     ■ :    ■:::■        "■  "■ 

l  :..  .!.  n  ~j  \i  ■■  ■.::> ..:    :.. :»  ~ .!.  \i  st ..!  :■  !.. :■  .J  :■     ..!  ..! 


Page  1 1  -44 


I  13  5   2  J    A  ] 
13  5   2  1 

:PCflR(flHS(lH 

Using  the  variable  X  to  represent  eigenvalues,  this  characteristic  polynomial  is 
to  be  interpreted  as  X  3-2X  2-22X  +2 1  =0. 

Function  EGVL 

Function  EGVL  (EiGenVaLues)  produces  the  eigenvalues  of  a  square  matrix. 
For  example,  the  eigenvalues  of  the  matrix  shown  below  are  calculated  in 
ALG  mode  using  function  EGVL: 


The  eigenvalues  X  =  [  -Vl  0,  Vl  0  ]. 


Note:  In  some  cases,  you  may  not  be  able  to  find  an  'exact'  solution  to  the 
characteristic  polynomial,  and  you  will  get  an  empty  list  as  a  result  when 
using  Function  EGVL.  If  that  were  to  happen  to  you,  change  the  calculation 
mode  to  Approx  in  the  CAS,  and  repeat  the  calculation. 


For  example,  in  exact  mode,  the  following  exercise  produces  an  empty  list  as 
the  solution: 


15  -2  lJ 

-2  11 

B 

-1  2 

-2  lJ 

:EGVL(flHS(lH 

Page  11-45 


Change  mode  to  Approx  and  repeat  the  entry,  to  get  the  following 
eigenvalues:  L  \  1 »  Jo ::  d  ■,. )     E-.  1  »  JH  —'d  !S  dd  )     5-.  —  1 »  r'b ;:  U  .i5 ..! . 

Function  EGV 

Function  EGV  (EiGenValues  and  eigenvectors)  produces  the  eigenvalues  and 
eigenvectors  of  a  square  matrix.  The  eigenvectors  are  returned  as  the 
columns  of  a  matrix,  while  the  corresponding  eigenvalues  are  the  components 
of  a  vector. 

For  example,  in  ALG  mode,  the  eigenvectors  and  eigenvalues  of  the  matrix 
listed  below  are  found  by  applying  function  EGV: 


±L>    1  ■  CJCJ 


-1.00  5.00  3.00 
1.00     3.00  4.00 


TTWTTWl 


3. 
4. 


00 
00] 


-1.00  5.00 
1.00  3.00 

:EGVCRHSC1.00i) 

^1.00     1.00  "0.031 
0.79  -0.51   1.00  [0 
-0 .91  0.65   0.34  J 

ta[fiiH[fiEEgiiigiqNi«tiiaj 


The  result  shows  the  eigenvalues  as  the  columns  of  the  matrix  in  the  result  list. 
To  see  the  eigenvalues  we  can  use:  GET(ANS(1  ),2),  i.e.,  get  the  second 
element  in  the  list  in  the  previous  result.  The  eigenvalues  are: 


 1  i.lh- — o.yy  4.uu| 

EGV(flNS(1.00H 
1.00    1.00  -0.03 
0.79  -0.51  1.00  |  C0.  H 
-0.91  0.65  0.84 
:GET(FINS(1.00),£.00) 

[0.29  3.  16  7.54] 
bi:tiaH^cc»ii.»oi.]«in(M] 


I 


In  summary, 


X}  =  0.29,  Xt  =  [  1. 00,0.79,-0.91  ]T, 
^2  =  3.16,  x2  =  [1.00,-0.51,  0.65] T, 
^3  =  7.54,  Xt  =  [-0.03,  1 .00,  0.84] T. 


Note:  A  symmetric  matrix  produces  all  real  eigenvalues,  and  its  eigenvectors 
are  mutually  perpendicular.  For  the  example  just  worked  out,  you  can  check 
that  Xt  »x2  =  0,  Xt  »x3  =  0,  and  x2  oc3  =  0. 


Page  1 1  -46 


Function  JORDAN 

Function  JORDAN  is  intended  to  produce  the  diagonalization  or  Jordan-cycle 
decomposition  of  a  matrix.  In  RPN  mode,  given  a  square  matrix  A,  function 
JORDAN  produces  four  outputs,  namely: 

•  The  minimum  polynomial  of  matrix  A  (stack  level  4) 

•  The  characteristic  polynomial  of  matrix  A  (stack  level  3) 

•  A  list  with  the  eigenvectors  corresponding  to  each  eigenvalue  of 
matrix  A  (stack  level  2) 

•  A  vector  with  the  eigenvectors  of  matrix  A  (stack  level  4) 


For  example,  try  this  exercise  in  RPN  mode: 

C C4, 1,-2], Li , 2, 


1 , 0]] 


JORDAN 


The  output  is  the  following: 


'XA3+-6*xA2+2*X+8' 

'XA3+-6*xA2+2*X+8' 

{} 

{} 


The  same  exercise,  in  ALG  mode,  looks  as  in  the  following  screen  shots: 


i-^  -i  o  J 

4  1-2 
1    2  -1 
-2-1  0 
JORDflHCflHSC133 
CX-A-3-6*XA2+2*X+83  X^3-6 
*XA2+2*X+B,0,0> 


I  1    2  -1| 

1-2  -1  0  J 


:  JORDFMflNS(l)) 
t  XA3-6*ttA2+2*X+S,  X^3-ej 
*XA2+2*X+B,OiO> 
4X-A-3-6*X^2+2*X+8!  X^3- 
6*XA2+2*X+B,0,0> 


Function  MAD 

This  function,  although  not  available  in  the  EIGEN  menu,  also  provides 
information  related  to  the  eigenvalues  of  a  matrix.  Function  MAD  is 
available  through  the  MATRICES  OPERATIONS  sub-menu  (^matrices  )  and  is 
intended  to  produce  the  adjoint  matrix  of  a  matrix. 


Page  11-47 


In  RPN  mode,  function  MAD  generate  a  number  of  properties  of  a  square 
matrix,  namely: 


•  the  determinant  (stack  level  4) 

•  the  formal  inverse  (stack  level  3), 

•  in  stack  level  2,  the  matrix  coefficients  of  the  polynomial  p(x)  defined 
by  (x-l-A)  •p(x)=m(x)-l/ 

•  the  characteristic  polynomial  of  the  matrix  (stack  level  1) 

Notice  that  the  equation  (x-l-A)-p(x)=m(x)-l  is  similar,  in  form,  to  the 
eigenvalue  equation  Ax  =  A,-x. 


As  an  example,  in  RPN  mode,  try: 


cc 4,  i,-2]  ci , --i  ::i      --:!.  mad 


The  result  is: 


-8. 

[[  0.1 3  -0.25  -0.38][-0.25  0.50  -0.25][-0.38  -0.25  -0.88]] 
{[[1  0  0][0  1  0][0  0  1]]  [[-2  1  -2][1  -4-l][-2-l  -6]  [[-1  2  3] [2 -4  2][3  2  7]]} 
'XA3+-6*xA2+2*X+8' 


The  same  exercise,  in  ALG  mode,  will  look  as  follows: 

[t  .  rj rj  a.  i  .  rjrj  i.  .  cjrj  ■ 
1-00     2.00  -1-001 
-2.00  -1.00   0.00  J 


:MRDCRHSC1.003) 

"0.13  -0.25-0.3 
-3.00  1-0.25  0.50  -0.> 
-0.33  -0.25  -0.3 


[[. 125,-. 25,-. 375] 
[-.25,. 5,-. 25] 
[-.375?-.25?-.S75]]? 

[[1,0,0] 
[0, 1,0] 


Matrix  factorization 

Matrix  factorization  or  decomposition  consists  of  obtaining  matrices  that  when 
multiplied  result  in  a  given  matrix.  We  present  matrix  decomposition  through 
the  use  of  Functions  contained  in  the  matrix  FACT  menu.    This  menu  is 
accessed  through  l<n  J  matrices  . 


Page  1 1-48 


HHTRIlEl"  HEHU 


i.  CREATE.. 
a.OFERRTIOnS.. 


3.FHCT0RIZflTI0n.. 


H .  QUADRATIC  FORH.. 
S .  LiriEHF;  SVSTEHS.. 
H.LiriEHF;  HFFL.. 


HflTRIK  FACTOR.  HERU 

a.m 
?-.ar, 
H.gr 
S .  l'lHUF; 

c.svd 

MhTRI!-!  FACTOR.  HERU 


3.GR 
H.qr 
S .  l'lHUF; 

c.svd 


Function  contained  in  this  menu  are:  LQ,  LU,  QR,SCHUR,  SVD,  SVL 


Function  LU 

Function  LU  takes  as  input  a  square  matrix  A,  and  returns  a  lower-triangular 
matrix  L,  an  upper  triangular  matrix  U,  and  a  permutation  matrix  P,  in  stack 
levels  3,  2,  and  1,  respectively.  The  results  L,  U,  and  P,  satisfy  the  equation 
PA  =  LU.  When  you  call  the  LU  function,  the  calculator  performs  a  Crout 
LU  decomposition  of  A  using  partial  pivoting. 

For  example,  in  RPN  mode:  C  C  - 1 ,  2 ,  5  "I  C  3 ,  1 ,  -2  "I  C  7 ,  6 ,  5  "I  J  LU 
produces: 

3:[[7  0  0][-l  2.86  0][3 -1.57-1] 
2:  [[1  0.86  0.71][0  1  2][0  0  1]] 
1:  [[0  0  1][1  0  0] [0  1  0]] 
In  ALG  mode,  the  same  exercise  will  be  shown  as  follows: 


3-00  1.00-2.00 
7.00   6.00  5.00 

LUCRHSCl .  00)3 

7.00  0.00  0.00 
-1.00   2. 36  0.00 

3.00  -1.57  -1.00J 


1. 

0. 
0. 


[[7. ?0. ?0. ] 

[-1. ?2. 85714285714,0 

[3. ?-l. 57142357143,- 
[[1. ? .357142357143, .7 

[0. ? 1. ?2. ] 

[0.  ,0.  ,1,]]  

taitiimtiCE'nii'Hqi'n«tuaj 


Orthogonal  matrices  and  singular  value  decomposition 

A  square  matrix  is  said  to  be  orthogonal  if  its  columns  represent  unit  vectors 
that  are  mutually  orthogonal.  Thus,  if  we  let  matrix  U  =  [V]  v2  ...  vn]  where 
the  Vj,  i  =  1 ,  2,       n,  are  column  vectors,  and  if  Vj«Vj  =  8^  where  SS|  is  the 
Kronecker's  delta  function,  then  U  will  be  an  orthogonal  matrix.  This 
conditions  also  imply  that  U-  UT  =  I. 


Page  11-49 


The  Singular  Value  Decomposition  (SVD)  of  a  rectangular  matrix  AmXn  consists 
in  determining  the  matrices  U,  S,  and  V,  such  that  AmXn  =  U  mXm  S  mXn  -V  T  nXn/ 
where  U  and  V  are  orthogonal  matrices,  and  S  is  a  diagonal  matrix.  The 
diagonal  elements  of  S  are  called  the  singular  values  of  A  and  are  usually 
ordered  so  that  s,  >si+lf  for  /  =  1 ,  2,       n-1 .  The  columns  [Uj]  of  U  and  [Vj]  of 
V  are  the  corresponding  singular  vectors. 

Function  SVD 

In  RPN,  function  SVD  (Singular  Value  Decomposition)  takes  as  input  a  matrix 
AnXm/  and  returns  the  matrices  UnXn/  VmXm,  and  a  vector  s  in  stack  levels  3,  2, 
and  1,  respectively.  The  dimension  of  vector  s  is  equal  to  the  minimum  of  the 
values  n  and  m.  The  matrices  U  and  V  are  as  defined  earlier  for  singular 
value  decomposition,  while  the  vector  s  represents  the  main  diagonal  of  the 
matrix  S  used  earlier. 

For  example,  in  RPN  mode:  C  C  5 ,  4 ,  - 1  ]  ,  C  2 ,  -3 ,  5 "! ,  C  7 ,  2 ,  8  "I  "I  SVD 

3:  [[-0.27  0.81  -0.53][-0.37  -0.59  -0.72][-0.89  3.09E-3  0.46]] 
2:  [[  -0.68  -0.14  -0.72][  0.42  0.73  -0.54][-0.60  0.67  0.44]] 
1:  [  12.15  6.88  1.42] 

Function  SVL 

Function  SVL  (Singular  VaLues)  returns  the  singular  values  of  a  matrix  AnXm  as 
a  vector  s  whose  dimension  is  equal  to  the  minimum  of  the  values  n  and  m. 
For  example,  in  RPN  mode,  C  C  5 ,  4 ,  - 1  ]  ,  L  2 ,  -3 ,  5  J  ,  C  7 ,  2 ,  8  "I  "I  SVL 
produces  [  12.15  6.88  1.42]. 

Function  SCHUR 

In  RPN  mode,  function  SCHUR  produces  the  Schur  decomposition  of  a  square 
matrix  A  returning  matrices  Q  and  T,  in  stack  levels  2  and  1,  respectively, 
such  that  A  =  Q  T  QT,  where  Q  is  an  orthogonal  matrix,  and  T  is  a  triangular 
matrix.    For  example,  in  RPN  mode, 

c  c  2 , 3 ,  ■■■■■  i  ::i  c  5 , 4 ,  ■-2  ::i  c  7 , 5 , 4  ::i  ::i  schur 

results  in: 

2:  [[0.66  -0.29  -0.70][-0.73  -0.01  -0.68][  -0.1 9  -0.96  0.21  ]] 


Page  1 1-50 


1 :  [[-1 .03  1 .02  3.86  ][  0  5.52  8.23  ][  0  -1 .82  5.52]] 
Function  LQ 

The  LQ  function  produces  the  LQ  factorization  of  a  matrix  AnXm  returning  a 
lower  LnXm  trapezoidal  matrix,  a  QmXm  orthogonal  matrix,  and  a  PnXn 
permutation  matrix,  in  stack  levels  3,  2,  and  1 .    The  matrices  A,  L,  Q  and  P 
are  related  by  PA  =  LQ.  (A  trapezoidal  matrix  out  of  an  nxm  matrix  is  the 
equivalent  of  a  triangular  matrix  out  of  an  nxn  matrix).  For  example, 
EC  1,  -2,  ill  2,  1,  -2]E  5,  -2,  ill  LQ 

produces 

3:  [[-5.48  0  0][-l  .1 0  -2.79  0][-l  .83  1 .43  0.78]] 

2:  [[-0.91  0.37-0.18]  [-0.36-0.50  0.79]  [-0.20-0.78  -0.59]] 

1:  [[0  0  1][0  1  0][1  0  0]] 

Function  QR 

In  RPN,  function  QR  produces  the  QR  factorization  of  a  matrix  AnXm  returning 
a  QnXn  orthogonal  matrix,  a  RnXm  upper  trapezoidal  matrix,  and  a  PmXm 
permutation  matrix,  in  stack  levels  3,  2,  and  1 .  The  matrices  A,  P,  Q  and  R 
are  related  by  A  P  =  Q  R.  For  example, 

EC  1,-2, ill  2, 1,-2]E  5,-2, ill  QR 
produces 

3:  [[-0.18  0.39  0.90][-0.37 -0.88  0.30][-0.91  0.28-0.30]] 
2:  [[  -5.48  -0.37  1 .83][  0  2.42  -2.20][0  0  -0.90]] 

1:  [[1  0  0][0  0  1][0  1  0]]  

Note:  Examples  and  definitions  for  all  functions  in  this  menu  are  available 
through  the  help  facility  in  the  calculator.  Try  these  exercises  in  ALG  mode  to 
see  the  results  in  that  mode. 


Matrix  Quadratic  Forms 

A  quadratic  form  from  a  square  matrix  A  is  a  polynomial  expression 
originated  from  x-AxT.  For  example,  if  we  use  A  =  [[2,1,-1][5,4,2][3,5,- 
1  ]],  and  x  =  [X  Y  Z]T,  the  corresponding  quadratic  form  is  calculated  as 


Page  11-51 


Finally, 


x  •  A  •  xr  =  [X    Y  Z] 


=  [X    Y  z\ 


x-AxT 


2  1  -1  X 
5    4    2   •  Y 

3  5-1  Z 
2X+Y-Z  ' 

5X  +  4Y  +  2Z 
3X  +  5Y-Z 
2X2+4Y2-Z2+6XY+2XZ+7ZY 


The  QUADF  menu 

The  HP  49  G  calculator  provides  the  QUADF  menu  for  operations  related  to 
QUADratic  Forms.  The  QUADF  menu  is  accessed  through  C+T) matrices  . 


matrices  hehu 


1.  CREATE.. 

2.  OPERATION.. 

3 .  FACTORIZATION. 


H .  QUADRATIC  FORH.. 


S .  LiriEHF;  SYSTEHS.. 

e.LinEAR  hffl.. 


HflTRIK  yUHC.  FORM  HERU 

3.GHUSS 

H .  GKR 

5. SYLVESTER 

fi.HHTRICES.. 

This  menu  includes  functions  AXQ,  CHOLESKY,  GAUSS,  QXA,  and 
SYLVESTER. 


Function  AXQ 

In  RPN  mode,  function  AXQ  produces  the  quadratic  form  corresponding  to  a 
matrix  AnXn  in  stack  level  2  using  the  n  variables  in  a  vector  placed  in  stack 
level  1 .  Function  returns  the  quadratic  form  in  stack  level  1  and  the  vector  of 
variables  in  stack  level  1 .  For  example, 

c  c  2 1 1 1  ■■■■■  i  ::i  i  c  5 1 4 , 2  ::i ,  c  3 , 5  *.  ■■■■■  1  ::i  ::i  ^ 

c 5  x 5 , 5  v 5 , 5  z 5  ::i  ^  rxq 

returns 

2:  '2*XA2+(6*Y+2*Z)*X+4*YA2+7*Z*y-ZA2' 
1 :  [X  'Y'  T] 


Page  11-52 


Function  QXA 

Function  QXA  takes  as  arguments  a  quadratic  form  in  stack  level  2  and  a 
vector  of  variables  in  stack  level  1,  returning  the  square  matrix  A  from  which 
the  quadratic  form  is  derived  in  stack  level  2,  and  the  list  of  variables  in  stack 
level  1 .     For  example, 

5  !::! 2 + Y ""■  — Z """  2 + s4  * !::! *  Y  —  1  6  * !::!  *  Z  5  [enter) 

c 5  x 5 , 5  y  5 , 5  z 5  ::i  [enter}  QXA 

returns 

2:  [[1  2-8][2  1  0][-8  0-1]] 
1 :  [X  'Y'  'Z'] 

Diagonal  representation  of  a  quadratic  form 

Given  a  symmetric  square  matrix  A,  it  is  possible  to  "diagonalize"  the  matrix 
A  by  finding  an  orthogonal  matrix  P  such  that  PT  A  P  =  D,  where  D  is  a 
diagonal  matrix.    If  Q  =  x  A  xT  is  a  quadratic  form  based  on  A,  it  is  possible 
to  write  the  quadratic  form  Q  so  that  it  only  contains  square  terms  from  a 
variable  y,  such  that  x  =  P  y,  by  using  Q  =  x  AxT  =  (P  y)  A  (P-y)T  = 
y(PTAP)y=yDyT 

Function  SYLVESTER 

Function  SYLVESTER  takes  as  argument  a  symmetric  square  matrix  A  and 
returns  a  vector  containing  the  diagonal  terms  of  a  diagonal  matrix  D,  and  a 
matrix  P,  so  that  PT  A  P  =  D.  For  example, 

c  c  2 ,  i ,  ■-■  i  ::i ,  c  i ,  4 , 2  ::i ,  c  ■■■■■  1 , 2 ,  ■■■■■  1  ::i  ::i  svl vester 

produces 

2:  [  1/2  2/7-23/7] 

1:  [[2  1  -1][0  7/2  5/2][0  0  1]] 

Function  GAUSS 

Function  GAUSS  returns  the  diagonal  representation  of  a  quadratic  form  Q  = 
x-A-xT  taking  as  arguments  the  quadratic  form  in  stack  level  2  and  the  vector 
of  variables  in  stack  level  1 .  The  result  of  this  function  call  is  the  following: 

•  An  array  of  coefficients  representing  the  diagonal  terms  of  D  (stack 
level  4) 

•  A  matrix  P  such  that  A  =  PT-D-P  (stack  level  3) 

•  The  diagonalized  quadratic  form  (stack  level  2) 


Page  1 1-53 


•  The  list  of  variables  (stack  level  1 ) 
For  example, 


c 5  x 5 , 5  v 5 , 5  z 5  ::i  ^  gruss 


returns 
4 
3 
2 
1 


[1  -0.333  20.333] 
[[1  2 -8] [0-3  16][0  0  1]] 

,61/3*ZA2+-l/3*(16*Z+-3*Y)A2+(-8*z+2*Y+X)A2/ 
[X  'Y'  T] 


Linear  Applications 


HflTRICES  HEHU 


r.FHCTORIIfiTIOn.. 
H.GURDRRTIC  FORH.. 
S .  LiriEHF:  SYSTEHS.. 

I  m  hi  1 1  ■  1 1 IM 

F.EIGEnVECTORS.. 
S .  VECTOR.. 


LiriEHF;  HFFL  METiU 

a.isoH 

?■ .  KEF; 
H.HKISOH 
5.  MATRICES.. 

menu  is  presented  below 

by  i 

calculator's  own  help  facility.  The  figures  show  the  help  facility  entry  and  the 
attached  examples. 


Function  IMAGE 


IMAGE: 

Image  of  a  linear  ap- 
plication of  matrix  M 
IMAGED  [1,2,  3],  [4,5,6] 

{[1    0]    [0  1]} 

See:  KER  BRSIS 


HELP 

™»fli§ID 


{[1  0]  [0  1]> 


Function  ISOM 


Finds  elements  of  a 
2-d  or  3-d  linear 
isometry 
ISOM<CC0,-13, Cli0]]> 

See:  MKISQM  


Page  11-54 


Function  KER 


Kernel  of  a  linear  ap- 
plication of  matrix  M 
<ER<CCl,2,3]i C4,5,6]]> 

See:  IMAGE 


Function  MKISOM 


MKISOM:  I 
Make  an  isometry  giuen 
its  elements 
MKISOM<tt,  1) 

[C-1,0], C0?-1]] 

See:  ISOM 


:  HELP 

{[-1  2  -1]> 

:HELP 

:MKIS0M(tt,1) 

Page  1 1-55 


Chapter  1 2 
Graphics 

In  this  chapter  we  introduce  some  of  the  graphics  capabilities  of  the  calculator. 
We  will  present  graphics  of  functions  in  Cartesian  coordinates  and  polar 
coordinates,  parametric  plots,  graphics  of  conies,  bar  plots,  scatterplots,  and 
a  variety  of  three-dimensional  graphs. 

Graphs  options  in  the  calculator 

To  access  the  list  of  graphic  formats  available  in  the  calculator,  use  the 
keystroke  sequence  LjnJi^  (GD  )    Please  notice  that  if  you  are  using  the 
RPN  mode  these  two  keys  must  be  pressed  simultaneously  to  activate  any  of 
the  graph  functions.  After  activating  the  2D/3D  function,  the  calculator  will 
produce  the  PLOT  SETUP  window,  which  includes  the  TYPE  field  as  illustrated 
below. 


woooooooooooooooooow  F  L  V  T  _■  E  T  U  F  .; 

TMD*:liH.L.UlJ.« 

Eft: 

Ind«p  =  K  _SiHUl.t 

^Connect 

H-TiCh:10.  Y-TiCh:10. 

Choos*  typ«  of  put 

Right  in  front  of  the  TYPE  field  you  will,  most  likely,  see  the  option  Function 
highlighted.    This  is  the  default  type  of  graph  for  the  calculator.  To  see  the 
list  of  available  graph  types,  press  the  soft  menu  key  labeled  BIHSE3.    This  will 
produce  a  drop  down  menu  with  the  following  options  (use  the  up-  and  down- 
arrow  keys  to  see  all  the  options): 


Page  12-1 


These  graph  options  are  described  briefly  next. 

Function:  for  equations  of  the  form  y  =  f(x)  in  plane  Cartesian  coordinates 
Polar:  for  equations  of  the  from  r  =  f(0)  in  polar  coordinates  in  the  plane 
Parametric:  for  plotting  equations  of  the  form  x  =  x(t),  y  =  y(t)  in  the  plane 
Diff  Eq:  for  plotting  the  numerical  solution  of  a  linear  differential  equation 
Conic:  for  plotting  conic  equations  (circles,  ellipses,  hyperbolas,  parabolas) 
Truth:  for  plotting  inequalities  in  the  plane 

Histogram:  for  plotting  frequency  histograms  (statistical  applications) 
Bar:  for  plotting  simple  bar  charts 

Scatter:  for  plotting  scatter  plots  of  discrete  data  sets  (statistical  applications) 
Slopefield:  for  plotting  traces  of  the  slopes  of  a  function  f(x,y)  =  0. 
Fast3D:    for  plotting  curved  surfaces  in  space 

Wireframe:  for  plotting  curved  surfaces  in  space  showing  wireframe  grids 

Ps-Contour:  for  plotting  contour  plots  of  surfaces 

Y-  Slice:  for  plotting  a  slicing  view  of  a  function  f(x,y). 

Gridmap:  for  plotting  real  and  imaginary  part  traces  of  a  complex  function 

Pr-Surface:  for  parametric  surfaces  given  by  x  =  x(u,v),  y  =  y(u,v),  z  =  z(u,v). 

Plotting  an  expression  of  the  form  y  =  f(x) 

In  this  section  we  present  an  example  of  a  plot  of  a  function  of  the  form  y  = 
f(x).  In  order  to  proceed  with  the  plot,  first,  purge  the  variable  x,  if  it  is 
defined  in  the  current  directory  (x  will  be  the  independent  variable  in  the 
calculator's  PLOT  feature,  therefore,  you  don't  want  to  have  it  pre-defined). 
Create  a  sub-directory  called  TPLOT'  (for  test  plot),  or  other  meaningful  name, 
to  perform  the  following  exercise. 
As  an  example,  let's  plot  the  function, 


•  First,  enter  the  PLOT  SETUP  environment  by  pressing,  CSDi^D  .  Make 
sure  that  the  option  Function  is  selected  as  the  TYPE,  and  that  'X'  is 
selected  as  the  independent  variable  (INDEP).     Press  C^^D i!!!ii!i!Ll:!iiiiii  to 


Page  1  2-2 


return  to  normal  calculator  display.  The  PLOT  SET  UP  window  should 
look  similar  to  this: 


wmmmrw  setups 

wmmm 

TyP"2:|i'liL-k+lJiB 

Eft: 

Ind«p  =  K  _SiHU(.t 

^Connect 

H-TiCh=iD.  V-TiCh:iD. 

^Pixels 

Choose  type  of  put 

•  Note:  You  will  notice  that  a  new  variable,  called  PPAR,  shows  up  in 
your  soft  menu  key  labels.  This  stands  for  Plot  PARameters.  To  see  its 
contents,  press  LrD  13233.  A  detailed  explanation  of  the  contents  of 
PPAR  is  provided  later  in  this  Chapter.  Press  L  +  J  to  drop  this  line 
from  the  stack. 


Enter  the  PLOT  environment  by  pressing  Ij^JJ^  (press  them 
simultaneously  if  in  RPN  mode).  Press  lul!!!  to  get  you  into  the 
equation  writer.  You  will  be  prompted  to  fill  the  right-hand  side  of  an 
equation  Yl  (x)  =  ■.  Type  the  function  to  be  plotted  so  that  the 
Equation  Writer  shows  the  following:  


Yl00=- 


J2T 


Press  (enter}  to  return  to  the  PLOT  SETUP  window.  The  expression 
'Yl(X)  =  EXP(-XA2/2)/V(2*7i)/  will  be  highlighted.  Press  to 
return  to  normal  calculator  display. 


Note:  Two  new  variables  show  up  in  your  soft  menu  key  labels,  namely 
EQ  and  Yl .  To  see  the  contents  of  EQ,  use  CSBSIli.  The  content  of  EQ 
is  simply  the  function  name  'Yl(X)'.  The  variable  EQ  is  used  by  the 
calculator  to  store  the  equation,  or  equations,  to  plot. 


To  see  the  contents  of  Yl  press  (j^J 
defined  as  the  program: 


You  will  get  the  function  Y1(X) 


Page  1  2-3 


«  'EXP  (-XA2/2)  /  V(2*7t)  '  ». 

Press  C3D/  twice,  to  drop  the  contents  of  the  stack. 


•     Enter  the  PLOT  WINDOW  environment  by  entering  CSDJ^L  (press 

them  simultaneously  if  in  RPN  mode).     Use  a  range  of  -4  to  4  for  H- 

VI EW,  then  press  EEQ3  to  generate  the  V-VIEW  automatically.  The 
PLOT  WINDOW  screen  looks  as  follows: 


^»FL0T  HiriDOH  - 

FunaiongBK 

H. 

Indcp  Loh:  DflFouU 

Hish: Default 

_PiX«  If 

Enter  HiniHUH  horizontal.  ual.u« 

•  Plot  the  graph:  Q3SH3  E33I]  (wait  till  the  calculator  finishes  the 
graphs) 

•  To  see  labels:     DEED  (nxT\  PI3II!!!  EHIILl 

•  To  recover  the  first  graphics  menu:  [nxtJ [nxtJ llllli! 

•  To  trace  the  curve:  IIIIiLiillilCllLEil  iSllllill!  .  Then  use  the  right-  and  left-arrow 
keys  (CD CD)  to  move  about  the  curve.  The  coordinates  of  the  points 
you  trace  will  be  shown  at  the  bottom  of  the  screen.  Check  that  for  x 
=  1.05  ,  y  =  0.231.  Also,  check  that  for  x  =  -1.48  ,  y  =  0.134. 
Here  is  picture  of  the  graph  in  tracing  mode: 


v:i.eoE-i 

•     To  recover  the  menu,  and  return  to  the  PLOT  WINDOW  environment, 

press  (NxTjBMa,  then  (W)B0i. 


Page  1  2-4 


Some  useful  PLOT  operations  for  FUNCTION  plots 

In  order  to  discuss  these  PLOT  options,  we'll  modify  the  function  to  force  it  to 
have  some  real  roots  (Since  the  current  curve  is  totally  contained  above  the  x 
axis,  it  has  no  real  roots.)  Press  OBI  to  list  the  contents  of  the  function 
Yl  on  the  stack:  «  ->X  /EXP(-XA2/2)/  V(2*ti)  '  ».  To  edit  this  expression 
use: 


(3D  (3D  (3D  OCX3QCX3 

[enter] 


Launches  the  line  editor 
Moves  cursor  to  the  end  of  the  line 
Modifies  the  expression 
Returns  to  calculator  display 


Next,  store  the  modified  expression  into  variable  y  by  using  C5D 
RPN  mode,  or  [jT]^  (TroT)  KB  in  ALG  mode. 


if  in 


1  x2 

The  function  to  be  plotted  is  now,  / (x)  =    .  exp(  )  -  0.1 

V2;r  2 


Enter  the  PLOT  WINDOW  environment  by  entering  CSDJ^L  (press  them 
simultaneously  if  in  RPN  mode.)  Keep  the  range  of  -4  to  4  for  H-VIEW, 
press  ^  HUH  to  generate  the  V-VIEW.  To  plot  the  graph,  press  EMI 


Once  the  graph  is  plotted,  press  iillll!!!  to  access  the  function  menu. 
With  this  menu  you  can  obtain  additional  information  about  the  plot 
such  as  intersects  with  the  x-axis,  roots,  slopes  of  the  tangent  line, 
area  under  the  curve,  etc. 

For  example,  to  find  the  root  on  the  left  side  of  the  curve,  move  the 
cursor  near  that  point,  and  press  III31I  You  will  get  the  result:  ROOT: 
-1.6635....   Press  [nxt)  to  recover  the  menu.      Here  is  the  result  of 


"553 

Page  1  2-5 


•  If  you  move  the  cursor  towards  the  right-hand  side  of  the  curve,  by 
pressing  the  right-arrow  key  ((T)),  and  press  IlliSlijl,  the  result  now  is 
ROOT:  1.6635...  The  calculator  indicated,  before  showing  the  root, 
that  it  was  found  through  SIGN  REVERSAL  Press  (W)  to  recover  the 
menu. 

•  Pressing  lllllllj!  will  give  you  the  intersection  of  the  curve  with  the  x- 
axis,  which  is  essentially  the  root.  Place  the  cursor  exactly  at  the  root 
and  press  You  will  get  the  same  message  as  before,  namely 
SIGN  REVERSAL,  before  getting  the  result  l-SECT:  1.6635....  The 
EllIS!  function  is  intended  to  determine  the  intersection  of  any  two 
curves  closest  to  the  location  of  the  cursor.  In  this  case,  where  only 
one  curve,  namely,  Y1(X),  is  involved,  the  intersection  sought  is  that  of 
f(x)  with  the  x-axis,  however,  you  must  place  the  cursor  right  at  the 
root  to  produce  the  same  result.  Press  LnxtJ  to  recover  the  menu. 

•  Place  the  cursor  on  the  curve  at  any  point  and  press  illlill^  to  get  the 
value  of  the  slope  at  that  point.  For  example,  at  the  negative  root, 
SLOPE:  0.16670....  Press  [nxT)  to  recover  the  menu. 

•  To  determine  the  highest  point  in  the  curve,  place  the  cursor  near  the 
vertex  and  press  BSE  The  result  is  EXTRM:  0..  Press  [nxtJ  to  recover 
the  menu. 

•  Other  buttons  available  in  the  first  menu  are  il3I!i!3  to  calculate  the 
area  under  the  curve,  and  EEEEB  to  shade  an  area  under  the  curve. 
Press  [nxt)  to  see  more  options.  The  second  menu  includes  one  button 
called  EHiiili  that  flashes  for  a  few  seconds  the  equation  plotted.  Press 
EBU  Alternatively,  you  can  press  the  button  DEEQ  (NEXt  eQuation)  to 
see  the  name  of  the  function  Yl  (x).  Press  jNxrj  to  recover  the  menu. 

•  The  button  gives  the  value  of  f(x)  corresponding  to  the  cursor 
position.  Place  the  cursor  anywhere  in  the  curve  and  press  The 
value  will  be  shown  in  the  lower  left  corner  of  the  display.  Press  [nxtJ 
to  recover  the  menu. 

•  Place  the  cursor  in  any  given  point  of  the  trajectory  and  press  TANL 
to  obtain  the  equation  of  the  tangent  line  to  the  curve  at  that  point. 
The  equation  will  be  displayed  on  the  lower  left  corner  of  the  display. 
Press  (nxt)  to  recover  the  menu. 

•  If  you  press  IB 'III  the  calculator  will  plot  the  derivative  function,  f'(x) 
=  df/dx,  as  well  as  the  original  function,  f(x).    Notice  that  the  two 


Page  1  2-6 


curves  intercept  at  two  points.  Move  the  cursor  near  the  left  intercept 
point  and  press  illl  [SSI,  to  get  l-SECT:  (-0.6834... ,0.21 585). 
Press  (nxt)  to  recover  the  menu. 

•  To  leave  the  FCN  environment,  press  iilliii  (or  InxtJ'MMm). 

•  Press  B3IHI  to  return  to  the  PLOT  WINDOW  environment.  Then, 
press  [nxtJ  iiiiii=!=i~i:iiiiiil  to  return  to  normal  calculator  display. 


Note:  the  stack  will  show  all  the  graph  operations  performed,  properly 
identified. 


•  Enter  the  PLOT  environment  by  pressing,  simultaneously  if  in  RPN 
mode,  (Jnjj^.  Notice  that  the  highlighted  field  in  the  PLOT 
environment  now  contains  the  derivative  of  Yl  (X).  Press  [nxtJ IIIIIElI:!!!!!!!  to 
return  to  return  to  normal  calculator  display. 

•  Press  Lr^J EI9I  to  check  the  contents  of  EQ.  You  will  notice  that  it 
contains  a  list  instead  of  a  single  expression.  The  list  has  as  elements 
an  expression  for  the  derivative  of  Y1(X)  and  Y1(X)  itself. 
Originally,  EQ  contained  only  Yl(x).   After  we  pressed  1111  III!  in  the 

environment,  the  calculator  automatically  added  the  derivative  of 
Yl  (x)  to  the  list  of  equations  in  EQ. 

Saving  a  graph  for  future  use 

If  you  want  to  save  your  graph  to  a  variable,  get  into  the  PICTURE 
environment  by  pressing  CD .  Then,  press  HHilW|(W)HHi-^.  This 
captures  the  current  picture  into  a  graphics  object.    To  return  to  the  stack, 

 »-::l:::»::;:  :s  ::"»:»■:=«:::■:■: 

DreSS  il.ili:.:!:::!!.!!  II=:ff.a,I.Lh::!.::I . 

In  level  1  of  the  stack  you  will  see  a  graphics  object  described  as  Graphic 
131  x  64.  This  can  be  stored  into  a  variable  name,  say,  PIC  1 . 

To  display  your  figure  again,  recall  the  contents  of  variable  PIC1  to  the  stack. 
The  stack  will  show  the  line:  Graphic  131  x  64.  To  see  the  graph,  enter 
the  PICTURE  environment,  by  pressing  GD  • 

Clear  the  current  picture,  IHiiiilil  [nxtJ B3JB3. 


Page  1  2-7 


Move  the  cursor  to  the  upper  left  corner  of  the  display,  by  using  the  GD 
and  keys. 

To  display  the  figure  currently  in  level  1  of  the  stack  press  jNxrj  REPL  . 
To  return  to  normal  calculator  function,  press  IllSjlul  E3B31. 


Note:  To  save  printing  space,  we  will  not  include  more  graphs  produced  by 
following  the  instructions  in  this  Chapter.  The  user  is  invited  to  produce  those 
graphics  on  his  or  her  own. 


Graphics  of  transcendental  functions 

In  this  section  we  use  some  of  the  graphics  features  of  the  calculator  to  show 
the  typical  behavior  of  the  natural  log,  exponential,  trigonometric  and 
hyperbolic  functions.  You  will  not  see  more  graphs  in  this  chapter,  instead 
the  user  should  see  them  in  the  calculator. 

Graph  of  ln(X) 

Press,  simultaneously  if  in  RPN  mode,  the  left-shift  key  C3D  and  the  J^d  (f7T) ) 
key  to  produce  the  PLOT  SETUP  window.  The  field  labeled  Type  will  be 
highlighted.  If  the  option  Function  is  not  already  selected  press  the  soft  key 
labeled  Ijliillllilll,  use  the  up  and  down  keys  to  select  Function,  and  press  IIEI3IIIII 
to  complete  the  selection.  Check  that  the  field  labeled  Indep:  contains  the 
variable  'X'.  If  that  is  not  so,  press  the  down  arrow  key  twice  until  the  indep 
field  is  highlighted,  press  the  soft  key  labeled  III3II!  and  modify  the  value  of 
the  independent  variable  to  read  'X'.  Press  iliilll  when  done.  Press 
L^ZlJ !!!!ii!i!Ll:iii!i!i  to  return  to  normal  calculator  display. 

Next,  well  resize  the  plot  window.  First,  press,  simultaneously  if  in  RPN 
mode,  the  left-shift  key  C3D  and  the  (CjD  )  key  to  produce  the  PLOT- 
FUNCTION  window.  If  there  is  any  equation  highlighted  in  this  window, 
press  llliill  as  needed  to  clear  the  window  completely.  When  the  PLOT- 
FUNCTION  window  is  empty  you  will  get  a  prompt  message  that  reads:  No 
Equ.,  Press  add.    Press  the  soft  key  labeled   1333  .  This  will  trigger  the 


Page  1  2-8 


equation  writer  with  the  expression  Y1(X)=<4  .  Type  LN(X).  Press  [enter]  to 
return  to  the  PLOT-FUNCTION  window.  Press  (W)KEB  to  return  to  normal 
calculator  display. 

The  next  step  is  to  press,  simultaneously  if  in  RPN  mode,  the  left-shift  key  C5D 
and  the  J^L  (GD  )  key  to  produce  the  PLOT  WINDOW  -  FUNCTION  window. 
Most  likely,  the  display  will  show  the  horizontal  (H-View)  and  vertical  (v-view) 
ranges  as:  H-View: -6.5       6.5,  V-View:-3.1  3.2 

These  are  the  default  values  for  the  x-  and  y-range,  respectively,  of  the  current 
graphics  display  window.  Next,  change  the  H-View  values  to  read:  H-View:  -1 
10  by  using  ODt+D^BMCDUDBSM.  Next,  press  the  soft  key  labeled 
EEH9  to  let  the  calculator  determine  the  corresponding  vertical  range.  After  a 
couple  of  seconds  this  range  will  be  shown  in  the  PLOT  WINDOW- 
FUNCTION  window.  At  this  point  we  are  ready  to  produce  the  graph  of 
ln(X).  Press  1333:33 13333  to  plot  the  natural  logarithm  function. 

To  add  labels  to  the  graph  press  B39(W)[!SI3I!.  Press  HHHD  to  remove  the 
menu  labels,  and  get  a  full  view  of  the  graph.  Press  jNxrj  to  recover  the 
current  graphic  menu.  Press  InxtJ  13333  to  recover  the  original  graphical  menu. 

To  determine  the  coordinates  of  points  on  the  curve  press  1133333  (the  cursor 
moves  on  top  of  the  curve  at  a  point  located  near  the  center  of  the  horizontal 
range).  Next,  press  (X,Y)  to  see  the  coordinates  of  the  current  cursor 
location.  These  coordinates  will  be  shown  at  the  bottom  of  the  screen.  Use 
the  right-  and  left-arrow  keys  to  move  the  cursor  along  the  curve.  As  you 
move  the  cursor  along  the  curve  the  coordinates  of  the  curve  are  displayed  at 
the  bottom  of  the  screen.  Check  that  when  Y:l  .00E0,  X:2.72E0.  This  is  the 
point  (e,  7),  since  ln(e)  =  7.  Press  LnxtJ  to  recover  the  graphics  menu. 

Next,  we  will  find  the  intersection  of  the  curve  with  the  x-axis  by  pressing 
iilllili  IImII  The  calculator  returns  the  value  Root:  l,  confirming  that  ln(l)  =  0. 
Press  [Nxr)[NXT)mm  mSm  to  return  to  the  PLOT  WINDOW  -  FUNCTION. 
Press  [enter)  to  return  to  normal  calculator  display.  You  will  notice  that  the  root 
found  in  the  graphics  environment  was  copied  to  the  calculator  stack. 


Page  1  2-9 


Note:  When  you  press  G^D  ,  your  variables  list  will  show  new  variables 
called  III!!!  and  illlilHI  .  Press  CBHB  to  see  the  contents  of  this 
variable.  You  will  get  the  program  «  -»  X  'LN(X)'  »  ,  which  you  will 
recognize  as  the  program  that  may  result  from  defining  the  function  'Yl(X)  = 
LN(X)'  by  using  ( jnj PE  .  This  is  basically  what  happens  when  you  1339  a 
function  in  the  PLOT  -  FUNCTION  window  (the  window  that  results  from 
pressing  <-i  Jh^,  simultaneously  if  in  RPN  mode),  i.e.,  the  function  gets 
defined  and  added  to  your  variable  list. 

Next,  press  L  r>  J 1151  to  see  the  contents  of  this  variable.  A  value  of  1 0.275  is 
placed  in  the  stack.  This  value  is  determined  by  our  selection  for  the 
horizontal  display  range.  We  selected  a  range  between  -1  and  10  for  X.  To 
produce  the  graph,  the  calculator  generates  values  between  the  range  limits 
using  a  constant  increment,  and  storing  the  values  generated,  one  at  a  time, 
in  the  variable  111!!!!!  as  the  graph  is  drawn.  For  the  horizontal  range  (  -1,1  0), 
the  increment  used  seems  to  be  0.275.  When  the  value  of  X  becomes  larger 
than  the  maximum  value  in  the  range  (in  this  case,  when  X  =  10.275),  the 
drawing  of  the  graph  stops.  The  last  value  of  X  for  the  graphic  under 
consideration  is  kept  in  variable  X.    Delete  X  and  Yl  before  continuing. 


Graph  of  the  exponential  function 

First,  load  the  function  exp(X),  by  pressing,  simultaneously  if  in  RPN  mode,  the 
left-shift  key  C5D  and  the  (C^D)  key  to  access  the  PLOT-FUNCTION 

window.    Press  illl!!!  to  remove  the  function  LN(X),  if  you  didn't  delete  Yl  as 

suggested  in  the  previous  note.  Press  1333  and  type  C5D^  [alpha}(x]  [enter]  to 

enter  EXP(X)  and  return  to  the  PLOT-FUNCTION  window.  Press  \JkT\WBM  to 
return  to  normal  calculator  display. 

Next,  press,  simultaneously  if  in  RPN  mode,  the  left-shift  key  LjnJ  and  the 
ML  (<JD)  key  to  produce  the  PLOT  WINDOW  -  FUNCTION  window. 
Change  the  H-View  values  to  read:  H-View:  -8  2 

by  using  (jTJ MaCZD MM.  Next,  press  EED9.  After  the  vertical  range 
is  calculated,  press  BjEEQ  13333  to  plot  the  exponential  function. 


Page  12-10 


To  add  labels  to  the  graph  press  IHJilil  [nxtJ I3EH=H.  Press  CHIE  to  remove  the 
menu  labels,  and  get  a  full  view  of  the  graph.  Press  Inxt j (jvxr j to 
return  to  the  PLOT  WINDOW  -  FUNCTION.  Press  [enter}  to  return  to  normal 
calculator  display. 

The  PPAR  variable 

Press  G^D  to  recover  your  variables  menu,  if  needed.  In  your  variables  menu 
you  should  have  a  variable  labeled  PPAR  .  Press  C3BH3  to  get  the 
contents  of  this  variable  in  the  stack.  Press  the  down-arrow  key,  ,  to  launch 
the  stack  editor,  and  use  the  up-  and  down-arrow  keys  to  view  the  full  contents 
of  PPAR.  The  screen  will  show  the  following  values: 


U-3. 10797263231)  12* 
RPL>  £ 

OS.  ,-1.  107972632B1) 
<2. ,7-38905609893}  X 
S.   <B.,0.>  FUNCTION  Y 

PPAR  stands  for  Plot  PARameters,  and  its  contents  include  two  ordered  pairs  of 
real  numbers,  (-8.,-!  .10797263281)  and  (2.,7.38905609893), 

which  represent  the  coordinates  of  the  lower  left  corner  and  the  upper  right 
corner  of  the  plot,  respectively.  Next,  PPAR  lists  the  name  of  the  independent 
variable,  X,  followed  by  a  number  that  specifies  the  increment  of  the 
independent  variable  in  the  generation  of  the  plot.  The  value  shown  here  is 
the  default  value,  zero  (0.),  which  specifies  increments  in  X  corresponding  to 
1  pixel  in  the  graphics  display.  The  next  element  in  PPAR  is  a  list  containing 
first  the  coordinates  of  the  point  of  intersection  of  the  plot  axes,  i.e.,  (0.,0.), 
followed  by  a  list  that  specifies  the  tick  mark  annotation  on  the  x-  and  y-axes, 
respectively  {#  lOd  #  10d}.  Next,  PPAR  lists  the  type  of  plot  that  is  to  be 
generated,  i.e.,  FUNCTION,  and,  finally,  the  y-axis  label,  i.e.,  Y. 

The  variable  PPAR,  if  non-existent,  is  generated  every  time  you  create  a  plot. 
The  contents  of  the  function  will  change  depending  on  the  type  of  plot  and  on 
the  options  that  you  select  in  the  PLOT  window  (the  window  generated  by  the 
simultaneous  activation  of  the  LjnJ  and  J^L  (GD  )  keys. 


Page  12-11 


Inverse  functions  and  their  graphs 

Let  y  =  f(x),  if  we  can  find  a  function  y  =  g(x),  such  that,  g(f(x))  =  x,  then  we 
say  that  g(x)  is  the  inverse  function  of  f(x).  Typically,  the  notation  g(x)  =  f  '](x) 
is  used  to  denote  an  inverse  function.  Using  this  notation  we  can  write:  if  y  = 
f(x),  then  x  =  f  ](y).    Also,  W  ^x))  =  x,  and  f  ](f(x))  =  x. 

As  indicated  earlier,  the  ln(x)  and  exp(x)  functions  are  inverse  of  each  other, 
i.e.,  In(exp(x))  =  x,  and  exp(ln(x))  =  x.  This  can  be  verified  in  the  calculator 
by  typing  and  evaluating  the  following  expressions  in  the  Equation  Writer: 
LN(EXP(X))  and  EXP(LN(X)).  They  should  both  evaluate  to  X. 

When  a  function  f(x)  and  its  inverse  f  ~](x)  are  plotted  simultaneously  in  the 
same  set  of  axes,  their  graphs  are  reflections  of  each  other  about  the  line  y  = 
x.  Let's  check  this  fact  with  the  calculator  for  the  functions  LN(X)  and  EXP(X) 
by  following  this  procedure: 

Press,  simultaneously  if  in  RPN  mode,  C53^  .  The  function  Y1(X)  =  EXP(X) 
should  be  available  in  the  PLOT  -  FUNCTION  window  from  the  previous 
exercise.  Press  IIII3SII  ,  and  type  the  function  Y200  LhKX).  Also,  load 
the  function  Y3(X)  =  X.  Press  C^lIHW  to  return  to  normal  calculator  display. 

Press,  simultaneously  if  in  RPN  mode,  CSJ^L,  and  change  the  H-View 
range  to  read:  H-View:  -8  8 

Press  EUEQ  to  generate  the  vertical  range.  Press  !l;!!jili§[=!!  ![y!i:!!i!H  to  produce  the 
graph  of  y  =  ln(x),  y  =  exp(x),  and  y  =x,  simultaneously  if  in  RPN  mode. 

You  will  notice  that  only  the  graph  of  y  =  exp(x)  is  clearly  visible.  Something 
went  wrong  with  the  EED3  selection  of  the  vertical  range.  What  happens  is 
that,  when  you  press  EEB3  in  the  PLOT  FUNCTION  -  WINDOW  screen,  the 
calculator  produces  the  vertical  range  corresponding  to  the  first  function  in  the 
list  of  functions  to  be  plotted.  Which,  in  this  case,  happens  to  be  Y1(X)  = 
EXP(X).  We  will  have  to  enter  the  vertical  range  ourselves  in  order  to  display 
the  other  two  functions  in  the  same  plot. 


Page  12-12 


Press  MSm  to  return  to  the  PLOT  FUNCTION  -  WINDOW  screen.  Modify 
the  vertical  and  horizontal  ranges  to  read:  H-View:  -8       8,  V-View:  -4  4 

By  selecting  these  ranges  we  ensure  that  the  scale  of  the  graph  is  kept  1 
vertical  to  1  horizontal.  Press  ESSEBEiEE!  and  you  will  get  the  plots  of  the 
natural  logarithm,  exponential,  and  y  =  x  functions.  It  will  be  evident  from  the 
graph  that  LN(X)  and  EXP(X)  are  reflections  of  each  other  about  the  line  y  =  X. 
Press  HEH!  to  return  to  the  PLOT  WINDOW  -  FUNCTION.  Press  (enter)  to 
return  to  normal  calculator  display. 

Summary  of  FUNCTION  plot  operation 

In  this  section  we  present  information  regarding  the  PLOT  SETUP,  PLOT- 
FUNCTION,  and  PLOT  WINDOW  screens  accessible  through  the  left-shift  key 
combined  with  the  soft-menu  keys  CjD  through  C2D  •  Based  on  the  graphing 
examples  presented  above,  the  procedure  to  follow  to  produce  a  FUNCTION 
plot  (i.e.,  one  that  plots  one  or  more  functions  of  the  form  Y  =  F(X)),  is  the 
following: 

C5Di^,  simultaneously  if  in  RPN  mode:  Access  to  the  PLOT  SETUP  window. 
If  needed,  change  type  to  function,  and  enter  the  name  of  the  independent 
variable. 
Settings: 

•  A  check  on  _simult  means  that  if  you  have  two  or  more  plots  in  the 
same  graph,  they  will  be  plotted  simultaneously  when  producing  the 
graph. 

•  A  check  on  _Connect  means  that  the  curve  will  be  a  continuous  curve 
rather  than  a  set  of  individual  points. 

•  A  check  on  Pixels  means  that  the  marks  indicated  by  H-Tick  and  v- 
Tick  will  be  separated  by  that  many  pixels. 

•  The  default  value  for  both  by  H-Tick  and  v-Tick  is  1 0. 
Soft  key  menu  options: 

•  Use  ESS  to  edit  functions  of  values  in  the  selected  field. 

•  Use  BlEEB  to  select  the  type  of  plot  to  use  when  the  Type :  field  is 
highlighted.  For  the  current  exercises,  we  want  this  field  set  to 
FUNCTION. 


Page  12-13 


Note:  the  soft  menu  keys  DEED  and  mmm  are  not  available  at  the  same 
time.  One  or  the  other  will  be  selected  depending  on  which  input  field  is 
highlighted. 


•  Press  the  AXES  soft  menu  key  to  select  or  deselect  the  plotting  of  axes  in 
the  graph.  If  the  option  'plot  axes'  is  selected,  a  square  dot  will  appear 
in  the  key  label:  EEEBb  .  Absence  of  the  square  dot  indicates  that  axes 
will  not  be  plotted  in  the  graph. 

•  Use  H3SEH  to  erase  any  graph  currently  existing  in  the  graphics  display 
window. 

•  Use  illilEIII  to  produce  the  graph  according  to  the  current  contents  of  PPAR 
for  the  equations  listed  in  the  PLOT-FUNCTION  window. 

•  Press  [nxtJ  to  access  the  second  set  of  soft  menu  keys  in  this  screen. 

•  Use  lllillil  to  reset  any  selected  field  to  its  default  value. 

•  Use  EEEH!  to  cancel  any  changes  to  the  PLOT  SETUP  window  and  return 
to  normal  calculator  display. 

•  Press  llliillll  to  save  changes  to  the  options  in  the  PLOT  SETUP  window  and 
return  to  normal  calculator  display. 

C5D_^,  simultaneously  if  in  RPN  mode:  .  Access  to  the  PLOT  window  (in 
this  case  it  will  be  called  PLOT  -FUNCTION  window). 

Soft  menu  key  options: 

•  Use  iilllilli!  to  edit  the  highlighted  equation. 

•  Use  EISII  to  add  new  equations  to  the  plot. 

Note:  IIEEI3II  or  IIIHil  will  trigger  the  equation  writer  EQW  that  you  can 
use  to  write  new  equations  or  edit  old  equations. 

•  Use  IIIIIII!!  to  remove  the  highlighted  equation. 

•  Use  BUSES  to  add  an  equation  that  is  already  defined  in  your  variables 
menu,  but  not  listed  in  the  PLOT  -  FUNCTION  window. 

•  Use  iLiiu:!!!!!!::!!:!!  to  erase  any  graph  currently  existing  in  the  graphics  display 
window. 

•  Use  13333  to  produce  the  graph  according  to  the  current  contents  of  PPAR 
for  the  equations  listed  in  the  PLOT-FUNCTION  window. 

•  Press  [nxtJ  to  activate  the  second  menu  list. 


Page  12-14 


•  Use  CEEQQI  and  CBEQQ  to  move  the  selected  equation  one  location  up  or 
down,  respectively. 

•  Use  EffiiEE!  if  you  want  to  clear  all  the  equations  currently  active  in  the 
PLOT  -  FUNCTION  window.  The  calculator  will  verify  whether  or  not 
you  want  to  clear  all  the  functions  before  erasing  all  of  them.  Select  YES, 
and  press  11311  to  proceed  with  clearing  all  functions.  Select  NO,  and 
press  III31I!!!!  to  de-activate  the  option  CLEAR. 

•  Press  iiiiii=i=ili:iiiiiil  when  done  to  return  to  normal  calculator  display. 

CSDJ^L,  simultaneously  if  in  RPN  mode:  Access  to  the  PLOT  WINDOW 
screen. 

Settings: 

•  Enter  lower  and  upper  limits  for  horizontal  view  (H-View)  and  vertical 
view  (V-View)  ranges  in  the  plot  window.  Or, 

•  Enter  lower  and  upper  limits  for  horizontal  view  (H-View),  and  press  II3IS13, 
while  the  cursor  is  in  one  of  the  V-View  fields,  to  generate  the  vertical 
view  (V-View)  range  automatically.  Or, 

•  Enter  lower  and  upper  limits  for  vertical  view  (V-View),  and  press  EEBEI, 
while  the  cursor  is  in  one  of  the  H-View  fields,  to  generate  the  horizontal 
view  (H-View)  range  automatically. 

•  The  calculator  will  use  the  horizontal  view  (H-View)  range  to  generate 
data  values  for  the  graph,  unless  you  change  the  options  Indep  Low, 
(Indep)  High,  and  (Indep)  Step.  These  values  determine,  respectively,  the 
minimum,  maximum,  and  increment  values  of  the  independent  variable  to 
be  used  in  the  plot.  If  the  option  Default  is  listed  in  the  fields  Indep  Low, 
(Indep)  High,  and  (Indep)  Step,  the  calculator  will  use  the  minimum  and 
maximum  values  determined  by  H-View. 

•  A  check  on  _Pixels  means  that  the  values  of  the  independent  variable 
increments  (Step:)  are  given  in  pixels  rather  than  in  plot  coordinates. 

Soft  menu  key  options: 

•  Use  BUS  to  edit  any  entry  in  the  window. 

•  Use  ll!i!l]3il!l  as  explained  in  Settings,  above. 


Page  12-15 


•  Use  H3SEH  to  erase  any  graph  currently  existing  in  the  graphics  display 
window. 

•  Use  13333  to  produce  the  graph  according  to  the  current  contents  of  PPAR 
for  the  equations  listed  in  the  PLOT-FUNCTION  window. 

•  Press  jNxrj  to  activate  the  second  menu  list. 

•  Use  OSS  to  reset  the  field  selected  (i.e.,  where  the  cursor  is  positioned) 
to  its  default  value. 

•  Use  IliiilSIiil  to  access  calculator  stack  to  perform  calculations  that  may  be 
necessary  to  obtain  a  value  for  one  of  the  options  in  this  window.  When 
the  calculator  stack  is  made  available  to  you,  you  will  also  have  the  soft 
menu  key  options  1133111  and  111311!  . 

•  Use  IEiil!i![i!l!jj[!!l  in  case  you  want  to  cancel  the  current  calculation  and  return  to 
the  PLOT  WINDOW  screen.  Or, 

•  Use  liliiilill  to  accept  the  results  of  your  calculation  and  return  to  the  PLOT 
WINDOW  screen. 

•  Use  llililli!  to  get  information  on  the  type  of  objects  that  can  be  used  in 
the  selected  option  field. 

•  Use  H3331  to  cancel  any  changes  to  the  PLOT  WINDOW  screen  and 
return  to  normal  calculator  display. 

•  Press  liiilll  to  accept  changes  to  the  PLOT  WINDOW  screen  and  return 
to  normal  calculator  display. 

[j^Jgraph  f  simultaneously  if  in  RPN  mode:  Plots  the  graph  based  on  the 
settings  stored  in  variable  PPAR  and  the  current  functions  defined  in  the 
PLOT  -  FUNCTION  screen.  If  a  graph,  different  from  the  one  you  are  plotting, 
already  exists  in  the  graphic  display  screen,  the  new  plot  will  be 
superimposed  on  the  existing  plot.  This  may  not  be  the  result  you  desire, 
therefore,  I  recommend  to  use  the  13333=11333]]  soft  menu  keys  available  in  the 
PLOT  SETUP,  PLOT-FUNCTION  or  PLOT  WINDOW  screens. 

Plots  of  trigonometric  and  hyperbolic  functions 

The  procedures  used  above  to  plot  LN(X)  and  EXP(X),  separately  or 
simultaneously,  can  be  used  to  plot  any  function  of  the  form  y  =  f(x).  It  is  left 
as  an  exercise  to  the  reader  to  produce  the  plots  of  trigonometric  and 
hyperbolic  functions  and  their  inverses.  The  table  below  suggests  the  values 
to  use  for  the  vertical  and  horizontal  ranges  in  each  case.   You  can  include 


Page  12-16 


the  function  Y=X  when  plotting  simultaneously  a  function  and  its  inverse  to 
verify  their  'reflection'  about  the  line  Y  =  X. 


H-View  range 

V-View  range 

Function 

Minimum 

Maximum 

Minimum 

Maximum 

SIN(X) 

-3.15 

3.15 

AUTO 

ASIN(X) 

■1.2 

1.2 

AUTO 

SIN  &  ASIN 

-3.2 

3.2 

-1.6 

1.6 

COS(X) 

-3.15 

3.15 

AUTO 

ACOS(X) 

■1.2 

1.2 

AUTO 

COS  &  ACOS 

-3.2 

3.2 

■1.6 

1.6 

TAN(X) 

-3.15 

3.15 

-10 

10 

ATAN(X) 

-10 

10 

-1.8 

1.8 

TAN  &  ATAN 

-2 

-2 

-2 

-2 

SINH(X) 

-2 

2 

AUTO 

ASINH(X) 

-5 

5 

AUTO 

SINH  &  ASINH 

-5 

5 

-5 

5 

COSH(X) 

-2 

2 

AUTO 

ACOSH(X) 

-1 

5 

AUTO 

COS  &  ACOS 

-5 

5 

-1 

5 

TANH(X) 

-5 

5 

AUTO 

ATANH(X) 

■1.2 

1.2 

AUTO 

TAN  &  ATAN 

-5 

5 

-2.5 

2.5 

Generating  a  table  of  values  for  a  function 

The  combinations  (GD  )  and  C3D  J^i  (C2D  )/  pressed  simultaneously 

if  in  RPN  mode,  let's  the  user  produce  a  table  of  values  of  functions.  For 
example,  we  will  produce  a  table  of  the  function  Y(X)  =  X/(X+10),  in  the 
range  -5  <  X  <  5  following  these  instructions: 

•  We  will  generate  values  of  the  function  f(x),  defined  above,  for  values  of 
x  from  -5  to  5,  in  increments  of  0.5.  First,  we  need  to  ensure  that  the 
graph  type  is  set  to  FUNCTION  in  the  PLOT  SETUP  screen  J^, ,  press 
them  simultaneously,  if  in  RPN  mode).  The  field  in  front  of  the  Type  option 


Page  12-17 


will  be  highlighted.  If  this  field  is  not  already  set  to  FUNCTION,  press  the 
soft  key  llii!;!;!;!;!;!!::!  and  select  the  FUNCTION  option,  then  press  111:1111. 

•  Next,  press  to  highlight  the  field  in  front  of  the  option  EQ,  and  type 
the  function  expression:  'X/(X+10)' 

•  To  accept  the  changes  made  to  the  PLOT  SETUP  screen  press  jNxrj  EI9I. 
You  will  be  returned  to  normal  calculator  display. 

•  The  next  step  is  to  access  the  Table  Set-up  screen  by  using  the  keystroke 
combinationCSDlMI  (i.e.,  soft  key  (jD  )  -  simultaneously  if  in  RPN  mode. 
This  will  produce  a  screen  where  you  can  select  the  starting  value  (Start) 
and  the  increment  (Step).  Enter  the  following:  L5JL!LJ  ISllI 
CBCTGDWM  CBCZJCS— (i.e.,  Zoom  factor  =  0.5).  Toggle 
the  IKIHiI!:!  soft  menu  key  until  a  check  mark  appears  in  front  of  the  option 
Small  Font  if  you  so  desire.  Then  press  I!!!!!!!!!!!:!!!!.  This  will  return  you  to 
normal  calculator  display. 

The  TPAR  variable 


After  finishing  the  table  set  up,  your  calculator  will  create  a  variable  called 
TPAR  (Table  PARameters)  that  store  information  relevant  to  the  table  that  is  to 
be  generated.  To  see  the  contents  of  this  variable,  press  C^iiililiEi. 


•  To  see  the  table,  press  LjnJi^i  (i.e./  soft  menu  key  CJL) )  ~  simultaneously 
if  in  RPN  mode.  This  will  produce  a  table  of  values  of  x  =  -5,  -4.5, 
and  the  corresponding  values  of  f(x),  listed  as  Yl  by  default.  You  can  use 
the  up  and  down  arrow  keys  to  move  about  in  the  table.  You  will  notice 
that  we  did  not  have  to  indicate  an  ending  value  for  the  independent 
variable  x.  Thus,  the  table  continues  beyond  the  maximum  value  for  x 
suggested  early,  namely  x  =  5. 

Some  options  available  while  the  table  is  visible  are  ES3S33,  SIIII:!!!,  and 

•  The  lililili],  when  selected,  shows  the  definition  of  the  independent  variable. 

•  The  IlllISi  key  simply  changes  the  font  in  the  table  from  small  to  big,  and 
vice  versa.  Try  it. 


Page  12-18 


•     The         key,  when  pressed,  produces  a  menu  with  the  options:  In,  Out, 
Decimal,  Integer,  and  Trig.  Try  the  following  exercises: 


•  With  the  option  In  highlighted,  press  I1I3I.  The  table  is  expanded  so 
that  the  x-increment  is  now  0.25  rather  than  0.5.  Simply,  what  the 
calculator  does  is  to  multiply  the  original  increment,  0.5,  by  the  zoom 
factor,  0.5,  to  produce  the  new  increment  of  0.25.  Thus,  the  zoom  in 
option  is  useful  when  you  want  more  resolution  for  the  values  of  x  in 
your  table. 

•  To  increase  the  resolution  by  an  additional  factor  of  0.5  press  EIMlI, 
select  In  once  more,  and  press  HA  The  x-increment  is  now  0.01  25. 

•  To  recover  the  previous  x-increment,  press  HIIIIII  to  select 
the  option  Un-zoom.  The  x-increment  is  increased  to  0.25. 

•  To  recover  the  original  x-increment  of  0.5  you  can  do  an  un-zoom 
again,  or  use  the  option  zoom  out  by  pressing  BM2  IEI3IIII. 

•  The  option  Decimal  in         produces  x-increments  of  0.10. 

•  The  option  Integer  in  BM2  produces  x-increments  of  1 . 

•  The  option  Trig  in  produces  increments  related  to  fractions  of  n,  thus 
being  useful  when  plotting  trigonometric  functions. 

•  To  return  to  normal  calculator  display  press  (enter}  . 

Plots  in  polar  coordinates 

First  of  all,  you  may  want  to  delete  the  variables  used  in  previous  examples 
(e.g.,  X,  EQ,  Yl,  PPAR)  using  function  PURGE  ((roa)  HHS).  By  doing  this,  all 
parameters  related  to  graphics  will  be  cleared.  Press  Cjjj  j  to  check  that  the 
variables  were  indeed  purged. 

We  will  try  to  plot  the  function  f(0)  =  2(l-sin(6)),  as  follows: 

•  First,  make  sure  that  your  calculator's  angle  measure  is  set  to  radians. 

•  Press  LjnJi^ ,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  Polar,  by  pressing  llliilllliliii  ^3?  mill. 

•  Press  ^3?  and  type: 


Page  12-19 


•  The  cursor  is  now  in  the  indep  field.  Press  CZD(^^CS(3  IIEIIll  to 
change  the  independent  variable  to  6. 

•  Press  [nxtJ liai III!  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
window  (in  this  case  it  will  be  called  PLOT  -POLAR  window). 

•  Change  the  H-VIEW  range  to  -8  to  8,  by  using 
CSJ^+tJMSMIZSJMM,  and  the  V-VIEW  range  to  -6  to  2  by  using 
r~6~)      llEillO-]  113111. 

Note:  the  H-VIEW  and  V-VIEW  determine  the  scales  of  the  display 
window  only,  and  their  ranges  are  not  related  to  the  range  of  values 
of  the  independent  variable  in  this  case. 

•  Change  the  indep  Low  value  to  0,  and  the  High  value  to  6.28  2n),  by 
using:  QD MM    (~6j CZD CD 11. 

•  Press  HSliEi  EEECI  to  plot  the  function  in  polar  coordinates.  The  result  is  a 
curve  shaped  like  a  hearth.  This  curve  is  known  as  a  cardiod  (cardios, 
Greek  for  heart). 


ftUilEffFlHIIfliM 

P"" 

T 1  HI  1  hi  H 

•  Press  E5n(W)  EHHDM]  to  see  the  graph  with  labels.  Press  ®  to 
recover  the  menu.  Press  [nxt)  Slljjy  to  recover  the  original  graphics  menu. 

•  Press  D33£i  iliull  to  trace  the  curve.  The  data  shown  at  the  bottom  of 
the  display  is  the  angle  0  and  the  radius  r,  although  the  latter  is  labeled  Y 
(default  name  of  dependent  variable). 

•  Press  IWlECnH!  to  return  to  the  PLOT  WINDOW  screen.  Press  (W)BEH 
to  return  to  normal  calculator  display. 

In  this  exercise  we  entered  the  equation  to  be  plotted  directly  in  the  PLOT 
SETUP  window.  We  can  also  enter  equations  for  plotting  using  the  PLOT 
window,  i.e.,  simultaneously  if  in  RPN  mode,  pressing  C5D_^ .  For 
example,  when  you  press  LjnJ^^    after  finishing  the  previous  exercise,  you 


Page  1  2-20 


will  get  the  equation  /2*(1-SIN(0))'  highlighted.  Let's  say,  we  want  to  plot 
also  the  function  '2*(1  -COS(6))'  along  with  the  previous  equation. 


•  Press  ESSM  ,  and  type  CDQDE]^  CDCEJ  C^^CSS^,  to 

enter  the  new  equation. 

•  Press  H333=!  EEjjE]  to  see  the  two  equations  plotted  in  the  same  figure.  The 
result  is  two  intersecting  cordioids.  Press  EEEH1  I  on]  to  return  to  normal 
calculator  display.  


Plotting  conic  curves 

The  most  general  form  of  a  conic  curve  in  the  x-y  plane  is: 
Ax2+By2+Cxy+Dx+Ey+F  =  0.  We  also  recognize  as  conic  equations  those 
given  in  the  canonical  form  for  the  following  figures: 


where  xQ/  yQ,  a,  b,  and  /Care  constant. 

The  name  conic  curves  follows  because  these  figures  (circles,  ellipses, 
parabolas  or  hyperbolas)  result  from  the  intersection  of  a  plane  with  a  cone. 
For  example,  a  circle  is  the  intersection  of  a  cone  with  a  plane  perpendicular 
to  the  cone's  main  axis. 

The  calculator  has  the  ability  of  plotting  one  or  more  conic  curves  by  selecting 
Conic  as  the  function  TYPE  in  the  PLOT  environment.  Make  sure  to  delete  the 
variables  PPAR  and  EQ  before  continuing.  For  example,  let's  store  the  list  of 
equations 


•  circle: 

•  ellipse: 

•  parabola: 

•  hyperbola: 


(x-x0)2+(y-y0)2  =  r2 
(x-x0)2/a2  +  (y-y0)2/b2=  1 
(y-b)2  =  K(x-a)  or  (x-a)2  =  K(y-b) 
(x-x0)2/a2  +  (y-y0)2/b2=  1  or  xy  =  K, 


Page  12-21 


{ '(X-1)A2+(Y-2)A2=3' ,  'XA2/4+YA2/3=T  } 


into  the  variable  EQ. 

These  equations  we  recognize  as  those  of  a  circle  centered  at  (1,2)  with 
radius  V3,  and  of  an  ellipse  centered  at  (0,0)  with  semi-axis  lengths  a  =  2  and 

b  =  V3. 

•  Enter  the  PLOT  environment,  by  pressing  CfDJ^,,  simultaneously  if  in 
RPN  mode,  and  select  Conic  as  the  TYPE.  The  list  of  equations  will  be 
listed  in  the  EQ  field. 

•  Make  sure  that  the  independent  variable  (indep)  is  set  to  'X'  and  the 
dependent  variable  (Depnd)  to  'Y'. 

•  Press  jNxr j lliljljliil  to  return  to  normal  calculator  display. 

•  Enter  the  PLOT  WINDOW  environment,  by  pressing  EDJ^L, 
simultaneously  if  in  RPN  mode. 

•  Change  the  range  for  H-VIEW  to  -3  to  3,  by  using 
C^C^l:!:^  Also,  change  the  V-VIEW  range  to  -1 .5  to  2  by 

using  en czd czd czd nacxi  ■• 

•  Change  the  Indep  Low:  and  High:  fields  to  Default  by  using  jNxrj  llliilllj 
while  each  of  those  fields  is  highlighted.  Select  the  option  Reset  value 
after  pressing  lllillll  Press  (ECU  to  complete  the  resetting  of  values. 
Press  jNxrj  to  return  to  the  main  menu. 

•  riot  the  graph:  . 


r  1 1  iii  n  ii^h 

Note:  The  H-view  and  v-view  ranges  were  selected  to  show  the  intersection 
of  the  two  curves.  There  is  no  general  rule  to  select  those  ranges,  except 
based  on  what  we  know  about  the  curves.  For  example,  for  the  equations 
shown  above,  we  know  that  the  circle  will  extend  from  -3+1  =  -2  to  3+1  =4 
in  x,  and  from  -3+2=-l  to  3+2=5  in  y.    In  addition,  the  ellipse,  which  is 


Page  1  2-22 


centered  at  the  origin  (0,0),  will  extend  from  -2  to  2  in  x,  and  from  -V3  to  V3 
in  y. 

Notice  that  for  the  circle  and  the  ellipse  the  region  corresponding  to  the  left 
and  right  extremes  of  the  curves  are  not  plotted.  This  is  the  case  with  all 
circles  or  ellipses  plotted  using  Conic  as  the  type. 


•     To  see  labels:  illlilllil  IE 


To  recover  the  menu:  C^C^IfflSi! 

To  estimate  the  coordinates  of  the  point  of  intersection,  press  the  11(313)1 
menu  key  and  move  the  cursor  as  close  as  possible  to  those  points  using 
the  arrow  keys.  The  coordinates  of  the  cursor  are  shown  in  the  display. 
For  example,  the  left  point  of  intersection  is  close  to  (-0.692,  1 .67),  while 
the  right  intersection  is  near  (1 .89,0.5). 


\  \ 

K:-6.53E-i 

1  J 

v=i.e?ED 

V  1 

■: :  1 .  SHEA 

1  J 

V:  .5 

•  To  recover  the  menu  and  return  to  the  PLOT  environment,  press  (JW7JEEEH!!. 

•  To  return  to  normal  calculator  display,  press  (jv^IIIEEIII. 


Parametric  plots 

Parametric  plots  in  the  plane  are  those  plots  whose  coordinates  are  generated 
through  the  system  of  equations  x  =  x(t)  and  y  =  y(t),  where  t  is  known  as  the 
parameter.  An  example  of  such  graph  is  the  trajectory  of  a  projectile,  x(t)  = 
x0  +  Vq-COS  90-t,  y(t)  =  y0  +  v0-sin  90-t  -  Vi-g-t2.  To  plot  equations  like  these, 
which  involve  constant  values  x0,  y0,  v0,  and  0O,  we  need  to  store  the  values  of 
those  parameters  in  variables.  To  develop  this  example,  create  a  sub- 
directory called  'PROJM'  for  PROJectile  Motion,  and  within  that  sub-directory 
store  the  following  variables:  X0  =  0,  Y0  =  1 0,  V0  =  1 0  ,  90  =  30,  and  g  = 
9.806.  Make  sure  that  the  calculator's  angle  measure  is  set  to  DEG.  Next, 
define  the  functions  (use  l<n  J?EF  ): 


Page  1  2-23 


X(t)  =  XO  +  VO*COS(GO)*t 
Y(t)  =  YO  +  VO*SIN(eO)*t  -  0.5*g*tA2 


which  will  add  the  variables 


and  II 


:  10^0 

:  10^0 

:30M0 

:9.S06^ 

:  DEFIHEf 

^X(t)=X0+V0-COS£E> 

HOVflL 

to  the  soft  menu  key  labels. 


■ lUFVU 

30^0 

:9.S0^Q 

DEF I  NEFXCt  )=X0+V0-COSteH 


DEF I  HE[  YCt  )= Y0+ V0-S I  NEE* 
HOVflL 


To  produce  the  graph  itself,  follow  these  steps: 

•  Press  CSDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  Parametric,  by  pressing  mmuB  ^P^PmMm. 

•  Press  ^3?  and  type  'X(t)  +  i*Y(t)/  llliiiilllll  to  define  the  parametric  plot  as 
that  of  a  complex  variable.  (The  real  and  imaginary  parts  of  the  complex 
variable  correspond  to  the  x-  and  y-coordinates  of  the  curve.) 

•  The  cursor  is  now  in  the  indep  field.  Press  CZZ3^™)QJ(3  HEM  to 
change  the  independent  variable  to  t. 

•  Press  (jv^lIIEEii!  to  return  to  normal  calculator  display. 

•  Press  LtlJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
window  (in  this  case  it  will  be  called  PLOT  -PARAMETRIC  window). 
Instead  of  modifying  the  horizontal  and  vertical  views  first,  as  done  for 
other  types  of  plot,  we  will  set  the  lower  and  upper  values  of  the 
independent  variable  first  as  follows: 

•  Select  the  indep  Low  field  by  pressing  ^3? ^3?.  Change  this  value  to 
f^OjMSM.  Then,  change  the  value  of  High  to  CTDKEl.  Enter  QDCZD 
CZZZ) ii!lii!!=Ll:iilil!!  for  the  step  value  (i.e.,  step  =  0.1). 


Note:  Through  these  settings  we  are  indicating  that  the  parameter  t  will 
take  values  of  t  =  0,  0.1,  0.2,      etc.,  until  reaching  the  value  of  2.0. 


Press  IISIIIl!!.  This  will  generate  automatic  values  of  the  H-View  and  V-View 
ranges  based  on  the  values  of  the  independent  variable  t  and  the 
definitions  of  X(t)  and  Y(t)  used.  The  result  will  be: 


Page  1  2-24 


^^PLOT  HinDOH  - 

PftRftHETRIC^^ 

H-'.'i>ZH:[j^^^| 

Indcp  Loh=  0. 

Hiqh:2. 

St«p:  .1 

_ Pixels 

inter  HiniHUH  hori 

2 ■:■  ri + i.  ualufl 

•  Press  iLiiLlllii!!!::!!:!!  iLliLlll!!!!!;!!  to  draw  the  parametric  plot. 

•  Press  IIIlMii (W)  EBjlD  to  see  the  graph  with  labels.    The  window 
parameters  are  such  that  you  only  see  half  of  the  labels  in  the  x-axis. 


+  \ 

— i — i — i — i — i — i — i — hi 

n 

17  -^nqflflfl?1?? 

•  Press  [nxtJ  to  recover  the  menu.  Press  (jwrj  13333  to  recover  the  original 
graphics  menu. 

•  Press  IEEB3  ll(:!:P!lll!ll)ll  to  determine  coordinates  of  any  point  on  the  graph. 
Use  (T)  and  GD  to  move  the  cursor  about  the  curve.  At  the  bottom  of  the 
screen  you  will  see  the  value  of  the  parameter  t  and  coordinates  of  the 
cursor  as  (X,Y). 

•  Press  (WjEHIH!  to  return  to  the  PLOT  WINDOW  environment.  Then, 
press   C^D  ,  or  (JW7JB3BB,  to  return  to  normal  calculator  display. 

A  review  of  your  soft  menu  key  labels  shows  that  you  now  have  the  following 
variables:  t,  EQ,  PPAR,  Y,  X,  g,  00,  V0,  Y0,  X0.  Variables  t,  EQ,  and  PPAR 
are  generated  by  the  calculator  to  store  the  current  values  of  the  parameter,  t, 
of  the  equation  to  be  plotted  EQ  (which  contains  'X(t)  +  l*Y(t)'),  and  the  plot 
parameters.  The  other  variables  contain  the  values  of  constants  used  in  the 
definitions  of  X(t)  and  Y(t). 

You  can  store  different  values  in  the  variables  and  produce  new  parametric 
plots  of  the  projectile  equations  used  in  this  example.  If  you  want  to  erase  the 
current  picture  contents  before  producing  a  new  plot,  you  need  to  access 
either  the  PLOT,  PLOT  WINDOW,  or  PLOT  SETUP  screens,  by  pressing, 
Ij^JJ^  ,  IJnJ  ML  ,  or  C5D  J^L  (the  two  keys  must  be  pressed  simultaneously 


Page  1  2-25 


if  in  RPN  mode).  Then,  press  GXSDXE].  Press  BXH1  to  return  to  the  PLOT, 
PLOT  WINDOW,  or  PLOT  SETUP  screen.    Press  C™],  or  (n^MM,  to  return 
to  normal  calculator  display. 

Generating  a  table  for  parametric  equations 

In  an  earlier  example  we  generated  a  table  of  values  (X,Y)  for  an  expression 
of  the  form  Y=f(X),  i.e.,  a  Function  type  of  graph.    In  this  section,  we  present 
the  procedure  for  generating  a  table  corresponding  to  a  parametric  plot.  For 
this  purpose,  well  take  advantage  of  the  parametric  equations  defined  in  the 
example  above. 

•  First,  let's  access  the  TABLE  SETUP  window  by  pressing  , 
simultaneously  if  in  RPN  mode.  For  the  independent  variable  change 
the  Starting  value  to  0.0,  and  the  Step  value  to  0.1 .    Press  Kill. 

•  Generate  the  table  by  pressing,  simultaneously  if  in  RPN  mode, 

w  .  The  resulting  table  has  three  columns  representing  the 
parameter  t,  and  the  coordinates  of  the  corresponding  points.  For 
this  table  the  coordinates  are  labeled  XI  and  Yl . 


.X1 

Yl 

— 

10 

. BbbOESH 

10 

H50R7 

.a 

1.73E051 

10 

B03BB 

.3 

E . 51B07G 

11 

05B73 

.  H 

3.H6H10E 

11 

E1SSE 

.  S 

H.3301EF 

11 

EFHE5 

0- 

•  Use  the  arrow  keys,  (3D  CD  ,  to  move  about  the  table. 

•  Press  I  on  j  to  return  to  normal  calculator  display. 

This  procedure  for  creating  a  table  corresponding  to  the  current  type  of  plot 
can  be  applied  to  other  plot  types. 

Plotting  the  solution  to  simple  differential  equations 

The  plot  of  a  simple  differential  equation  can  be  obtained  by  selecting  Diff 
Eq  in  the  TYPE  field  of  the  PLOT  SETUP  environment  as  follows:  suppose  that 
we  want  to  plot  x(t)  from  the  differential  equation  dx/dt  =  exp(-t2),  with  initial 
conditions:  x  =  0  at  t  =  0.  The  calculator  allows  for  the  plotting  of  the  solution 


Page  1  2-26 


of  differential  equations  of  the  form  Y'(T)  =  F(T,Y).  For  our  case,  we  let  Y->x 
and  T-H  therefore,  F(T,Y)-»f(t,x)  =  exp(-t2). 

Before  plotting  the  solution,  x(t),  for  t  =  0  to  5,  delete  the  variables  EQ  and 
PPAR. 

•  Press  GDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  Dif  f  Eq. 

•  Press  ^3?  and  type  CD  ED  ^^{alpha}  CSDSCZDCZDKi™. 

•  The  cursor  is  now  in  the  H-var  field.  It  should  show  H-var :  0  and  also  v- 
vanl.  This  is  the  code  used  by  the  calculator  to  identify  the  variables  to 
be  plotted.  H-var:0  means  the  independent  variable  (to  be  selected  later) 
will  be  plotted  in  the  horizontal  axis.       Also,  v-var:i  means  the 
dependent  variable  (default  name  'Y')  will  be  plotted  in  the  vertical  axis. 

•  Press  ^3?  .  The  cursor  is  now  in  the  indep  field.  Press  L  r>  JL  1  J{alpha) 
LjnJ 0  113111  to  change  the  independent  variable  to  t. 

•  Press  [nxtJ llllllij!!!!:!!!!!!!  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
window  (in  this  case  it  will  be  called  PLOT  WINDOW  -  DIFF  EQ). 

•  Change  the  H-VIEW  and  V-VIEW  parameters  to  read:  H-VIEW:  -1  5, 
V-VIEW:-1  1.5 

•  Change  the  init  value  to  0,  and  the  Final  value  to  5  by  using:  L0J HHi 

L  5  J  iiiiii=:=i-=:iiiiiii  . 

•  The  values  Step  and  Tol  represent  the  step  in  the  independent  variable 
and  the  tolerance  for  convergence  to  be  used  by  the  numerical  solution. 
Let's  leave  those  values  with  their  default  settings  (if  the  word  default  is 
not  shown  in  the  Step:  field,  use  [nxtJ  liliilljl  to  reset  that  value  to  its 
default  value.  Press  jNxrj  to  return  to  the  main  menu.)    Press  . 

•  The  Init-Soln  value  represents  the  initial  value  of  the  solution  to  start  the 
numerical  result.  For  the  present  case,  we  have  for  initial  conditions  x(0) 
=  0,  thus,  we  need  to  change  this  value  to  0.0,  by  using  LfijBBBIH. 

•  Press  lEIGIilH  HEElD  to  plot  the  solution  to  the  differential  equation. 

•  Press  llliililii  ®  IE ilil !!!  DUD  to  see  the  graph  with  labels. 


Page  1  2-27 


•  Press  [nxtJ  to  recover  the  menu.  Press  [nxtJ mSm  to  recover  the  original 
graphics  menu. 

•  When  we  observed  the  graph  being  plotted,  you'll  notice  that  the  graph 
is  not  very  smooth.  That  is  because  the  plotter  is  using  a  time  step  that  is 
too  large.  To  refine  the  graph  and  make  it  smoother,  use  a  step  of  0.1 . 
Try  the  following  keystrokes:  BXH1  ^^^CZDQ^!^ 

The  plot  will  take  longer  to  be  completed,  but  the  shape  is  definitely 
smoother  than  before. 

•  Press  G3BB  (W)  EE!  EHUD,  to  see  axes  labels  and  range.  Notice  that 
the  labels  for  the  axes  are  shown  as  0  (horizontal)  and  1  (vertical).  These 
are  the  definitions  for  the  axes  as  given  in  the  PLOT  WINDOW  screen 
(see  above),  i.e.,  H-VAR  (t):  0,  and  V-VAR(x):  1 . 


•  Press  [NxT)[NxfJmmi  to  recover  menu  and  return  to  PICT  environment. 

•  Press  (:!:li!il)  to  determine  coordinates  of  any  point  on  the  graph.  Use 
(T)  and  GD  to  move  the  cursor  in  the  plot  area.  At  the  bottom  of  the 
screen  you  will  see  the  coordinates  of  the  cursor  as  (X,Y).  The  calculator 
uses  X  and  Y  as  the  default  names  for  the  horizontal  and  vertical  axes, 
respectively. 

•  Press  IWHSnH!  to  return  to  the  PLOT  WINDOW  environment.  Then, 
press   ISEJ  to  return  to  normal  calculator  display. 

More  details  on  using  graphical  solutions  of  differential  equations  are 
presented  in  Chapter  16. 


,i 


-i. 


5. 


-i. 


Page  1  2-28 


Truth  plots 

Truth  plots  are  used  to  produce  two-dimensional  plots  of  regions  that  satisfy  a 
certain  mathematical  condition  that  can  be  either  true  or  false.    For  example, 
suppose  that  you  want  to  plot  the  region  for  XA2/36  +  YA2/9  <  1 , 
proceed  as  follows: 

•  Press  CSDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Truth. 

•  Press  ^3?  and  type  {'(XA2/36+YA2/9  <  1  )','(XA2/1 6+YA2/9  >  1)'} 
iiiiiiiliLl:!!!!!!!  to  define  the  conditions  to  be  plotted. 

•  The  cursor  is  now  in  the  indep  field.   Leave  that  as  'X'  if  already  set  to 
that  variable,  or  change  it  to  'X'  if  needed. 

•  Press  Q^IIIEEIII  to  return  to  normal  calculator  display. 

•  Press  (jij ML  ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
window  (in  this  case  it  will  be  called  PLOT  WINDOW  -  TRUTH  window). 
Let's  keep  the  default  value  for  the  window's  ranges:  H-view:  -6.5  6.5, 
v-view :  -3  .  l    3  . 2  (To  reset  them  use  [nxt)  iiliiiiui  (select  Reset  all)  IIIll 
(nxt)). 


Note:  if  the  window's  ranges  are  not  set  to  default  values,  the  quickest  way 
to  reset  them  is  by  using  ®0BD  (select  Reset  oil)  Ul® . 


•  Press  EEEH3E333  to  draw  the  truth  plot.  Because  the  calculator  samples 
the  entire  plotting  domain,  point  by  point,  it  takes  a  few  minutes  to 
produce  a  truth  plot.  The  present  plot  should  produce  a  shaded  ellipse  of 
semi-axes  6  and  3  (in  x  and  y,  respectively),  centered  at  the  origin. 

•  Press  BHEiilWl  lEiiliniini  to  see  the  graph  with  labels.  The  window 
parameters  are  such  that  you  only  see  half  of  the  labels  in  the  x-axis. 
Press  [nxtJ  to  recover  the  menu.  Press  (jwrj  15333  to  recover  the  original 
graphics  menu. 

•  Press  (:!l!i!i!)  to  determine  coordinates  of  any  point  on  the  graph.  Use  the 
arrow  keys  to  move  the  cursor  about  the  region  plotted.  At  the  bottom  of 
the  screen  you  will  see  the  value  of  the  coordinates  of  the  cursor  as  (X,Y). 

•  Press  IWHSnH!  to  return  to  the  PLOT  WINDOW  environment.  Then, 
press   I  on]  ,  or  jNxr j MM,  to  return  to  normal  calculator  display. 


Page  1  2-29 


You  can  have  more  than  one  condition  plotted  at  the  same  time  if  you  multiply 
the  conditions.  For  example,  to  plot  the  graph  of  the  points  for  which  X2/36 
+  Y2/9  <  1 ,  and  X2/ 1  6  +  Y2/9  >  1 ,  use  the  following: 

•  Press  CSDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Press  ^  and  type  /(XA2/36+YA2/9  <  !)•  (XA2/1 6+YA2/9  >  1  )H 
to  define  the  conditions  to  be  plotted. 

•  Press  B3333EEEI;]  to  draw  the  truth  plot.  Again,  you  have  to  be  patient 
while  the  calculator  produces  the  graph.  If  you  want  to  interrupt  the  plot, 
press  tjwj  ,  once.  Then  press  i[;!!Sii[;i!!j  . 

Plotting  histograms,  bar  plots,  and  scatter  plots 

Histograms,  bar  plots  and  scatter  plots  are  used  to  plot  discrete  data  stored  in 
the  reserved  variable  ZDAT.  This  variable  is  used  not  only  for  these  types  of 
plots,  but  also  for  all  kind  of  statistical  applications  as  will  be  shown  in 
Chapter  18.  As  a  matter  of  fact,  the  use  of  histogram  plots  is  postponed  until 
we  get  to  that  chapter,  for  the  plotting  of  a  histogram  requires  to  perform  a 
grouping  of  data  and  a  frequency  analysis  before  the  actual  plot.  In  this 
section  we  will  show  how  to  load  data  in  the  variable  ZDAT  and  how  to  plot 
bar  plots  and  scatter  plots. 

We  will  use  the  following  data  for  plotting  bar  plots  and  scatter  plots: 


x       y  z 

"3J    2.1  1.1 

3.6    3.2  2.2 

4.2    4.5  3.3 

4.5    5.6  4.4 

4.9    3.8  5.5 

5.2    2.2  6.6 

Bar  plots 

First,  make  sure  your  calculator's  CAS  is  in  Exact  mode.  Next,  enter  the 
data  shown  above  as  a  matrix,  i.e., 


Page  1  2-30 


[[3.1,2.1,1 .1],[3.6,3.2,2.2],[4.2,4.5,3.3], 
[4.5,5.6,4.4],[4.9,3.8,5.5],[5. 2,2.2,6.6]^  [enter} 


to  store  it  in  ZDAT,  use  the  function  STOZ  (available  in  the  function  catalog, 
Lr>J  cat  ).  Press  VAR  to  recover  your  variables  menu.  A  soft  menu  key 
labeled  ZDAT  should  be  available  in  the  stack.  The  figure  below  shows  the 
storage  of  this  matrix  in  ALG  mode: 


4.2  4.5  3.3 

4.5  5.6  4.4 

4.9  3.3  5.5 

.5.2  2.2  6.6. 

:ST02(FlNSm) 

HOVflL 

To  produce  the  graph: 

•  Press  GDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  Bar. 

•  A  matrix  will  be  shown  at  the  SDAT  field.  This  is  the  matrix  we  stored 
earlier  into  SDAT. 

•  Highlight  the  Col:  field.  This  field  lets  you  choose  the  column  of  SDAT 
that  is  to  be  plotted.  The  default  value  is  1 .  Keep  it  to  plot  column  1  in 
ZDAT. 

•  Press  (j^lIIEEIII  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Change  the  V-View  to  read,  v-view:  0  5. 

•  Press  EEMi  OlS  to  draw  the  bar  plot. 


Page  12-31 


•  Press  H3IH1  to  return  to  the  PLOT  WINDOW  environment.    Then,  press 
I  on]  ,  or  jNxr j MB,  to  return  to  normal  calculator  display. 

The  number  of  bars  to  be  plotted  determines  the  width  of  the  bar.  The  H-  and 
V-VIEW  are  set  to  1 0,  by  default.  We  changed  the  V-VIEW  to  better 
accommodate  the  maximum  value  in  column  1  of  ZDAT.  Bar  plots  are  useful 
when  plotting  categorical  (i.e.,  non-numerical)  data. 

Suppose  that  you  want  to  plot  the  data  in  column  2  of  the  SDAT  matrix: 

•  Press  GDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Press  to  highlight  the  Col:  field  and  type  2  MB,  followed  by 

•  Press  CSDJ^L,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  V-View  to  read  v-view:  0  6 

•  Press  EEMi  IMD. 


•  Press  BiEHl  to  return  to  the  PLOT  WINDOW  screen,  then  Q^J   to  return 
to  normal  calculator  display. 

Scatter  plots 

We  will  use  the  same  ZDAT  matrix  to  produce  scatter  plots.  First,  we  will  plot 
the  values  of  y  vs.  x,  then  those  of  y  vs.  z,  as  follows: 

•  Press  CSDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Scatter. 


Page  12-32 


•  Press  ^?^?  to  highlight  the  Cols:  field.  Enter  UJMM  fZZjMM  to 
select  column  1  as  X  and  column  2  as  Y  in  the  Y-vs.-X  scatter  plot. 

•  Press  (jw^IIIEEII!  to  return  to  normal  calculator  display. 

•  Press  (jnj ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Change  the  plot  window  ranges  to  read:  H-View:  0  6,  V-View:  0  6. 

•  Press  EmM  Em  to  draw  the  bar  plot.  Press  BSS  ®  JESS  EMM  to  see 
the  plot  unencumbered  by  the  menu  and  with  identifying  labels  (the  cursor 
will  be  in  the  middle  of  the  plot,  however): 


v 


!■! 


•  Press  (^QL^IIIliiji  to  leave  the  EDIT  environment. 

•  Press  B3331  to  return  to  the  PLOT  WINDOW  environment.    Then,  press 
ioNj  f  or  (jwrJIEM,  to  return  to  normal  calculator  display. 

To  plot  y  vs.  z,  use: 

•  Press  GDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Press  to  highlight  the  Cols:  field.  Enter  CD!      ^JJIW'JM  to 
select  column  3  as  X  and  column  2  as  Y  in  the  Y-vs.-X  scatter  plot. 

•  Press  (jv^lIEEIIII  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Change  the  plot  window  ranges  to  read:  H-View:  0  7,  V-View:  0  7. 

•  Press         Em  to  draw  the  bar  plot.    Press  HBO  ®  0333!  EMM  to  see 
the  plot  unencumbered  by  the  menu  and  with  identifying  labels. 

-v 


!■! 


Page  12-33 


•  Press  (jwf7j(W)|JHi  to  leave  the  EDIT  environment. 

•  Press  B3331  to  return  to  the  PLOT  WINDOW  environment.    Then,  press 
ISEJ  /  or  IjjXTjWESM,  to  return  to  normal  calculator  display. 

Slope  fields 

Slope  fields  are  used  to  visualize  the  solutions  to  a  differential  equation  of  the 
form  y'  =  f(x,y).  Basically,  what  is  presented  in  the  plot  are  segments 
tangential  to  the  solution  curves,  since  y'  =  dy/dx,  evaluated  at  any  point  (x,y), 
represents  the  slope  of  the  tangent  line  at  point  (x,y). 

For  example,  to  visualize  the  solution  to  the  differential  equation  y'  =  f(x,y)  = 
x+y,  use  the  following: 

•  Press  GDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  siopefieid. 

•  Press  ^r?  and  type  'X+Y'  11. 

•  Make  sure  that  'X'  is  selected  as  the  indep:  and  'Y'  as  the  Depnd: 
variables. 

•  Press  InxtJ ECEIII  to  return  to  normal  calculator  display. 

•  Press  CSDJI^L,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Change  the  plot  window  ranges  to  read:  X-Left:-5,  X-Right:5,  Y-Near:-5,  Y-Far: 
5 

•  Press  OQEBaOOm  to  draw  the  slope  field  plot.       Press  HHi  |W| 

!IIIIi!ffl  to  see  the  plot  unencumbered  by  the  menu  and  with 
identifying  labels. 


V  /   1  II 

•///ii 

\  \  \  \  V 

\     \  \ 

A  \  ----- 

•     Press  [Nxf^^NxfJmmi  to  leave  the  EDIT  environment. 


Page  1  2-34 


•  Press  B3331  to  return  to  the  PLOT  WINDOW  environment.    Then,  press 
ioNj  f  or  C^iElli,  to  return  to  normal  calculator  display. 

If  you  could  reproduce  the  slope  field  plot  in  paper,  you  can  trace  by  hand 
lines  that  are  tangent  to  the  line  segments  shown  in  the  plot.  This  lines 
constitute  lines  of  y(x,y)  =  constant,  for  the  solution  of  y'  =  f(x,y).  Thus,  slope 
fields  are  useful  tools  for  visualizing  particularly  difficult  equations  to  solve. 

Try  also  a  slope  field  plot  for  the  function  y'  =  f(x,y)  =  -  (y/x)2,  by  using: 

•  Press  C3Di^,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  siopefieid. 

•  Press  ^?  and  type  '-  (Y/X)A2'  HI. 

•  Press  0333  to  draw  the  slope  field  plot.     Press  BSD  ® 

lull!!!!!!  to  see  the  plot  unencumbered  by  the  menu  and  with 
identifying  labels. 


-  -  \  \  i: 

Yi    \  \  - 

-    -  \    'l  -5., 

■  'l     \      -  -— - 
.  'l     'l  \ 

•  Press  (^QL^Iilllijl  to  leave  the  EDIT  environment. 

•  Press  B333I  to  return  to  the  PLOT  WINDOW  environment.    Then,  press 
{jmJ  f  or  (j^rjIEM,  to  return  to  normal  calculator  display. 

Fast  3D  plots 

Fast  3D  plots  are  used  to  visualize  three-dimensional  surfaces  represented  by 
equations  of  the  form  z  =  f(x,y).  For  example,  if  you  want  to  visualize  z  = 
f(x,y)  =  x2+y2,  we  can  use  the  following: 

•  Press  CSDi^,,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Fast3D. 

•  Press  ^3?  and  type  /XA2+YA2/  III!™ 


Page  1  2-35 


•  Make  sure  that  'X'  is  selected  as  the  Indep:  and  'Y'  as  the  Depnd:  variables. 

•  Press  InxtJ I!!!!!!:!!!!:!!!!!!!  to  return  to  normal  calculator  display. 

•  Press  EDJ^L/  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Keep  the  default  plot  window  ranges  to  read:  X-Left:-1,  X-Right:  1 ,  Y-Near:-1, 
Y-Far:  1,  Z-Low:  -1,  Z-High:  1,  Step  Indep:  10,  Depnd:  8 

Note:  The  Step  Indep:  and  Depnd:  values  represent  the  number  of 
gridlines  to  be  used  in  the  plot.  The  larger  these  number,  the  slower  it 
is  to  produce  the  graph,  although,  the  times  utilized  for  graphic 
generation  are  relatively  fast.  For  the  time  being  well  keep  the  default 
values  of  1 0  and  8  for  the  Step  data. 


Press  IHSlsji  EEECI  to  draw  the  three-dimensional  surface.  The  result  is  a 
wireframe  picture  of  the  surface  with  the  reference  coordinate  system 
shown  at  the  lower  left  corner  of  the  screen.  By  using  the  arrow  keys 
(GO CD  <\7? )  you  can  change  the  orientation  of  the  surface.  The 
orientation  of  the  reference  coordinate  system  will  change  accordingly. 
Try  changing  the  surface  orientation  on  your  own.    The  following  figures 


ram 

When  done,  press  153111 

Press  H3IH!  to  return  to  the  PLOT  WINDOW  environment. 
Change  the  Step  data  to  read:  step  indep:  20    Depnd:  16 
Press  IHSEEsB  E3»D  to  see  the  surface  plot.  Sample  views: 


Page  1  2-36 


•  When  done,  press  ISillllil. 

•  Press  BEH1  to  return  to  PLOT  WINDOW. 

•  Press  I^onJ  ,  or  (jv^JBBBB,  to  return  to  normal  calculator  display. 
Try  also  a  Fast  3D  plot  for  the  surface  z  =  f(x,y)  =  sin  (x2+y2) 

•  Press  C3D  J^, ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT  SETUP 
window. 

•  Press  ^?  and  type  'SIN(XA2+YA2)'  MM. 

•  Press  EEMi  IEI1S  to  draw  the  plot. 

•  When  done,  press  iilliliil 

•  Press  mm  to  return  to  PLOT  WINDOW. 

•  Press  I^onJ  ,  or  (jv^jKEBI,  to  return  to  normal  calculator  display. 


Wireframe  plots 

Wireframe  plots  are  plots  of  three-dimensional  surfaces  described  by  z  =  f(x,y). 
Unlike  Fast  3D  plots,  wireframe  plots  are  static  plots.  The  user  can  choose  the 
viewpoint  for  the  plot,  i.e.,  the  point  from  which  the  surface  is  seen.  For 
example,  to  produce  a  wireframe  plot  for  the  surface  z  =  x  +  2y  -3,  use  the 
following: 

•  Press   LjnJi^D  f  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Wireframe. 

•  Press  ^?  and  type  'X+2*Y-3'  EB. 

•  Make  sure  that  'X'  is  selected  as  the  indep:  and  'Y'  as  the  Depnd: 
variables. 

•  Press  [nxtJ to  return  to  normal  calculator  display. 

•  Press    LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Keep  the  default  plot  window  ranges  to  read:  X-Left:-1,  X-Right:  1 ,  Y-Near:-1, 
Y-Far:  1,  Z-Low:  -1,  Z-High:  1 ,  XE:0,YE:-3,  ZE:0,  Step  Indep:  10  Depnd:  8 


Page  1  2-37 


The  coordinates  XE,  YE,  ZE,  stand  for  "eye  coordinates/'  i.e.,  the  coordinates 
from  which  an  observer  sees  the  plot.  The  values  shown  are  the  default  values. 
The  Step  Indep:  and  Depnd:  values  represent  the  number  of  gridlines  to  be 
used  in  the  plot.  The  larger  these  number,  the  slower  it  is  to  produce  the 
qraph.  For  the  time  beinq  well  keep  the  default  values  of  1  0  and  8  for  the 
Step  data. 

•  Press  iLiLi!!!!!!!;:!!;!!  IIIi:!!i!E;3  to  draw  the  three-dimensional  surface.  The  result  is  a 
wireframe  picture  of  the  surface. 

•  Press  Em  (W)  HUB  EMI  to  see  the  graph  with  labels  and  ranges.  This 
particular  version  of  the  graph  is  limited  to  the  lower  part  of  the  display. 
We  can  change  the  viewpoint  to  see  a  different  version  of  the  graph. 


.5  v 

•  Press  (W)(W)IIfil  E3IH1  to  return  to  the  PLOT  WINDOW  environment. 

•  Change  the  eye  coordinate  data  to  read  :    xe  :  0       ye  :  -3       ze  :  3 

•  Press  [SEES  I333II  to  see  the  surface  plot. 

•  Press  113111!  to  see  the  graph  with  labels  and  ranges. 


a. s  v 


This  version  of  the  graph  occupies  more  area  in  the  display  than  the 
previous  one.  We  can  change  the  viewpoint,  once  more,  to  see  another 
version  of  the  graph. 

•  Press  [NxrJ[NXTjtSm  MEM  to  return  to  the  PLOT  WINDOW  environment. 

•  Change  the  eye  coordinate  data  to  read  :    xe  :  3       ye  :  3       ze  :  3 


Page  12-38 


•     Press  H3E=!  EEEE]  to  see  the  surface  plot.  This  time  the  bulk  of  the  plot  is 
located  towards  the  right  -hand  side  of  the  display. 


•  Press  BEH1  to  return  to  the  PLOT  WINDOW  environment. 

•  Press  L^J  /  or  (j^IIIEBIII,  to  return  to  normal  calculator  display. 

Try  also  a  Wireframe  plot  for  the  surface  z  =  f(x,y)  =  x2+y2 

•  Press  C3Di^,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT  SETUP 
window. 

•  Press  ^?  and  type  'X^+rZH. 

•  Press  EEm  EEM  to  draw  the  slope  field  plot.    Press  B3S  IWUBflD  D311I 
to  see  the  plot  unencumbered  by  the  menu  and  with  identifying  labels. 


•  Press  [NxT)[Nxr)iMSM  to  leave  the  EDIT  environment. 

•  Press  BEEQ  to  return  to  the  PLOT  WINDOW  environment.    Then,  press 
QwJ  r  or  (j^JIEEB,  to  return  to  normal  calculator  display. 

Ps-Contour  plots 

Ps-Contour  plots  are  contour  plots  of  three-dimensional  surfaces  described  by 
z  =  f(x,y).  The  contours  produced  are  projections  of  level  surfaces  z  = 
constant  on  the  x-y  plane.  For  example,  to  produce  a  Ps-Contour  plot  for  the 
surface  z  =  x2+y2,  use  the  following: 


Page  1  2-39 


•  Press  CSDi^,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Ps-Contour. 

•  Press  ^3?  and  type  'XA2+YA2'  HI. 

•  Make  sure  that  'X'  is  selected  as  the  indep:  and  'Y'  as  the  Depnd: 
variables. 

•  Press  WBM  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Keep  the  default  plot  window  ranges  to  read:  X-Left:-2,  X-Right:2,  Y-Near:-1 
Y-Far:  1,  Step  Indep:  10,  Depnd:  8 

•  Press  !L:!Li:!!i!!!::L:j!  iLiiLi:!!!!!!;!!  to  draw  the  contour  plot.  This  operation  will  take  some 
time,  so,  be  patient.  The  result  is  a  contour  plot  of  the  surface.  Notice 
that  the  contour  are  not  necessarily  continuous,  however,  they  do  provide 
a  good  picture  of  the  level  surfaces  of  the  function. 

•  Press  H3HKW)  lllllijilll  lulilll  to  see  the  graph  with  labels  and  ranges. 


m 

\  \  \ 

\ 

\ 

\  \ 

/  /  / 

•  Press  (W)(W)IIfiH!nH!  to  return  to  the  PLOT  WINDOW  environment. 

•  Press  L^J  /  or       111311,  to  return  to  normal  calculator  display. 

Try  also  a  Ps-Contour  plot  for  the  surface  z  =  f(x,y)  =  sin  x  cos  y. 


•  Press  CSDi^,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT  SETUP 
window. 

•  Press  ^7  and  type  ySIN(X)*COS(Y)'  MM. 


Press  IlEililllLMEllEji  irjimiiTM  to  draw  the  slope  field  plot.  Press  E3ffl(W)l!SHI!  HBBH 
to  see  the  plot  unencumbered  by  the  menu  and  with  identifying  labels. 


— -— i. 

V1 

■    /  — - 

-— —  y  / 

■  \  \ 

.  \  \  "^-.— — - 

Page  1  2-40 


•  Press  [NxfJ[NxT)mMI  to  leave  the  EDIT  environment. 

•  Press  to  return  to  the  PLOT  WINDOW  environment.  Then,  press 
ioNj  ,  or  (jv^jIBM,  to  return  to  normal  calculator  display. 

Y-Slice  plots 

Y-Slice  plots  are  animated  plots  of  z-vs.-y  for  different  values  of  x  from  the 
function  z  =  f(x,y).  For  example,  to  produce  a  Y-Slice  plot  for  the  surface  z  = 
x3-xy3,  use  the  following: 

•  Press  LjnJi^D  f  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Y-Slice. 

•  Press  ^?  and  type  /XA3+X*YA3MIII™ 

•  Make  sure  that  'X'  is  selected  as  the  Indep:  and  'Y'  as  the  Depnd:  variables. 

•  Press  [nxtJ I!!!!!!:!!!!:!!!!!!!  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Keep  the  default  plot  window  ranges  to  read:  X-Left:-1,  X-Right:  1 ,  Y-Near:-1, 
Y-Far:  1,  Z-Low:-l,  Z-High:l,  Step  Indep:  10  Depnd:  8 

•  Press  B3j]3jjl  ItEiH  to  draw  the  three-dimensional  surface.  You  will  see  the 
calculator  produce  a  series  of  curves  on  the  screen,  that  will  immediately 
disappear.  When  the  calculator  finishes  producing  all  the  y-slice  curves, 
then  it  will  automatically  go  into  animating  the  different  curves.  One  of 
the  curves  is  shown  below. 


•  Press   LonJ    to  stop  the  animation.    Press  B3331  to  return  to  the  PLOT 
WINDOW  environment. 

•  Press  IsaJ  ,  or  (jv^jKEBI,  to  return  to  normal  calculator  display. 
Try  also  a  Ps-Contour  plot  for  the  surface  z  =  f(x,y)  =  (x+y)  sin  y. 


Page  12-41 


•  Press  C3D  ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT  SETUP 
window. 

•  Press  ^7  and  type  '(X+Y)*SIN(Y)'  KB. 

•  Press  r32EE3=!  ESSE!  to  produce  the  Y-Slice  animation. 

•  Press  I  on]  to  stop  the  animation. 

•  Press  BiEHi  to  return  to  the  PLOT  WINDOW  environment.  Then,  press 
Loa/J  ,  or  C^DiEM,  to  return  to  normal  calculator  display. 

Gridmap  plots 

Gridmap  plots  produce  a  grid  of  orthogonal  curves  describing  a  function  of  a 
complex  variable  of  the  form  w  =f(z)  =  f(x+iy),  where  z  =  x+iy  is  a  complex 
variable.  The  functions  plotted  correspond  to  the  real  and  imaginary  part  of 
w  =  0(x,y)  +  i^x^),  i.e.,  they  represent  curves  ®(x,y)  ^constant,  and  ^(x^) 
=  constant.  For  example,  to  produce  a  Gridmap  plot  for  the  function  w  = 
sin(z),  use  the  following: 

•  Press  (JnJ  joflo  ,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  Gridmap. 

•  Press  ^?  and  type  'SIN(X+i*Y)'  WSM. 

•  Make  sure  that  'X'  is  selected  as  the  Indep:  and  'Y'  as  the  Depnd:  variables. 

•  Press  InxtJ lEIl!  to  return  to  normal  calculator  display. 

•  Press  LjnJ ML ,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

•  Keep  the  default  plot  window  ranges  to  read:  X-Left:-1,  X-Right:  1 ,  Y-Near:-1 
Y-Far:  1,  XXLeft:-!  XXRight:l,  YYNean-1,  yyFar:  1,  Step  Indep:  10  Depnd:  8 

•  Press  EEEEsB  E33D  to  draw  the  gridmap  plot.  The  result  is  a  grid  of 
functions  corresponding  to  the  real  and  imaginary  parts  of  the  complex 
function. 


•     Press  C^C^IM  BXH1  to  return  to  the  PLOT  WINDOW  environment. 


Page  1  2-42 


•  Press  I  on  J  ,  or  (JW7JB9IIB,  to  return  to  normal  calculator  display. 

Other  functions  of  a  complex  variable  worth  trying  for  Gridmap  plots  are: 
(1 )  SIN((X,Y))     i.e.,  F(z)  =  sin(z)  (2)(X,Y)A2         i.e.,  F(z)  =  z2 

(3)  EXP((X,Y))     i.e.,  F(z)  =  ez  (4)  SINH((X,Y))  i.e.,  F(z)  =  sinh(z) 

(5)  TAN((X,Y))    i.e.,  F(z)  =  tan(z)  (6)  ATAN((X,Y))  i.e.,  F(z)  =  tan  ^z) 

(7)  (X,Y)A3        i.e.,  F(z)  =  z3  (8)  1/(X,Y)        i.e.,  F(z)  =  1/z 

(9)  V  (X,Y)         i.e.,  F(z)  =  i}n 

Pr-Surface  plots 

Pr-Surface  (parametric  surface)  plots  are  used  to  plot  a  three-dimensional 
surface  whose  coordinates  (x,y,z)  are  described  by  x  =  x(X,Y),  y  =  y(X,Y), 
z=z(X,Y),  where  X  and  Y  are  independent  parameters. 

Note:  The  equations  x  =  x(X,Y),  y  =  y(X,Y),  z=z(X,Y)  represent  a  parametric 
description  of  a  surface.  X  and  Y  are  the  independent  parameters.  Most 
textbooks  will  use  (u,v)  as  the  parameters,  rather  than  (X,Y).  Thus,  the 
parametric  description  of  a  surface  is  given  as  x  =  x(u,v),  y  =  y(u,v),  z=z(u,v). 

For  example,  to  produce  a  Pr-Surface  plot  for  the  surface  x  =  x(X,Y)  =  X  sin  Y, 
y  =  y(X,Y)  =  x  cos  Y,  z=z(X,Y)=X,  use  the  following: 

•  Press   C5DJ^,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 
Change  TYPE  to  Pr-Surface. 

Press  ^?  and  type  /{X*SIN(Y),  X*COS(Y),  X}'  EH. 
Make  sure  that  'X'  is  selected  as  the  Indep:  and  'Y'  as  the  Depnd: 
variables. 

Press  Q^IIIEEIII  to  return  to  normal  calculator  display. 
Press  C5DJ^,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
WINDOW  screen. 

Keep  the  default  plot  window  ranges  to  read:  X-Left:-1,  X-Right:  1 ,  Y-Near:-1, 
Y-Far:  1,  Z-Low:  -1,  Z-High:l,  XE:  0,  YE:-3,  zE:0,  Step  Indep:  10,  Depnd:  8 
Press  IH?EEB  E33II  to  draw  the  three-dimensional  surface. 
Press  IIIjHil(W3  DEEH!  EHEEI  to  see  the  graph  with  labels  and  ranges. 


Page  1  2-43 


•  Press  (W)(W)iafil  H3IH!  to  return  to  the  PLOT  WINDOW  environment. 

•  Press  I  on]  ,  or  to  return  to  normal  calculator  display. 

The  VPAR  variable 

The  VPAR  (Volume  Parameter)  variable  contains  information  regarding  the 
"volume"  used  to  produce  a  three  dimensional  graph.  Therefore,  you  will  see 
it  produced  whenever  you  create  a  three  dimensional  plot  such  as  Fast3D, 
Wireframe,  or  Pr-Surface. 


Interactive  drawing 

Whenever  we  produce  a  two-dimensional  graph,  we  find  in  the  graphics 
screen  a  soft  menu  key  labeled  illilllli!.  Pressing  lEIIIIii  produces  a  menu  that 
include  the  following  options  (press  InxtJ  to  see  additional  functions): 


V 

-6.5 

6.5 

liHaiiMfllHiH 

a: 

V 

-6.5 

6.5 

liMfiHF7MTiV111 

DEL  |EF;hl"E|  HEhU 

a: 

V 

-6.5 

6.5 

SUE  |  REPL  |PICT-»j 

KjV^I         |  PICT 

Through  the  examples  above,  you  have  the  opportunity  to  try  out  functions 
LABEL,  MENU,  PICT->,  and  REPL.  Many  of  the  remaining  functions,  such  as 
DOT+,  DOT-,  LINE,  BOX,  CIRCL,  MARK,  DEL,  etc.,  can  be  used  to  draw 


Page  1  2-44 


points,  lines,  circles,  etc.  on  the  graphics  screen,  as  described  below.  To  see 
how  to  use  these  functions  we  will  try  the  following  exercise: 

First,  we  get  the  graphics  screen  corresponding  to  the  following  instructions: 

•  Press  LjnJi^D  /  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  type  to  Function,  if  needed 

•  Change  EQ  to  'X' 

•  Make  sure  that  Indep:  is  set  to  'X'  also 

•  Press  [nxtJ iiiiii!i!G:liiiiii  to  return  to  normal  calculator  display. 

•  Press  C^DJm^L,  simultaneously  if  in  RPN  mode,  to  access  the  PLOT 
window  (in  this  case  it  will  be  called  PLOT  -POLAR  window). 

•  Change  the  H-VIEW  range  to  -10  to  10,  by  using 
CDUD&*mCDUDJB?m,  and  the  V-VIEW  range  to  -5  to  5  by 
using  CD CD KM  CD BM. 

•  Press  EEMi  MEM  to  plot  the  function. 

•  Press  IIIlMlQ^  IIIIHiiOII  to  add  labels  to  the  graph.  Press  Q^C*D  (or 
{J^Jprev_  )  to  recover  the  original  EDIT  menu. 

Next,  we  illustrate  the  use  of  the  different  drawing  functions  on  the  resulting 
graphics  screen.  They  require  use  of  the  cursor  and  the  arrow  keys 
(C3DCD^>^3?)  to  move  the  cursor  about  the  graphics  screen. 

DOT+  and  DOT- 

When  DOT+  is  selected,  pixels  will  be  activated  wherever  the  cursor  moves 
leaving  behind  a  trace  of  the  cursor  position.  When  DOT-  is  selected,  the 
opposite  effect  occurs,  i.e.,  as  you  move  the  cursor,  pixels  will  be  deleted. 

For  example,  use  the  CD^b  keys  to  move  the  cursor  somewhere  in  the 
middle  of  the  first  quadrant  of  the  x-y  plane,  then  press  lllillillllll.  The  label  will 
be  selected  (IlililMi).  Press  and  hold  the  CD  key  to  see  a  horizontal  line  being 
traced.  Now,  press  MSB,  to  select  this  option  (  IIMIlHl!  ).  Press  and  hold 
the  CT)  key  to  see  the  line  you  just  traced  being  erased.  Press  IIMII,  when 
done,  to  deselect  this  option. 


Page  1  2-45 


MARK 

This  command  allows  the  user  to  set  a  mark  point  which  can  be  used  for  a 
number  of  purposes,  such  as: 

•  Start  of  line  with  the  LINE  or  TLINE  command 

•  Corner  for  a  BOX  command 

•  Center  for  a  CIRCLE  command 

Using  the  MARK  command  by  itself  simply  leaves  an  x  in  the  location  of  the 
mark  (Press  {jwrj uMB  to  see  it  in  action). 

LINE 

This  command  is  used  to  draw  a  line  between  two  points  in  the  graph.  To  see 
it  in  action,  position  the  cursor  somewhere  in  the  first  quadrant,  and  press 
(j^Jprev_  HJJQ.  A  MARK  is  placed  over  the  cursor  indicating  the  origin  of  the 
line.  Use  the  CD  key  to  move  the  cursor  to  the  right  of  the  current  position, 
say  about  1  cm  to  the  right,  and  press  EES.  A  line  is  draw  between  the  first 
and  the  last  points. 

Notice  that  the  cursor  at  the  end  of  this  line  is  still  active  indicating  that  the 
calculator  is  ready  to  plot  a  line  starting  at  that  point.  Press  ^3?  to  move  the 
cursor  downwards,  say  about  another  cm,  and  press  EES  again.  Now  you 
should  have  a  straight  angle  traced  by  a  horizontal  and  a  vertical  segments. 
The  cursor  is  still  active.  To  deactivate  it,  without  moving  it  at  all,  press  EES. 
The  cursor  returns  to  its  normal  shape  (a  cross)  and  the  LINE  function  is  no 
longer  active. 

TUNE 

(Toggle  LINE)  Move  the  cursor  to  the  second  quadrant  to  see  this  function  in 
action.  Press  DEES.  A  MARK  is  placed  at  the  start  of  the  toggle  line.  Move 
the  cursor  with  the  arrow  keys  away  from  this  point,  and  press  EBES.  A  line 
is  drawn  from  the  current  cursor  position  to  the  reference  point  selected  earlier. 
Pixels  that  are  on  in  the  line  path  will  be  turned  off,  and  vice  versa.  To 
remove  the  most  recent  line  traced,  press  EBES  again.  To  deactivate  TLINE, 
move  the  cursor  to  the  original  point  where  TLINE  was  activated,  and  press 


Page  1  2-46 


BOX 

This  command  is  used  to  draw  a  box  in  the  graph.  Move  the  cursor  to  a 
clear  area  of  the  graph,  and  press  This  highlights  the  cursor.    Move  the 

cursor  with  the  arrow  keys  to  a  point  away,  and  in  a  diagonal  direction,  from 
the  current  cursor  position.  Press  IIIIIII  again.  A  rectangle  is  drawn  whose 
diagonal  joins  the  initial  and  ending  cursor  positions.  The  initial  position  of 
the  box  is  still  marked  with  and  x.  Moving  the  cursor  to  another  position  and 
pressing  lllillil  will  generate  a  new  box  containing  the  initial  point.  To  deselect 
BOX,  move  the  cursor  to  the  original  point  where  BOX  was  activated,  then 

press  mm  mm. 


CIRCL 

This  command  produces  a  circle.  Mark  the  center  of  the  circle  with  a  MARK 
command,  then  move  the  cursor  to  a  point  that  will  be  part  of  the  periphery  of 
the  circle,  and  press  IS339.    To  deactivate  CIRCL,  return  the  cursor  to  the 
MARK  position  and  press  IIIIiB. 

Try  this  command  by  moving  the  cursor  to  a  clear  part  of  the  graph,  press 
CESS.  Move  the  cursor  to  another  point,  then  press  IsjEHS.  A  circle  centered 
at  the  MARK,  and  passing  through  the  last  point  will  be  drawn. 


LABEL 

Pressing  IIII3II!!!  places  the  labels  in  the  x-  and  y-axes  of  the  current  plot.  This 
feature  has  been  used  extensively  through  this  chapter. 

DEL 

This  command  is  used  to  remove  parts  of  the  graph  between  two  MARK 
positions.  Move  the  cursor  to  a  point  in  the  graph,  and  press  CESS.  Move 
the  cursor  to  a  different  point,  press  ICESEI  again.  Then,  press  lliiE!!!!!.  The 
section  of  the  graph  boxed  between  the  two  marks  will  be  deleted. 


Page  1  2-47 


ERASE 

The  function  ERASE  clears  the  entire  graphics  window.  This  command  is 
available  in  the  PLOT  menu,  as  well  as  in  the  plotting  windows  accessible 
through  the  soft  menu  keys. 


MENU 

Pressing  ILiliiilili  will  remove  the  soft  key  menu  labels  to  show  the  graphic 
unencumbered  by  those  labels.    To  recover  the  labels,  press  [nxt)  . 


SUB 

Use  this  command  to  extract  a  subset  of  a  graphics  object.  The  extracted 
object  is  automatically  placed  in  the  stack.  Select  the  subset  you  want  to 
extract  by  placing  a  MARK  at  a  point  in  the  graph,  moving  the  cursor  to  the 
diagonal  corner  of  the  rectangle  enclosing  the  graphics  subset,  and  press 
IliOIl.   This  feature  can  be  used  to  move  parts  of  a  graphics  object  around  the 
graph. 


REPL 

This  command  places  the  contents  of  a  graphic  object  currently  in  stack  level 
1  at  the  cursor  location  in  the  graphics  window.  The  upper  left  corner  of  the 
graphic  object  being  inserted  in  the  graph  will  be  placed  at  the  cursor 
position.    Thus,  if  you  want  a  graph  from  the  stack  to  completely  fill  the 
graphic  window,  make  sure  that  the  cursor  is  placed  at  the  upper  left  corner 
of  the  display. 


PICT^ 

This  command  places  a  copy  of  the  graph  currently  in  the  graphics  window 
on  to  the  stack  as  a  graphic  object.  The  graphic  object  placed  in  the  stack 
can  be  saved  into  a  variable  name  for  storage  or  other  type  of  manipulation 


This  command  copies  the  coordinates  of  the  current  cursor  position,  in  user 
coordinates,  in  the  stack. 


Page  1  2-48 


Zooming  in  and  out  in  the  graphics  display 

Whenever  you  produce  a  two-dimensional  FUNCTION  graphic  interactively, 
the  first  soft-menu  key,  labeled  BSIiEl,  lets  you  access  functions  that  can  be 
used  to  zoom  in  and  out  in  the  current  graphics  display.  The  ZOOM  menu 
includes  the  following  functions  (press  Lnxt J  to  move  to  the  next  menu): 


We  present  each  of  these  functions  following.  You  just  need  to  produce  a 
graph  as  indicated  in  Chapter  1  2,  or  with  one  of  the  programs  listed  earlier 
in  this  Chapter. 


ZFACT,  ZIN,  ZOUT,  and  ZLAST 

Pressing  BEES  produces  an  input  screen  that  allows  you  to  change  the  current 
X-  and  Y-Factors.  The  X-  and  Y-Factors  relate  the  horizontal  and  vertical  user- 
defined  unit  ranqes  to  their  corresponding  pixel  ranqes.  Chanqe  the  H- 
Fadorjoread  e!  and  press  5L  change  the  vLtor  to  read  2^ 
press  lllllMll:lllll.  Check  off  the  option  ^Recenter  on  cursor,  and  press  IIIiJM 

Back  in  the  graphics  display,  press  EE!  .  The  graphic  is  re-drawn  with  the 
new  vertical  and  horizontal  scale  factors,  centered  at  the  position  where  the 
cursor  was  located,  while  maintaining  the  original  PICT  size  (i.e.,  the  original 
number  of  pixels  in  both  directions).  Using  the  arrow  keys,  scroll 

horizontally  or  vertically  as  far  as  you  can  of  the  zoomed-in  graph. 

To  zoom  out,  subjected  to  the  H-  and  V-Factors  set  with  ZFACT,  press  B333 
BEES.  The  resulting  graph  will  provide  more  detail  than  the  zoomed-in  graph. 
You  can  always  return  to  the  very  last  zoom  window  by  using  BES1. 


Page  1  2-49 


BOXZ 

Zooming  in  and  out  of  a  given  graph  can  be  performed  by  using  the  soft- 
menu  key  BOXZ.  With  BOXZ  you  select  the  rectangular  sector  (the  "box") 
that  you  want  to  zoom  in  into.  Move  the  cursor  to  one  of  the  corners  of  the 
box  (using  the  arrow  keys),  and  press  III!!!:!:!:!!!.  Using  the  arrow  keys  once 
more,  move  the  cursor  to  the  opposite  corner  of  the  desired  zoom  box.  The 
cursor  will  trace  the  zoom  box  in  the  screen.  When  desired  zoom  box  is 
selected,  press  EEEd.  The  calculator  will  zoom  in  the  contents  of  the  zoom 
box  that  you  selected  to  fill  the  entire  screen. 

If  you  now  press  il:;il!il!3iji,  the  calculator  will  zoom  out  of  the  current  box  using 
the  H-  and  V-Factors,  which  may  not  recover  the  graph  view  from  which  you 
started  the  zoom  box  operation. 

ZDFLT,  ZAUTO 

Pressing  iliiiiSiillj!  re-draws  the  current  plot  using  the  default  x-  and  y-ranges,  i.e., 
-6.5  to  6.5  in  x,  and  -3.1  to  3.1  in  y.  The  command  l!:;:l!l!i]3ili!l,  on  the  other 
hand,  creates  a  zoom  window  using  the  current  independent  variable  (x) 
range,  but  adjusting  the  dependent  variable  (y)  range  to  fit  the  curve  (as  when 
you  use  the  function  MM  in  the  PLOT  WINDOW  input  form  (CSDJ^L, 
simultaneously  in  RPN  mode). 

HZIN,  HZOUT,  VZIN  and  VZOUT 

These  functions  zoom  in  and  out  the  graphics  screen  in  the  horizontal  or 
vertical  direction  according  to  the  current  H-  and  V-Factors. 

CNTR 

Zooms  in  with  the  center  of  the  zoom  window  in  the  current  cursor  location. 
The  zooming  factors  used  are  the  current  H-  and  V-Factors. 

ZDECI 

Zooms  the  graph  so  as  to  round  off  the  limits  of  the  x-interval  to  a  decimal 
value. 


Page  1  2-50 


ZINTG 

Zooms  the  graph  so  that  the  pixel  units  become  user-define  units.  For  example, 
the  minimum  PICT  window  has  131  pixels.  When  you  use  ZINTG,  with  the 
cursor  at  the  center  of  the  screen,  the  window  gets  zoomed  so  that  the  x-axis 
extends  from  -64.5  to  65.5. 


ZSQR 

Zooms  the  graph  so  that  the  plotting  scale  is  maintained  at  1:1  by  adjusting 
the  x  scale,  keeping  the  y  scale  fixed,  if  the  window  is  wider  than  taller.  This 
forces  a  proportional  zooming. 

ZTRIG 

Zooms  the  graph  so  that  the  x  scale  incorporates  a  range  from  about  — 3ti  to 
+37i,  the  preferred  range  for  trigonometric  functions. 

Note:  None  of  these  functions  are  programmable.  They  are  only  useful  in 
an  interactive  way.  Do  not  confuse  the  command  EQESQ  in  the  ZOOM  menu 
with  the  function  ZFACTOR,  which  is  used  for  gas  dynamic  and  chemistry 
applications  (see  Chapter  3). 

The  SYMBOLIC  menu  and  graphs 

The  SYMBOLIC  menu  is  activated  by  pressing  the  Lsymb J  key  (fourth  key  from 
the  left  in  fourth  row  from  the  top  of  the  keyboard).  This  menu  provides  a  list 
of  menus  related  to  the  Computer  Algebraic  System  or  CAS,  these  are: 


SYHBOLIC  HEHU 


2 .  HRITHHETK.. 

3.  CALCULUS.. 
H.  GRAPH.. 

S .  SOLVER.. 
G.TRIGOnOHETRIC. 


SYHBOLIC  HEHU 


2 .  HRITHHETK.. 

3 .  CALCULUS.. 
H .  GRAPH.. 

5.  SOLVER.. 
C.TRIGOnOHETRIC. 


All  but  one  of  these  menus  are  available  directly  in  the  keyboard  by  pressing 
the  appropriate  keystroke  combination  as  follows.  The  Chapter  of  the  user 
manual  where  the  menus  are  described  is  also  listed: 


Page  12-51 


ALGEBRA.. 

(the 

(3D 

key) 

Ch.  5 

ARITHMETIC. 

(the 

CD 

key) 

Ch.  5 

CALCULUS.. 

(J=T)CALC_ 

(the 

CD 

key) 

Ch.  13 

SOLVER.. 

(the 

CD 

key) 

Ch.  6 

TRIGONOMETRIC. 

(the 

CD 

key) 

Ch.  5 

EXP&LN.. 

(the 

CD 

key) 

Ch.  5 

The  SYMB/GRAPH  menu 

The  GRAPH  sub-menu  within  the  SYMB  menu  includes  the  following  functions: 


SYHBOLIC  GRAPH  HEHU 


a.GROP-ftDD 
3. PLOT 
H . PLuTflDD 
5.PUt  setup.. 
G .  L"  I  'j  Tl  T  H  E= 


SYHBOLIC  GRHPH  HEHU 


H.PLOTflDD 

5. PUt  setup.. 

G.SIGnTflE 

?.TflP-VflL 

E .  T  h  E= '.'  H  F; 


3  .  L"  i  M  E=  N  L I . . 


DEFINE:  same  as  the  keystroke  sequence  l<n  JPEF     (the  L  2  j  key) 
GROBADD:  pastes  two  GROBs  first  over  the  second  (See  Chapter  22) 
PLOT(function):  plots  a  function,  similar  to  LjnJi^D 

PLOTADD(function):  adds  this  function  to  the  list  of  functions  to  plot,  similar  to 

(J^J  2D/3D^ 

Plot  setup..:  same  as  C3D  J^, 

SIGNTAB(function):  sign  table  of  given  function  showing  intervals  of  positive 
and  negative  variation,  zero  points  and  infinite  asymptotes 
TABVAL:  table  of  values  for  a  function 
TABVAR:  variation  table  of  a  function 

Examples  of  some  of  these  functions  are  provided  next. 

PLOT(XA2-l )  is  similar  to  EDi^D  with  EQ:  XA2  -1 .  Using  EEMi  Em 
produces  the  plot: 


X"X"X"X"X"X"X"X"X"X"X"  F  L  V  T       E  T  U  F  ■X"X"X"X"X"X"X"X"X"X"X 

TMD*:liH.L.UlJ.« 

E4:xa-i 

In44p:X  SiHULt 
H-TicR : 10 .  V-TicR:iD. 
Choose  type  of  plot 

^Connect 
^Fixe  Is 

Page  1  2-52 


PLOTADD(XA2-X)  is  similar  to       J^£.  but  adding  this  function  to  EQ:  XA2 

I  kinn  ll-ili-lMiiiiiiliriliT;!:]  nmr\nre 


l;  :  F;  ■  j 

Ind-sp  =  !■:  SiHuLt 
H-TiCh:iD.  V-TiCh=iD. 

Chooj*  type  of  plot 

^Connect 
^Pixels 

IijiiM  |  i::,  Vj  |TF;hlE|  FCn  I  EDIT  IlhnlL 


TABVAL(XA2-1  ,{1 ,  3})  produces  a  list  of  {min  max}  values  of  the  function  in  the 
interval  {1,3},  while  SIGNTAB(XA2-1 )  shows  the  sign  of  the  function  in  the 
interval  (-oo,+),  with  f(x)  >  0  in  (-co,-l),  f(x)  <0,  in  (-1,1),  and  f(x)  >  0  in  (1,+  oo). 

F  HLLK   J  ~ 


TRBVRI_(x2-l,U  3>) 

{x2-l  {{1  3>  {0  S» 

signtrb(x2-i) 

{-■:■:■  +  -1  -  1  +  +*> 


TABVAR(LN(X)/X)  produces  the  following  table  of  variation: 

L  -r ■■-■■-■I  LNL4J  I  I 


TRBVRR^  x 

C   'LhKXVX'  C  C 

f?'    '0+0'  +  'EXPaV  - 

'+«■'   >  £   '?'    '?'  t 

■l^EXPav  i  +:  0  y  y 

Graphic  113  x  131  > 


A  detailed  interpretation  of  the  table  of  variation  is  easier  to  follow  in  RPN 
mode: 


F=: 


F'  = 


X 

X^-LN(X) 


SQ(X) 

-II  klfV^i-ll  .  

Wari 


F'=: 


->: 


X^-LN(X) 

SQ(X) 
-(LH(X)-l) 


I  toumananaaiamBanHniH 


at  ion  table: 
1 


-w  ?  0+0  +  e  -  +w  X 
?  ?   *  t-j-l-+:0F 


Page  1  2-53 


The  output  is  in  a  graphical  format,  showing  the  original  function,  F(X),  the 
derivative  F'(X)  right  after  derivation  and  after  simplification,  and  finally  a 
table  of  variation.  The  table  consists  of  two  rows,  labeled  in  the  right-hand 
side.  Thus,  the  top  row  represents  values  of  X  and  the  second  row  represents 
values  of  F.  The  question  marks  indicates  uncertainty  or  non-definition.  For 
example,  for  X<0,  LN(X)  is  not  defined,  thus  the  X  lines  shows  a  question  mark 
in  that  interval.  Right  at  zero  (0+0)  F  is  infinite,  for  X  =  e,  F  =  1/e.  F 
increases  before  reaching  this  value,  as  indicated  by  the  upward  arrow,  and 
decreases  after  this  value  (X=e)  becoming  slightly  larger  than  zero  (+:0)  as  X 
goes  to  infinity.  A  plot  of  the  graph  is  shown  below  to  illustrate  these 
observations: 


-K- 


Function  DRAW3DMATRIX 

This  function  takes  as  argument  a  nxm  matrix,  Z,  =  [  Zjj],  and  minimum  and 
maximum  values  for  the  plot.  You  want  to  select  the  values  of  vmin  and  vmax  so 
that  they  contain  the  values  listed  in  Z.  The  general  call  to  the  function  is, 
therefore,  DRAW3DMATRIX(Z,vmin,vmax).  To  illustrate  the  use  of  this  function 
we  first  generate  a  6x5  matrix  using  RANM({6,5}),  and  then  call  function 
DRAW3DMATRIX,  as  shown  below: 


[-6  -3  -6  -6  3  - 
7  -5  0    0  3 
5  -6  -H  -3  -3 

-3  H    7    3  6 
5  -i  H  -H  i 

.7  -H  -5  7  -?. 

EDIT  |  VIEH  |  RCL  |  £Tut  |PURGE|CLERR 

-h,  -c-  -h.  -h,  3 

7-5  0    0  3 

5  -6  -H  -3  -3 

-3  H    7    3  6 

5  -i  H  -H  i 

.7  -H  -5  7  -7. 

:DRAH3DHATRIK(HnS(i).-i0.iO) 

HOVRL 

EDIT  |  VIEH  |  RCL  |  3 

TO*  |F  UF;'jE|lLEhF; 

The  plot  is  in  the  style  of  a  FAST3DPLOT.  Different  views  of  the  plot  are  shown 
below: 

T 


Page  1  2-54 


Chapter  1 3 
Calculus  Applications 

In  this  Chapter  we  discuss  applications  of  the  calculator's  functions  to 
operations  related  to  Calculus,  e.g.,  limits,  derivatives,  integrals,  power  series, 
etc. 

The  CALC  (Calculus)  menu 

Many  of  the  functions  presented  in  this  Chapter  are  contained  in  the 
calculator's  CALC  menu,  available  through  the  keystroke  sequence  {^Jcalc_ 
(associated  with  the  LU  key).  The  CALC  menu  shows  the  following  entries: 


CALC  METlU 

li.DERIY.  &  HUE'] . ..  I 

2. LIMIT:-  SERIES.. 

3 .  DIFFERENTIAL  EGH.S.. 

H. GRAPH.. 

5.DERVK 

fi.inTVK 

I       !       1       icsncLl  ok 

The  first  four  options  in  this  menu  are  actually  sub-menus  that  apply  to  (1) 
derivatives  and  integrals,  (2)  limits  and  power  series,  (3)  differential 
equations,  and  (4)  graphics.  The  functions  in  entries  (1)  and  (2)  will  be 
presented  in  this  Chapter.  Differential  equations,  the  subject  of  item  (3),  are 
presented  in  Chapter  16.  Graphic  functions,  the  subject  of  item  (4),  were 
presented  at  the  end  of  Chapter  12.  Finally,  entries  5.  DERVX  and  6.INTVX 
are  the  functions  to  obtain  a  derivative  and  a  indefinite  integral  for  a  function 
of  the  default  CAS  variable  (typically,  'X').  Functions  DERVX  and  INTVX  are 
discussed  in  detail  later. 

Limits  and  derivatives 

Differential  calculus  deals  with  derivatives,  or  rates  of  change,  of  functions 
and  their  applications  in  mathematical  analysis.  The  derivative  of  a  function 
is  defined  as  a  limit  of  the  difference  of  a  function  as  the  increment  in  the 
independent  variable  tends  to  zero.  Limits  are  used  also  to  check  the 
continuity  of  functions. 


Page  13-1 


Function  lim 

The  calculator  provides  function  lim  to  calculate  limits  of  functions.  This 
function  uses  as  input  an  expression  representing  a  function  and  the  value 
where  the  limit  is  to  be  calculated.  Function  lim  is  available  through  the 
command  catalog  (0_w  {alpha}[^J(l}  )  or  through  option  2.  LIMITS  & 
SERIES...  of  the  CALC  menu  (see  above). 


Note:  The  functions  available  in  the  LIMITS  &  SERIES  menu  are  shown  next: 


LIMIT:"      SERIES  HERU 

a.  Uh 
3. SERIES 
H . TflYLORO 

5 .  TflVLR 

6.  CALCULUS.. 

Function  DIVPC  is  used  to  divide  two  polynomials  producing  a  series 
expansion.  Functions  DIVPC,  SERIES,  TAYLORO,  and  TAYLOR  are  used  in 
series  expansions  of  functions  and  discussed  in  more  detail  in  this  Chapter. 


Function  lim  is  entered  in  ALG  mode  as  1  x=a)  to  calculate  the 

limit  lim  f{x)  .  In  RPN  mode,  enter  the  function  first,  then  the  expression 

x^>a 

'x=a',  and  finally  function  lim.  Examples  in  ALG  mode  are  shown  next, 
including  some  limits  to  infinity.  The  keystrokes  for  the  first  example  are  as 
follows  (using  Algebraic  mode,  and  system  flag  1  1  7  set  to  CHOOSE  boxes): 

(jT)  calc_  (X)  BBI CX3  IISl  (gGDCDCB  i  (g  CB  ^    CD  (enter) 


:  limCX+1) 

X+l 

2 

:  lim(x£-2) 

X+l 

-1 

CftSCM  HELP 


The  infinity  symbol  is  associated  with  the  L  0  J  key,  i.e..,      j00  . 


Page  1  3-2 


Derivatives 

The  derivative  of  a  function  f(x)  at  x  =  a  is  defined  as  the  limit 
g.A,)-lin/<'  +  *>-/M 

dx  h->0  h 

Some  examples  of  derivatives  using  this  limit  are  shown  in  the  following 
screen  shots: 


x+h  x 

:  lim 

e  -e 

1      h  J 

X 

e 

Functions  DERIV  and  DERVX 

The  function  DERIV  is  used  to  take  derivatives  in  terms  of  any  independent 
variable,  while  the  function  DERVX  takes  derivatives  with  respect  to  the  CAS 
default  variable  VX  (typically  X).  While  function  DERVX  is  available  directly 
in  the  CALC  menu,  both  functions  are  available  in  the  DERIV.&INTEG  sub- 
menu within  the  CALCL  menu  (  C3D$^-  )• 

Function  DERIV  requires  a  function,  say  f(t),  and  an  independent  variable,  say, 
t,  while  function  DERVX  requires  only  a  function  of  VX.  Examples  are  shown 
next  in  ALG  mode.  Recall  that  in  RPN  mode  the  arguments  must  be  entered 
before  the  function  is  applied. 


DERIvtt-eSt) 
DERIV(SIH(s),s) 


t  .  t 
COS(s)l 


:  DERVXtJX -X) 


-[[-l+2-Jx)-Jx) 


2-X 


The  DERIV&INTEG  menu 

The  functions  available  in  this  sub-menu  are  listed  below: 


Page  1  3-3 


DERI1.1 .  ft  IP.TE'] .  HEnU 

3.DERYK  1 
H.DIY 

5 .  FOURIER 

6.  HESS 

PER  I'.' .      iriTEG.  HEHU 


S.inTVK 
5 . LflPL 
iLi .  FRE'.'hL 
ii.RISCH 
ia.SIGHfl 


DERI'.' .      iriTEG.  HERU 


3.  LflPL 
ill .  FRE'.'hL 
ii.RISCH 
ia.SIGHfl 

i? .  l'ighh1.1:-: 


Out  of  these  functions  DERIV  and  DERVX  are  used  for  derivatives.  The  other 
functions  include  functions  related  to  anti-derivatives  and  integrals  (IBP,  INTVX, 
PREVAL,  RISCH,  SIGMA,  and  SIGMAVX),  to  Fourier  series  (FOURIER),and  to 
vector  analysis  (CURL,  DIV,  HESS,  LAPL).  Next  we  discuss  functions  DERIV 
and  DERVX,  the  remaining  functions  are  presented  either  later  in  this  Chapter 
or  in  subsequent  Chapters. 

Calculating  derivatives  with  d 

The  symbol  is  available  as   §  (the  jcosj  key).  This  symbol  can  be  used 

to  enter  a  derivative  in  the  stack  or  in  the  Equation  Writer  (see  Chapter  2).  If 
you  use  the  symbol  to  write  a  derivative  into  the  stack,  follow  it  immediately 
with  the  independent  variable,  then  by  a  pair  of  parentheses  enclosing  the 
function  to  be  differentiated.  Thus,  to  calculate  the  derivative  d(sin(r),r),  use, 
in  ALG  mode:  C3  ^S™)QDOED^  C^@^GD(3(£wir) 


In  RPN  mode,  this  expression  must  be  enclosed  in  quotes  before  entering  it 
into  the  stack.  The  result  in  ALG  mode  is: 


COSCr) 

In  the  Equation  Writer,  when  you  press  LxU  §  ,  the  calculator  provides  the 

following  expression: 


Page  1  3-4 


The  insert  cursor  (^)  will  be  located  right  at  the  denominator  awaiting  for  the 
user  to  enter  an  independent  variable,  say,  s:  [alpha)     }(s\  .  Then,  press  the 
right-arrow  key  (CD)  to  move  to  the  placeholder  between  parentheses: 


Next,  enter  the  function  to  be  differentiated,  say,  s*ln(s): 


RlTTlgJTmi:WrP|^J;T|J;T4f.]i^j:iJ 

To  evaluate  the  derivative  in  the  Equation  Writer,  press  the  up-arrow  key  <^>, 
four  times,  to  select  the  entire  expression,  then,  press  HEED.  The  derivative  will 
be  evaluated  in  the  Equation  Writer  as: 


en 


Note:  The  symbol  d  is  used  formally  in  mathematics  to  indicate  a  partial 
derivative,  i.e.,  the  derivative  of  a  function  with  more  than  one  variable. 
However,  the  calculator  does  not  distinguish  between  ordinary  and  partial 


Page  1  3-5 


derivatives,  utilizing  the  same  symbol  for  both.  The  user  must  keep  this 
distinction  in  mind  when  translating  results  from  the  calculator  to  paper. 


The  chain  rule 

The  chain  rule  for  derivatives  applies  to  derivatives  of  composite  functions.  A 
general  expression  for  the  chain-rule  is  d{f[g(x)]}/dx  =  (df/dg)-  (dg/dx). 
Using  the  calculator,  this  formula  results  in: 


dlg(x>dlf[g(x) 


EaaEnaoaiBEiMfliHEBini  EaaEnaBgaBanMHiHEBini 

The  terms  dl  in  front  of  g(x)  and  f(g(x))  in  the  expression  above  are 
abbreviations  the  calculator  uses  to  indicate  a  first  derivative  when  the 
independent  variable,  in  this  case  x,  is  clearly  defined.  Thus,  the  latter  result  is 
interpreted  as  in  the  formula  for  the  chain  rule  shown  above.  Here  is  another 
example  of  a  chain  rule  application: 


EDIT   CURL"   GIG  -  E'.'hL  FACTO  SIHP  M  EDIT   CURS   RIG  ■  E'.'HL  FACTO  SIHP 


Derivatives  of  equations 

You  can  use  the  calculator  to  calculate  derivatives  of  equations,  i.e., 

expressions  in  which  derivatives  will  exist  in  both  sides  of  the  equal  sign. 
Some  examples  are  shown  below: 


dix(t)=2-(siH(e(tH-die(t)] 

:  ^(yCx)=x2-3-x) 

dly(x)=2-x-3 

:DERIv(h(t)=LH(t  -lit) 

dlhCti=-|r^ 
t  -1 

Page  1  3-6 


DERVX(Y(X)=TfMXH 


f  2 
^  ^  — |-  _ — _  1 1  .■  i  j  ■.  . 


dlVCXHTRHCXr+1 
:  DERVX(G(X)=X-LN(XH 

dlGCXJ-CLHCXHll 


Notice  that  in  the  expressions  where  the  derivative  sign  (d)  or  function  DERIV 
was  used,  the  equal  sign  is  preserved  in  the  equation,  but  not  in  the  cases 
where  function  DERVX  was  used.  In  these  cases,  the  equation  was  re-written 
with  all  its  terms  moved  to  the  left-hand  side  of  the  equal  sign.  Also,  the  equal 
sign  was  removed,  but  it  is  understood  that  the  resulting  expression  is  equal  to 
zero. 


Implicit  derivatives 

Implicit  derivatives  are  possible  in  expressions  such  as: 


Bx(t>dix(t)=a.(i+x(t».dixO) 


EDIT   CURS   GIG  ■  E'.'hL  FACTO  SIHP  M  EDIT   CURS    BIG    E'.'HL  FACTO  SIHP 


Application  of  derivatives 

Derivatives  can  be  used  for  analyzing  the  graphs  of  functions  and  for 
optimizing  functions  of  one  variable  (i.e.,  finding  maxima  and  minima). 
Some  applications  of  derivatives  are  shown  next. 

Analyzing  graphics  of  functions 

In  Chapter  1  1  we  presented  some  functions  that  are  available  in  the  graphics 
screen  for  analyzing  graphics  of  functions  of  the  form  y  =  f(x).  These 
functions  include  (X,Y)  and  TRACE  for  determining  points  on  the  graph,  as 
well  as  functions  in  the  ZOOM  and  FCN  menu.  The  functions  in  the  ZOOM 
menu  allow  the  user  to  zoom  in  into  a  graph  to  analyze  it  in  more  detail. 
These  functions  are  described  in  detail  in  Chapter  1  2.  Within  the  functions  of 
the  FCN  menu,  we  can  use  the  functions  SLOPE,  EXTR,  F',  and  TANL  to 
determine  the  slope  of  a  tangent  to  the  graph,  the  extrema  (minima  and 


Page  1  3-7 


maxima)  of  the  function,  to  plot  the  derivative,  and  to  find  the  equation  of  the 
tangent  line. 


Try  the  following  example  for  the  function  y  =  tan(x). 

•  Press  CfDJ^i,  simultaneously  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 

•  Change  TYPE  to  FUNCTION,  if  needed,  by  using  [ECEGB]. 

•  Press  ^3?  and  type  in  the  equation  /TAN(X)/. 

•  Make  sure  the  independent  variable  is  set  to  'X'. 

•  Press  [nxtJ  i!ii!!>M-I:Siiliii  to  return  to  normal  calculator  display. 

•  Press  LtiJ ML  ,  simultaneously,  to  access  the  PLOT  window 

•  Change  H-VIEW  range  to  -2  to  2,  and  V-VIEW  range  to  -5  to  5. 

•  Press  iLiiLi:!!!!!!:!!!:!!  EEEIII  to  plot  the  function  in  polar  coordinates. 
The  resulting  plot  looks  as  follows: 


/ 

rcimnnfifflff 

/ 
an 

•  Notice  that  there  are  vertical  lines  that  represent  asymptotes.  These 
are  not  part  of  the  graph,  but  show  points  where  TAN(X)  goes  to  ±  oo 
at  certain  values  of  X. 

•  Press  MSm  11(311)11,  and  move  the  cursor  to  the  point  X:  1 .08E0,  Y: 
1.86E0.  Next,  press  (W)fflieHB.  The  result  is  Slope: 
4.45010547846. 

•  Press  [nx^[nxt)MSM.  This  operation  produces  the  equation  of  the 
tangent  line,  and  plots  its  graph  in  the  same  figure.  The  result  is 
shown  in  the  figure  below: 


/ 

Ton 

Lin*:  Y=H.H50j 

/ 

3HS 

Page  1  3-8 


•  Press  InxtJ  lllilu!  EEKHS  IonJ  to  return  to  normal  calculator  display. 
Notice  that  the  slope  and  tangent  line  that  you  requested  are  listed  in 
the  stack. 


Function  DOMAIN 

Function  DOMAIN,  available  through  the  command  catalog  (LxD_^z;  ), 
provides  the  domain  of  definition  of  a  function  as  a  list  of  numbers  and 
specifications.  For  example, 


DOMfllHCLH(X)) 


{-w  ?  0  +  +«} 


indicates  that  between  -go  and  0,  the  function  LN(X)  is  not  defined  (?),  while 
from  0  to  +oo,  the  function  is  defined  (+).  On  the  other  hand, 


domfiin(Ji-x£) 

<-»?-!  +  !?  +■»> 


indicates  that  the  function  is  not  defined  between  -oo  and  -1,  nor  between  1 
and  +oo.  The  domain  of  this  function  is,  therefore,  -1  <X<1 . 


Function  TABVAL 

This  function  is  accessed  through  the  command  catalog  or  through  the  GRAPH 
sub-menu  in  the  CALC  menu.  Function  TABVAL  takes  as  arguments  a  function 
of  the  CAS  variable,  f(X),  and  a  list  of  two  numbers  representing  a  domain  of 
interest  for  the  function  f(X).  Function  TABVAL  returns  the  input  values  plus  the 
range  of  the  function  corresponding  to  the  domain  used  as  input.  For 
example, 


1 

rRBVRL 

{<->{##}}} 

Page  1  3-9 


This  result  indicates  that  the  range  of  the  function  f(X)  = 


\4i  V26] 

corresponding  to  the  domain  D  =  { -1 ,5  }  is  R  =  ^  f  • 

Function  SIGNTAB 

Function  SIGNTAB,  available  through  the  command  catalog  (Lr>J  cat  )f 

provides  information  on  the  sign  of  a  function  through  its  domain.  For 
example,  for  the  TAN(X)  function, 


:SIGHTRBCTflHCX)3 

{-«■  ?  -|  -  0  +  ^  ?  +«■} 


SIGNTAB  indicates  that  TAN(X)  is  negative  between  -n/2  and  0,  and  positive 
between  0  and  n/2.  For  this  case,  SIGNTAB  does  not  provide  information 
(?)  in  the  intervals  between  -00  and  -n  /2,  nor  between  +71  /2  and  00.  Thus, 
SIGNTAB,  for  this  case,  provides  information  only  on  the  main  domain  of 
TAN(X),  namely  -n  /2  <  X  <  +71  /2. 


A  second  example  of  function  SIGNTAB  is  shown  below: 


SIGNTAB 


X+l 

 1  +  +.v.> 


For  this  case,  the  function  is  negative  for  X<-1  and  positive  for  X>  -1 . 

Function  TABVAR 

This  function  is  accessed  through  the  command  catalog  or  through  the  GRAPH 
sub-menu  in  the  CALC  menu.  It  uses  as  input  the  function  f(VX),  where  VX  is 
the  default  CAS  variable.  The  function  returns  the  following,  in  RPN  mode: 


•     Level  3:  the  function  f(VX) 


Page  13-10 


•  Two  lists,  the  first  one  indicates  the  variation  of  the  function  (i.e., 
where  it  increases  or  decreases)  in  terms  of  the  independent  variable 
VX,  the  second  one  indicates  the  variation  of  the  function  in  terms  of 
the  dependent  variable. 

•  A  graphic  object  showing  how  the  variation  table  was  computed. 

Example:  Analyze  the  function  Y  =  X3-4X2-1  1X+30,  using  the  function 
TABVAR.  Use  the  following  keystrokes,  in  RPN  mode: 

5  x-'-3  4*XA2-1 1*X+3S  5  (^CB^  (select  TABVAR)  MM 

This  is  what  the  calculator  shows  in  stack  level  1 : 

l:  Graphic  113  x  95 
F=:(x3-4-X£-ll-X+30) 

F'=:(3-X£-4-2-X-ll) 

->:((3-X-llXX+lX) 
 ■■■■  -.L-  ■;      ■;        4-  -.I-,  i .-.  ■  


This  is  a  graphic  object.  To  be  able  to  the  result  in  its  entirety,  press  ^3? . 
The  variation  table  of  the  function  is  shown  as  follows: 


Variation  table: 
+  -1  -   4^   +  +«  X 

-»  t  36  i  Z|P  t  +»  F 

-mm1 

Press  L^J  to  recover  normal  calculator  display.  Press  to  drop  this  last 
result  from  the  stack. 

Two  lists,  corresponding  to  the  top  and  bottom  rows  of  the  graphics  matrix 
shown  earlier,  now  occupy  level  1 .  These  lists  may  be  useful  for 
programming  purposes.    Press  CS  to  drop  this  last  result  from  the  stack. 


Page  13-11 


The  interpretation  of  the  variation  table  shown  above  is  as  follows:  the 
function  F(X)  increases  for  X  in  the  interval  (-00,  -1),  reaching  a  maximum  equal 
to  36  at  X  =  -1.  Then,  F(X)  decreases  until  X=  1  1/3,  reaching  a  minimum  of 
-400/27.  After  that  F(X)  increases  until  reaching  +c».  Also,  at  X  =  ±00,  F(X)= 
±00. 


Using  derivatives  to  calculate  extreme  points 

"Extreme  points,"  or  extrema,  is  the  general  designation  for  maximum  and 
minimum  values  of  a  function  in  a  given  interval.  Since  the  derivative  of  a 
function  at  a  given  point  represents  the  slope  of  a  line  tangent  to  the  curve  at 
that  point,  then  values  of  x  for  which  f'(x)  =0  represent  points  where  the  graph 
of  the  function  reaches  a  maximum  or  minimum.  Furthermore,  the  value  of 
the  second  derivative  of  the  function,  f"(x),  at  those  points  determines  whether 
the  point  is  a  relative  or  local  maximum  [f"(x)<0]  or  minimum  [f"(x)>0].  These 
ideas  are  illustrated  in  the  figure  below. 


In  this  figure  we  limit  ourselves  to  determining  extreme  points  of  the  function  y 
=  f(x)  in  the  x-interval  [a,b].  Within  this  interval  we  find  two  points,  x  =  xm 
and  x  =  xM,  where  f'(x)=0.  The  point  x  =  xm,  where  f"(x)>0,  represents  a  local 
minimum,  while  the  point  x  =  xM,  where  f"(x)<0,  represents  a  local  maximum. 
From  the  graph  of  y  =  f(x)  it  follows  that  the  absolute  maximum  in  the  interval 
[a,b]  occurs  at  x  =  a,  while  the  absolute  minimum  occurs  at  x  =  b. 


Page  13-12 


For  example,  to  determine  where  the  critical  points  of  function  'XA3-4*XA2- 


x3- 

-ll-X+30 

:flNS(l)*F 

-ll-X+36 

:  DERVXCF) 

(3-X- 

11MX+1) 

X  -4-X  -1  l-X+3^ 

DERVXCF) 

(3-X-llMX+l)| 
S0LVEVXCRHSCD3 


We  find  two  critical  points,  one  at  x  =  11/3  and  one  at  x 
the  second  derivative  at  each  point  use: 

1=  L'LKV.".!.!- J  I 

C3-X-11HX+1) 
:SOLVEVX(flNS(lH 


-1 .  To  evaluate 


:  DERVXCDERVXtF^FPP 

C3-X-4)-a 


:  SUESt(fPPjX=-^] 


:+HUM(flHS(lH 


14. 


The  last  screen  shows  that  f"(ll/3)  =  14,  thus,  x  =  11/3  is  a  relative 


-  3  - 

:+NUMtflNStlH 

14. 

:SUBST(FPP,X=-1) 

6-1-8 

:+NUMtflNStlH 

-14. 

This  result  indicates  that  f"(-l)  =  -14,  thus,  x  =  -1  is  a  relative  maximum. 
Evaluate  the  function  at  those  points  to  verify  that  indeed  f(-l)  >  f(l  1/3). 


+NUM[SUBST[F 

-14.8148148151 
+NUM(SUBST(F,X=-1H 

36 


Higher  order  derivatives 

Higher  order  derivatives  can  be  calculated  by  applying  a  derivative  function 
several  times,  e.g., 


Page  13-13 


s^[^(*SIN(X))] 

COS(X)+COS(X)+X--SIN(X) 

2-3 


Anti-derivatives  and  integrals 

An  anti-derivative  of  a  function  f(x)  is  a  function  F(x)  such  that  f(x)  =  dF/dx. 
For  example,  since  d(x3)  /dx  =  3x2,  an  anti-derivative  of  f(x)  =  3x2  is  F(x)  =  x3 
+  C,  where  C  is  a  constant.  One  way  to  represent  an  anti-derivative  is  as  a 

indefinite  integral  i.e.,  ^f(x)dx  =  F(x)  +  C,  if  and  only  if,  f(x)  =  dF/dx, 
and  C  =  constant. 


Functions  INT,  INTVX,  RISCH,  SIGMA  and  SIGMAVX 

The  calculator  provides  functions  INT,  INTVX,  RISCH,  SIGMA  and  SIGMAVX 
to  calculate  anti-derivatives  of  functions.  Functions  INT,  RISCH,  and  SIGMA 
work  with  functions  of  any  variable,  while  functions  INTVX,  and  SIGMAVX 
utilize  functions  of  the  CAS  variable  VX  (typically,  V).    Functions  INT  and 
RISCH  require,  therefore,  not  only  the  expression  for  the  function  being 
integrated,  but  also  the  independent  variable  name.  Function  INT,  requires 
also  a  value  of  x  where  the  anti-derivative  will  be  evaluated.  Functions  INTVX 
and  SIGMAVX  require  only  the  expression  of  the  function  to  integrate  in  terms 
of  VX.  Some  examples  are  shown  next  in  ALG  mode: 


INTVX 


UeX) 


CX-13-e 

:  IHTVXCRSIHCX2L 

Jl-SQCX3+X-flSIHCX) 


SIGMFlVX(£X-3M 


SIGMA 


X! 
X-0 


X-l  „ 
X-2  ^ 


2 

3 

 3  *     2  * 

IE: F"  linTYKI  LflPL 

|PREYA|RISCH|SIGHA 

SIGMFl(s-s!,s) 


Page  13-14 


Please  notice  that  functions  SIGMAVX  and  SIGMA  are  designed  for 
integrands  that  involve  some  sort  of  integer  function  like  the  factorial  (!) 
function  shown  above.  Their  result  is  the  so-called  discrete  derivative,  i.e., 
one  defined  for  integer  numbers  only. 

Definite  integrals 

In  a  definite  integral  of  a  function,  the  resulting  anti-derivative  is  evaluated  at 
the  upper  and  lower  limit  of  an  interval  (a,b)  and  the  evaluated  values 

subtracted.  Symbolically,     f(x)dx  =  F(b)  -  F(a),  where  f(x)  =  dF/dx. 


The  PREVAL(f(x),a,b)  function  of  the  CAS  can  simplify  such  calculation  by 
returning  f(b)-f(a)  with  x  being  the  CAS  variable  VX. 


70 


PREVFlLl3-X^-X,0,5J 

PREVflL(X-LN(XU,5) 

5-LHC5! 

UiaMHn^iffia«iaa^iflfeM:iMw;rr 


To  calculate  definite  integrals  the  calculator  also  provides  the  integral  symbol 

as  the  keystroke  combination  CS  (  (associated  with  the  (jmJ  key).  The 

simplest  way  to  build  an  integral  is  by  using  the  Equation  Writer  (see  Chapter 

2  for  an  example).  Within  the  Equation  Writer,  the  symbol  Ir^J  ( 

produces  the  integral  sign  and  provides  placeholders  for  the  integration  limits 
(a,b),  for  the  function,  f(x),  and  for  the  variable  of  integration  (x).  The 
following  screen  shots  show  how  to  build  a  particular  integral.  The  insert 
cursor  is  first  located  in  the  lower  limit  of  integration,  enter  a  value  and  press 
the  right-arrow  key  (Q} )  to  move  to  the  upper  limit  of  integration.  Enter  a 
value  in  that  location  and  press  CD  again  to  move  to  the  integrand  location. 
Type  the  integrand  expression,  and  press  once  more  to  move  to  the 
differential  place  holder,  type  the  variable  of  integration  in  that  location  and 
the  integral  is  ready  to  be  calculated. 


EDIT   CURL"   GIG  ■  EVflL  FACTO  SIHP  H  EDIT  CUR 


aegiHEBffiHBBiai 


Page  13-15 


At  this  point,  you  can  press  [enter]  to  return  the  integral  to  the  stack,  which  will 
show  the  following  entry  (ALG  mode  shown): 


This  is  the  general  format  for  the  definite  integral  when  typed  directly  into  the 
stack,  i.e.,  {(lower  limit,  upper  limit,  integrand,  variable  of  integration) 

Pressing  (enter)  at  this  point  will  evaluate  the  integral  in  the  stack: 


^5 

s2-lds 

2 

36 

The  integral  can  be  evaluated  also  in  the  Equation  Writer  by  selecting  the 
entire  expression  an  using  the  soft  menu  key  BZEH. 

Step-by-step  evaluation  of  derivatives  and  integrals 

With  the  Step/Step  option  in  the  CAS  MODES  windows  selected  (see  Chapter 
1 ),  the  evaluation  of  derivatives  and  integrals  will  be  shown  step  by  step.  For 
example,  here  is  the  evaluation  of  a  derivative  in  the  Equation  Writer: 


llllllS 


Page  13-16 


Notice  the  application  of  the  chain  rule  in  the  first  step,  leaving  the  derivative 
of  the  function  under  the  integral  explicitly  in  the  numerator.  In  the  second 
step,  the  resulting  fraction  is  rationalized  (eliminating  the  square  root  from  the 
denominator),  and  simplified.  The  final  version  is  shown  in  the  third  step. 
Each  step  is  shown  by  pressing  the  EEEQ  menu  key,  until  reaching  the  point 
where  further  application  of  function  EVAL  produce  no  more  changes  in  the 
expression. 

The  following  example  shows  the  evaluation  of  a  definite  integral  in  the 
Equation  Writer,  step-by-step: 


EaaEnaoaiBEiMfliHEBini 


:  +1 
Square  root 


s2+l 

Rat  i  ona 1  f r act  i  on 

9 


A. 

Rat  ional 

f ract  ion 

Rat  ional 

f ract  ion 

2 

2-s 

Notice  that  the  step-by-step  process  provides  information  on  the  intermediate 
steps  followed  by  the  CAS  to  solve  this  integral.  First,  CAS  identifies  a  square 
root  integral,  next,  a  rational  fraction,  and  a  second  rational  expression,  to 
come  up  with  the  final  result.    Notice  that  these  steps  make  a  lot  of  sense  to 
the  calculator,  although  not  enough  information  is  provided  to  the  user  on  the 
individual  steps. 


Page  13-17 


Integrating  an  equation 

Integrating  an  equation  is  straightforward,  the  calculator  simply  integrates 
both  sides  of  the  equation  simultaneously,  e.g., 


Techniques  of  integration 

Several  techniques  of  integration  can  be  implemented  in  the  calculators,  as 
shown  in  the  following  examples. 


Substitution  or  change  of  variables 


Suppose  we  want  to  calculate  the 


i  f2  x 
mteqral  . — 

JoVT^ 


-dx  .  If  we  use  step-by- 


step  calculation  in  the  Equation  Writer,  this  is  the  sequence  of  variable 
substitutions: 


P9iiiinfrai?poifflniifiii(gniiii 

This  second  step  shows  the  proper  substitution  to  use,  u  =  x2-1 1 . 


Int[u'*F(u>]  with  u= 

Square  root 
Fx1 


Rat  i  ona 1  f r act  i  on 

9 

Rat  i  ona 1  f r act  i  on 
2  1 


2-x 


2-x 


lint  Lu  '  *h  ^.l^  J  with  u= 

Square  root 
Fx1 

Rational  fraction 

9 


aaaBnaiagiEEaaiBEBini 


Page  13-18 


The  last  four  steps  show  the  progression  of  the  solution:  a  square  root, 
followed  by  a  fraction,  a  second  fraction,  and  the  final  result.  This  result  can 
be  simplified  by  using  function  illlilill,  to  read: 


Integration  by  parts  and  differentials 

A  differential  of  a  function  y  =  f(x),  is  defined  as  dy  =  f'(x)  dx,  where  f'(x)  is 
the  derivative  of  f(x).  Differentials  are  used  to  represent  small  increments  in 
the  variables.  The  differential  of  a  product  of  two  functions,  y  =  u(x)v(x),  is 
given  by  dy  =  u(x)dv(x)  +du(x)v(x),  or,  simply,  d(uv)  =  udv  -  vdu.  Thus,  the 

integral  of  udv  =  d(uv)  -  vdu,  is  written  as  udv  =  jd(uv)  —  jvdu  .  Since  by 
the  definition  of  a  differential,  Jdy  =  y,  we  write  the  previous  expression  as 


judv  =  uv  -  jvdu  . 


This  formulation,  known  as  integration  by  parts,  can  be  used  to  find  an 
integral  if  dv  is  easily  integrable.  For  example,  the  integral  jxexdx  can  be 
solved  by  integration  by  parts  if  we  use  u  =  x,  dv  =  exdx,  since,  v  =  ex.  With 
du  =  dx,  the  integral  becomes  jxexdx  =  judv  =  uv  -  jvdu  =  xex  -  jexdx  =  xex  -  ex. 

The  calculator  provides  function  IBP,  under  the  CALC/DERIV&INTG  menu,  that 
takes  as  arguments  the  original  function  to  integrate,  namely,  u(X)*v'(X),  and 
the  function  v(X),  and  returns  u(X)*v(X)  and  -v(X)*u'(X).  In  other  words, 
function  IBP  returns  the  two  terms  of  the  right-hand  side  in  the  integration  by 
parts  equation.  For  the  example  used  above,  we  can  write  in  ALG  mode: 


-e*} 

IE: F"  |inTYK|  LflPL  |PRE' 

.'hIRIl'lH 

Page  13-19 


Thus,  we  can  use  function  IBP  to  provide  the  components  of  an  integration  by 
parts.  The  next  step  will  have  to  be  carried  out  separately. 


It  is  important  to  mention  that  the  integral  can  be  calculated  directly  by  using, 
for  example, 


IHTVXl 


X 


Integration  by  partial  fractions 

Function  PARTFRAC,  presented  in  Chapter  5,  provides  the  decomposition  of  a 
fraction  into  partial  fractions.  This  technique  is  useful  to  reduce  a  complicated 
fraction  into  a  sum  of  simple  fractions  that  can  then  be  integrated  term  by 
term.  For  example,  to  integrate 


X5  +5 


X4  +2X3  +X 


-dX 


we  can  decompose  the  fraction  into  its  partial  component  fractions,  as  follows: 


PRRTFRRC 


X5+5 


„  2,  5    IS,     4  ,^3 


: PARTFRAC 


H  3  2 
JT+a-JT+JT 


K  CH+15 


13 
K+i 


:inTYK(AnStitt 

^■Ka-a.K+-|-iD.Ln(K5+-^+i3.Ln(K^ 


■nariit^i^iiii^iffmiiaiMi  bnariit^i^iiiifl^niffmiiaaiiii 


The  direct  integration  produces  the  same  result,  with  some  switching  of  the 
terms  (Rigorous  mode  set  in i  the  CAS  -  see  Chapter  2): 


inTYJUAnsatt 

|.Ka-a.K+-|-iD.LnaKi)+-^+i3«Lrp:i^ 


H  3  2 
JT+a-JT+JT 


4-Ka-a.K+i3.LnaK+ii)+ 


-tio-LntiKm-^ 


BitiiiHitiiEC'HII'HO'HlliijaJ 


Page  1  3-20 


Improper  integrals 

These  are  integrals  with  infinite  limits  of  integration.  Typically,  an  improper 
integral  is  dealt  with  by  first  calculating  the  integral  as  a  limit  to  infinity,  e.g., 

r^dx     t.     ^  dx 
—  =  lim    — . 

Using  the  calculator,  we  proceed  as  follows: 


J  i 


7« 


iz1 


 1 


F 


1 


limCnHSC133 


e-1 


Alternatively,  you  can  evaluate  the  integral  to  infinity  from  the  start,  e.g., 


-^dx 

x 


1 


Integration  with  units 

An  integral  can  be  calculated  with  units  incorporated  into  the  limits  of 
integration,  as  in  the  example  shown  below  that  uses  ALG  mode,  with  the 
CAS  set  to  Approx  mode.  The  left-hand  side  figure  shows  the  integral  typed 
in  the  line  editor  before  pressing  [enter]  .    The  right-hand  figure  shows  the  result 


x  dx 

. 333333333333_mm 

Page  13-21 


If  you  enter  the  integral  with  the  CAS  set  to  Exact  mode,  you  will  be  asked  to 
change  to  Approx  mode,  however,  the  limits  of  the  integral  will  be  shown  in  a 
different  format  as  shown  here: 


-mm 

2  . 

x  dx 

0-1 

-mm 

. 333333333333_nn3 

These  limits  represent  1x1  _mm  and  Oxl_mm,  which  is  the  same  as  l_mm  and 
0_mm,  as  before.  Just  be  aware  of  the  different  formats  in  the  output. 

Some  notes  in  the  use  of  units  in  the  limits  of  integrations: 
1  -  The  units  of  the  lower  limit  of  integration  will  be  the  ones  used  in  the  final 
result,  as  illustrated  in  the  two  examples  below: 


l_min 


J  1_; 


t2"dt 


71999.  6666667_s"" 


2  -  Upper  limit  units  must  be  consistent  with  lower  limit  units.  Otherwise,  the 
calculator  simply  returns  the  unevaluated  integral.  For  example, 


3  -  The  integrand  may  have  units  too    For  example: 


l_yr 


l_ni 


r  dr 


f1, 

le. 


X'l_sdx 


.5_s 


4  -  If  both  the  limits  of  integration  and  the  integrand  have  units,  the  resulting 
units  are  combined  according  to  the  rules  of  integration.  For  example, 


Page  1  3-22 


fl0_S 


J  1-9 


s 


£.33333333333_(93-s2. 


J0_S 

 35U-CTM 

■■■TlBiliMMDdlilllB 


Infinite  series 


00 


An  infinite  series  has  the  form  ^/z(/?)(x  —  d)n  .  The  infinite  series  typically 

77=0,1 

starts  with  indices  n  =  0  or  n  =  1 .    Each  term  in  the  series  has  a  coefficient 
h(n)  that  depends  on  the  index  n. 

Taylor  and  Maclaurin's  series 

A  function  f(x)  can  be  expanded  into  an  infinite  series  around  a  point  x=x0  by 
using  a  Taylor's  series,  namely, 


where  f(n)(x)  represents  the  n-th  derivative  of  f(x)  with  respect  to  x,  f(0)(x)  =  f(x). 
If  the  value  x0  is  zero,  the  series  is  referred  to  as  a  Maclaurin's  series,  i.e., 


Taylor  polynomial  and  reminder 

In  practice,  we  cannot  evaluate  all  terms  in  an  infinite  series,  instead,  we 
approximate  the  series  by  a  polynomial  of  order  k,  Pk(x),  and  estimate  the 
order  of  a  residual,  Rk(x),  such  that 


Page  1  3-23 


i.e., 


f(x)  =  Pk(x)  +  Rk(x). 


The  polynomial  Pk(x)  is  referred  to  as  Taylor's  polynomial.    The  order  of  the 
residual  is  estimated  in  terms  of  a  small  quantity  h  =  x-x0/  i.e.,  evaluating  the 
polynomial  at  a  value  of  x  very  close  to  x0.  The  residual  if  given  by 


Rk(x)  = 


hK 


where  \  is  a  number  near  x  =  x0.  Since  \  is  typically  unknown,  instead  of  an 
estimate  of  the  residual,  we  provide  an  estimate  of  the  order  of  the  residual  in 
reference  to  h,  i.e.,  we  say  that  Rk(x)  has  an  error  of  order  hn+1,  or  R  «  0(hk+1). 
If  h  is  a  small  number,  say,  h«l,  then  hk+1  will  be  typically  very  small,  i.e., 


« 


hk«  ...«  h  «  1 .  Thus,  for  x  close  to  x0,  the  larger  the  number  of 


elements  in  the  Taylor  polynomial,  the  smaller  the  order  of  the  residual. 

Functions  TAYLR,  TAYLRO,  and  SERIES 

Functions  TAYLR,  TAYLRO,  and  SERIES  are  used  to  generate  Taylor 
polynomials,  as  well  as  Taylor  series  with  residuals.   These  functions  are 
available  in  the  CALC/LIMITS&SERIES  menu  described  earlier  in  this  Chapter. 

Function  TAYLORO  performs  a  Maclaurin  series  expansion,  i.e.,  about  X  =  0, 
of  an  expression  in  the  default  independent  variable,  VX  (typically  X).  The 
expansion  uses  a  4-th  order  relative  power,  i.e.,  the  difference  between  the 
highest  and  lowest  power  in  the  expansion  is  4.  For  example, 


:  TflYLORote^l 

:TflYLOR0£SIH(XH 

DIYPCI  Uh  |SERIE|TrtYLO|TAYLR|  CflLC 

Page  1  3-24 


Function  TAYLR  produces  a  Taylor  series  expansion  of  a  function  of  any 
variable  x  about  a  point  x  =  a  for  the  order  k  specified  by  the  user.  Thus,  the 
function  has  the  format  TAYLR(f(x-a),x,k).  For  example, 


:TflYLR(siN(s-^],s,6] 

1    ,t5+   1   lt4+  1  lt3++ 

DIYPCI  l.iH  |SERIE|TflVLO|TflVLR|  CflLC 

DIVPC|  Uh  |SERIE|THVLO|TflVLR|  CflLC 

Function  SERIES  produces  a  Taylor  polynomial  using  as  arguments  the  function 
f(x)  to  be  expanded,  a  variable  name  alone  (for  Maclaurin's  series)  or  an 
expression  of  the  form  Variable  =  value'  indicating  the  point  of  expansion  of 
a  Taylor  series,  and  the  order  of  the  series  to  be  produced.    Function  SERIES 
returns  two  output  items  a  list  with  four  items,  and  an  expression  for  h  =  x  -  a, 
if  the  second  argument  in  the  function  call  is  'x=a',  i.e.,  an  expression  for  the 
increment  h.  The  list  returned  as  the  first  output  object  includes  the  following 
items: 

1  -  Bi-directional  limit  of  the  function  at  point  of  expansion,  i.e.,  lim  f(x) 

2  -  An  equivalent  value  of  the  function  near  x  =  a 

3  -  Expression  for  the  Taylor  polynomial 

4  -  Order  of  the  residual  or  remainder 

Because  of  the  relatively  large  amount  of  output,  this  function  is  easier  to 
handle  in  RPN  mode.    For  example: 


jLimit :  1  Equiu:  1  Exp* 
1=  h=X-f 

Drop  the  contents  of  stack  level  1  by  pressing  j  +  J,  and  then  enter  Leval J f  to 
decompose  the  list.  The  results  are  as  follows: 


Page  1  3-25 


4: 
3: 


l: 


Expans! 


-1 


Limit: l 
EquiuM 

Remain:  (h^ 


afflaMHaHMHffiHBnaBaBnH 


4Expans:  '-W20*hA6+ 
1^24*hA4+-1^2*hA2+l ' 


In  the  right-hand  side  figure  above,  we  are  using  the  line  editor  to  see  the 
series  expansion  in  detail. 


Page  1  3-26 


Chapter  14 

Multi-variate  Calculus  Applications 

Multi-variate  calculus  refers  to  functions  of  two  or  more  variables.  In  this 
Chapter  we  discuss  the  basic  concepts  of  multi-variate  calculus  including 
partial  derivatives  and  multiple  integrals. 


Multi-variate  functions 

A  function  of  two  or  more  variables  can  be  defined  in  the  calculator  by  using 
the  DEFINE  function  (LjnJ^L_  ).  To  illustrate  the  concept  of  partial  derivative, 
we  will  define  a  couple  of  multi-variate  functions,  f(x,y)  =  x  cos(y),  and  g(x,y,z) 
=  (x2+y2)1/2sin(z),  as  follows: 


DEFIHEC,fCx,y)=x-C0SCy),3 
HOVflL 

DEFIHe(  9(x,y,z)=Jx2+y2-£ 
HOVRL 

aiHJHiHCE'HII'HO-HllEia 


We  can  evaluate  the  functions  as  we  would  evaluate  any  other  calculator 
function,  e.g., 


2'C0S(3) 


J5.f 


Graphics  of  two-dimensional  functions  are  possible  using  Fast3D,  Wireframe, 
Ps-Contour,  Y-Slice,  Gridmap,  and  Pr-Surface  plots  as  described  in  Chapter 
12. 


Partial  derivatives 

Consider  the  function  of  two  variables  z  =  f(x,y),  the  partial  derivative  of  the 
function  with  respect  to  x  is  defined  by  the  limit 


Page  14-1 


Similarly, 


¥_  =  limf(x  +  h,y)-f(x,y) 
dx    *-*>  h 

£-lim 

3y  *-><> 


f{x,y  +  k)-f{x,y) 


We  will  use  the  multi-variate  functions  defined  earlier  to  calculate  partial 
derivatives  using  these  definitions.  Here  are  the  derivatives  of  f(x,y)  with 
respect  to  x  and  y,  respectively: 


=  lim[^x+h^-f(x^ 

h+0l  h 

,   COSCy] 

^■■■I.liHiiiriiiiffiiiiHJIImu 


I  UUSh^ 

ilin|f'^+k'-fh|tf' 
k+01  k 

-Cx-SIHCy)] 
■■■■ff.hHiiiriiiiffiiiiHJIImi 


Notice  that  the  definition  of  partial  derivative  with  respect  to  x,  for  example, 
requires  that  we  keep  y  fixed  while  taking  the  limit  as  h->0.  This  suggest  a 
way  to  quickly  calculate  partial  derivatives  of  multi-variate  functions:  use  the 
rules  of  ordinary  derivatives  with  respect  to  the  variable  of  interest,  while 
considering  all  other  variables  as  constant.  Thus,  for  example, 

■|-  (x  cosO))  =  cos( y\  -|-  (x  cos(y))  =  -x  sin(y) , 

ox  oy 

which  are  the  same  results  as  found  with  the  limits  calculated  earlier. 
Consider  another  example, 


dx 


(yx2  +  y2)=  2yx  +  0  =  2xy 


In  this  calculation  we  treat  y  as  a  constant  and  take  derivatives  of  the 
expression  with  respect  to  x. 

Similarly,  you  can  use  the  derivative  functions  in  the  calculator,  e.g.,  DERVX, 
DERIV,  d  (described  in  detail  in  Chapter  1  3)  to  calculate  partial  derivatives. 
Recall  that  function  DERVX  uses  the  CAS  default  variable  VX  (typically,  X), 


Page  14-2 


therefore,  with  DERVX  you  can  only  calculate  derivatives  with  respect  to  X. 
Some  examples  of  first-order  partial  derivatives  are  shown  next: 


COS(y) 


 x— SINty]    I   k^x  +y_  


x-SIN(y) 


^(9(x,y,z)) 


2±L 


■SIHCz) 


:  DERVx(x-Y2-V2) 


DERVXCX-SIHCY+XH 

COS(X+Y)-X+S I HCX+ YJ 


DERIvts-t^-e^t) 


s-2-t-e 


■I'liiiigiitwi'Jiiiiiii'iwiM'iauii^  ■I'liiii'Jiitwi'Jiiiiiii'iMiM'iauii^ 


Higher-order  derivatives 

The  following  second-order  derivatives  can  be  defined 


d^£_d_(df^  d2f  _  d 
dx 


dx 


d2f 
dydx 


ydxj 


dy\dx  j 


dy1  dy 


d2f 


'  dxdy 


The  last  two  expressions  represent  cross-derivatives,  the  partial  derivatives 
signs  in  the  denominator  shows  the  order  of  derivation.  In  the  left-hand  side, 
the  derivation  is  taking  first  with  respect  to  x  and  then  with  respect  to  y,  and  in 
the  right-hand  side,  the  opposite  is  true.    It  is  important  to  indicate  that,  if  a 
function  is  continuous  and  differentiable,  then 


d2f  =  d2f 
dydx  dxdy 


Page  1 4-3 


Third-,  fourth-,  and  higher  order  derivatives  are  defined  in  a  similar  manner. 


To  calculate  higher  order  derivatives  in  the  calculator,  simply  repeat  the 
derivative  function  as  many  times  as  needed.  Some  examples  are  shown 
below: 

— 7?  '_■ '_'  ■_' '.  M  J 


0 


,   x-COSCy) 


-SIHCy) 


-SIHCy) 

^■iawi.]^ai-ji.]^i-iHB.n-jiMli:^i:i«d 


The  chain  rule  for  partial  derivatives 

Consider  the  function  z  =  f(x,y),  such  that  x  =  x(t),  y  =  y(t).  The  function  z 
actually  represents  a  composite  function  of  t  if  we  write  it  as  z  =  f[x(t),y(t)]. 
The  chain  rule  for  the  derivative  dz/dt  for  this  case  is  written  as 


dz  _  dz  dx  dz  dy 
dv     dx  dv    dy  dv 


To  see  the  expression  that  the  calculator  produces  for  this  version  of  the  chain 
rule  use: 


:^-(z(x(Uya)H 
|dly(Ud2z(x(Uy(tH+dlx(tH 


The  result  is  given  by  d  1  y(t)-d2z(x(t),y(t))+d  1  x(t)-d  1  z(x(y),y(t)).  The  term  d  1  y(t) 
is  to  be  interpreted  as  "the  derivative  of  y(t)  with  respect  to  the  1st  independent 
variable,  i.e.,  t",  or  dly(t)  =  dy/dt.  Similarly,  dlx(t)  =  dx/dt.  On  the  other 
hand,  dl  z(x(t),y(t))  means  "the  first  derivative  of  z(x,y)  with  respect  to  the  first 
independent  variable,  i.e.,  x",  or  d  1  z(x(t),y(t))  =  dz/dx.  Similarly, 
d2z(x(t),y(t))  =  dz/dy.  Thus,  the  expression  above  is  to  be  interpreted  as: 

dz/dt  =  (dy/dt).(dz/dy)  +  (dx/dt)-  (dz/dx). 


Page  1 4-4 


Total  differential  of  a  function  z  =  z(x,y) 

From  the  last  equation,  if  we  multiply  by  dt,  we  get  the  total  differential  of  the 
function  z  =  z(x,y),  i.e.,  dz  =  (dz/dx)dx  +  (dz/dy)dy. 


A  different  version  of  the  chain  rule  applies  to  the  case  in  which  z  =  f(x,y),  x 
=  x(u,v),  y  =  y(u,v),  so  that  z  =  f[x(u,v),  y(u,v)].  The  following  formulas 
represent  the  chain  rule  for  this  situation: 

dz  _  dz  dx  dz  dy  dz  _  dz  dx  dz  dy 
du     dx  du    dy  du '       dv     dx  dv    dy  dv 


Determining  extrema  in  functions  of  two  variables 

In  order  for  the  function  z  =  f(x,y)  to  have  an  extreme  point  (extrema)  at  (x0,y0), 
its  derivatives  d\/dx  and  d\/dy  must  vanish  at  that  point.  These  are  necessary 
conditions.  The  sufficient  conditions  for  the  function  to  have  an  extreme  at 
point  (x0,y0)  are  d\/dx  =  0,  Sf/5y  =  0,  and  A  =  {d2\/dx2)-  [d2\/dy2)-[d2\/dxdy]2 
>  0.  The  point  (x0,y0)  is  a  relative  maximum  if  d2{/dx2<  0,  or  a  relative 
minimum  if  &\/dx2>  0.  The  value  A  is  referred  to  as  the  discriminant. 

If  A  =  (d2{/dx2)-  [d2\/  dy2)-[d2\/  dxdy]2  <  0,  we  have  a  condition  known  as  a 
saddle  point,  where  the  function  would  attain  a  maximum  in  x  if  we  were  to 
hold  y  constant,  while,  at  the  same  time,  attaining  a  minimum  if  we  were  to 
hold  x  constant,  or  vice  versa. 


Example  1  -  Determine  the  extreme  points  (if  any)  of  the  function  f(X,Y)  =  X3- 
3X-Y2+5.  First,  we  define  the  function  f(X,Y),  and  its  derivatives  fX(X,Y)  = 
dt/dX,  fY(X,Y)  =  ai/fff.  Then,  we  solve  the  equations  fX(X,Y)  =  0  and  fY(X,Y) 
=  0,  simultaneously: 


DEFinE(,FtK,VJ=K3-3.K-Va+El) 


HOVflL 


3>X  -3 


■     u      i  ^^^  i  ■■  =  

:^(F(K,Y"*FY 

-O.Y3 

:S0LYE(CFK  FYDj'EK  YTJ 

 ■[□■!=!  v= li 3  □■:= 

-i  "i" = Li □ : 

Page  1 4-5 


We  find  critical  points  at  (X,Y)  =  (1,0),  and  (X,Y)  =  (-1,0).  To  calculate  the 
discriminant,  we  proceed  to  calculate  the  second  derivatives,  fXX(X,Y)  = 
d2\/dX2f  fXY(X,Y)  =  d2{/dX/DY,  and  fYY(X,Y)  =  d2{/dY2. 


■  J-ULYLU-T  A  r  1  Ji'Li'i  IJ'J 

£CK=i  v=li: 

CK=-i  v=li:] 

^(RCLCFK'UfFKK 

:^(RCL('FY'UfFYY 

-2 

'. r. '_ l '. ■  r  i  "j.1  ft  i  j- 


^(RCLCFK'UfFKY 


:  FKKiFYV-FKV  fD 


0 


The  last  result  indicates  that  the  discriminant  is  A  =  -1  2X,  thus,  for  (X,Y)  =  (1,0), 
A  <0  (saddle  point),  and  for  (X,Y)  =  (-1,0),  A>0  and  d2f/dX2<0  (relative 
maximum).  The  figure  below,  produced  in  the  calculator,  and  edited  in  the 
computer,  illustrates  the  existence  of  these  two  points: 


Using  function  HESS  to  analyze  extrema 

Function  HESS  can  be  used  to  analyze  extrema  of  a  function  of  two  variables 
as  shown  next.  Function  HESS,  in  general,  takes  as  input  a  function  of  n 
independent  variables  §(x]f  x2,  ...,xn),  and  a  vector  of  the  functions  ['x/ 
'x2'...'xn'].  Function  HESS  returns  the  Hessian  matrix  of  the  function  defined 
as  the  matrix  H  =  [hjj]  =  [d^/dx-fix-^,  the  gradient  of  the  function  with  respect 
to  the  n-variables,  grad  f  =  [  dty/dxu  d$/dx2  ,  ...  d$/dxn],  and  the  list  of 
variables  ['x/V.-.V]. 


Page  1 4-6 


Applications  of  function  HESS  are  easier  to  visualize  in  the  RPN  mode. 
Consider  as  an  example  the  function  (])(X,Y,Z)  =  X2  +  XY  +  XZ,  we'll  apply 
function  HESS  to  function  §  in  the  following  example.  The  screen  shots  show 
the  RPN  stack  before  and  after  applying 


X  +X-Y+X-Z 
[X  Y  Z] 


unction  HESS. 

4: 

3: 

12  1  11 

1  0  0 

Li  e  @J 

2: 

[2-X+Y+Z  X  X] 

l: 

[X  Y  Z] 

When  applied  to  a  function  of  two  variables,  the  gradient  in  level  2,  when 
made  equal  to  zero,  represents  the  equations  for  critical  points,  i.e.,  d^/dx^  - 
0,  while  the  matrix  in  level  3  represent  second  derivatives.  Thus,  the  results 
from  the  HESS  function  can  be  used  to  analyze  extrema  in  functions  of  two 
variables.  For  example,  for  the  function  f(X,Y)  =  X3-3X-Y2+5,  proceed  as 
follows  in  RPN  mode: 


'XA3-3*X-YA2+5'  (enter)  ['X'/Y']  (enter) 

HESS 

SOLVE 

[EVAL) 

'si'  (2°t)  's2'  (W) 


Enter  function  and  variables 
Apply  function  HESS 
Find  critical  points 
Decompose  vector 
Store  critical  points 


The  variables  si  and  s2,  at  this  point,  contain  the  vectors  ['X=-l '/Y=0]  and 
['X=l  ','Y=0],  respectively.  The  Hessian  matrix  is  at  level  1  at  this  point. 


Store  Hessian  matrix 
Substitute  si  into  H 


The  resulting  matrix  A  has  an  elements  an  =  d2$/dX2  =  -6.,  a22=  d2<\>/dX2 
2.,  and  a12  =  021  =  d2^/dXSY  =  0.  The  discriminant,  for  this  critical  point 
si  (-1 ,0)  is  A  =  (d2{/dx2)-  (d2{/dy2)-[d2{/dxdy]2  =  (-6.)(-2.)  =  1  2.0  >  0.  Since 
d2$/dX2  <0,  point  si  represents  a  relative  maximum. 

Next,  we  substitute  the  second  point,  s2,  into  H: 

C^D  II  II  SUBST  CB^f  Substitute  s2  into  H 


Page  1 4-7 


The  resulting  matrix  has  elements  an  =  d2$/dX2  =  6.,  a22  =  d2$/dX2  =  -2., 
anda12=a21  =  82§/dXdY  =  0.  The  discriminant,  for  this  critical  point  s2(l  ,0) 
is  A  =  (d2{/dx2y  (dVdy2)-[d2t/dxdy]2  =  (6.)(-2.)  =  -1  2.0  <  0,  indicating  a 
saddle  point. 

Multiple  integrals 

A  physical  interpretation  of  an  ordinary  integral,     f(x)dx,  is  the  area 

Ja 

under  the  curve  y  =  f(x)  and  abscissas  x  =  a  and  x  =  b.    The  generalization 
to  three  dimensions  of  an  ordinary  integral  is  a  double  integral  of  a  function 
f(x,y)  over  a  region  R  on  the  x-y  plane  representing  the  volume  of  the  solid 
body  contained  under  the  surface  f(x,y)  above  the  region  R.    The  region  R 
can  be  described  as  R  =  {a<x<b,  f(x)<y<g(x)}  or  as  R  =  {c<y<d,  r(y)<x<s(y)}. 
Thus,  the  double  integral  can  be  written  as 

J*b  rg(x)  rd  rs(y) 

</>(x,y)dydx  =  0(x9y)dydx 
a  J f(x)  Jc  Jr(y) 

R 


Calculating  a  double  integral  in  the  calculator  is  straightforward.  A  double 
integral  can  be  built  in  the  Equation  Writer  (see  example  in  Chapter  2).  An 
example  follows.  This  double  integral  is  calculated  directly  in  the  Equation 
Writer  by  selecting  the  entire  expression  and  using  function  HEH.  The  result 
is  3/2.    Step-by-step  output  is  possible  by  setting  the  Step/Step  option  in  the 
CAS  MODES  screen. 


2 

^x+ydy  dx4 

1 

EDIT  |  CURS 

I  GIG  -|  EVflL  |FACT0| SIHP 

Page  1 4-8 


3-x  2-x+l 

2  2 
Rat  i  ona 1  f  ract  i  on 


EDIT  |  CURS  |  GIG  -|  EVflL  |FflCT0|  SIHP 

3-x~  2-x+ 

A. 
± 

2  2 

Rat  ional 

f ract  ion 

2 

X 

Rat  ional 

f ract  ion 

X 

Jacobian  of  coordinate  transformation 

Consider  the  coordinate  transformation  x  =  x(u,v),  y  =  y(u,v).  The  Jacobian  of 
this  transformation  is  defined  as 


J  \=  det(J)  =  det 


( 'dx 

du 

dv 

dy 

dy 

dv  j 

When  calculating  an  integral  using  such  transformation,  the  expression  to  use 
is  ^(/){x,y)dydx  =  ||^[x(i/,v),^(w,v)]  |  J  \  dudv ,  where  R'  is  the  region  R 

R  R' 

expressed  in  (u,v)  coordinates. 

Double  integral  in  polar  coordinates 

To  transform  from  polar  to  Cartesian  coordinates  we  use  x(r,0)  =  r  cos  0,  and 
y(r,  0)  =  r  sin  0.  Thus,  the  Jacobian  of  the  transformation  is 

dx  dx 

'~  dy    dy_  ~  sin(/9)     r  •  cos(6>) 
dr  dO 

With  this  result,  integrals  in  polar  coordinates  are  written  as 


=  r 


Page  14-9 


\\<f>{r,6)dA  =  [  Q<t>{r,0)rdrd6 


where  the  region  R'  in  polar  coordinates  is  R'  =  {a  <  0  <  p,  f(9)  <  r  <  g(0)}. 


Double  integrals  in  polar  coordinates  can  be  entered  in  the  calculator,  making 
sure  that  the  Jacobian  |  J  |  =  r  is  included  in  the  integrand.  The  following  is  an 
example  of  a  double  integral  calculated  in  polar  coordinates,  shown  step-by- 
step: 


r 

4  IT 

■ 

I 

2 

fQ  T  KJf"fl  "1 

1 

■ 

O  1  .J 

I 

Je 

m 

8 

■ 

9-r 

Rational  fraction 

r 


Linearizing 


■ 

2 

TT   +2  -1 

3216 

H 

Page  14-10 


Chapter  1 5 

Vector  Analysis  Applications 

In  this  Chapter  we  present  a  number  of  functions  from  the  CALC  menu  that 
apply  to  the  analysis  of  scalar  and  vector  fields.   The  CALC  menu  was 
presented  in  detail  in  Chapter  13.  In  particular,  in  the  DERIV&INTEG  menu 
we  identified  a  number  of  functions  that  have  applications  in  vector  analysis, 
namely,  CURL,  DIV,  HESS,  LAPL.  For  the  exercises  in  this  Chapter,  change 
your  angle  measure  to  radians. 


Definitions 

A  function  defined  in  a  region  of  space  such  as  (|)(x,y,z)  is  known  as  a  scalar 
field,  examples  are  temperature,  density,  and  voltage  near  a  charge.  If  the 
function  is  defined  by  a  vector,  i.e.,  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,  it 
is  referred  to  as  a  vector  field. 


The  following  operator,  referred  to  as  the  'del'  or  'nabla'  operator,  is  a  vector- 
based  operator  that  can  be  applied  to  a  scalar  or  vector  function: 

v[  ]-!•!-[  ]+*—[] 

ox  oy  oz 

When  this  operator  is  applied  to  a  scalar  function  we  can  obtain  the  gradient 
of  the  function,  and  when  applied  to  a  vector  function  we  can  obtain  the 
divergence  and  the  curl  of  that  function.  A  combination  of  gradient  and 
divergence  produces  another  operator,  called  the  Laplacian  of  a  scalar 
function.  These  operations  are  presented  next. 

Gradient  and  directional  derivative 

The  gradient  of  a  scalar  function  (|)(x,y,z)  is  a  vector  function  defined  by 

grade/)  =  V(f)  =  i  •  —  +  j-  —  +  k-  — 

dx        dy  dz 

The  dot  product  of  the  gradient  of  a  function  with  a  given  unit  vector 
represents  the  rate  of  change  of  the  function  along  that  particular  vector.  This 
rate  of  change  is  called  the  directional  derivative  of  the  function,  Du(|)(x,y,z)  = 
u*Vf 


Page  15-1 


At  any  particular  point,  the  maximum  rate  of  change  of  the  function  occurs  in 
the  direction  of  the  gradient,  i.e.,  along  a  unit  vector  u  =  V(])/ 1  V(|)  | . 

The  value  of  that  directional  derivative  is  equal  to  the  magnitude  of  the 
gradient  at  any  point  Dmax(|)(x,y,z)  =  V<\>  •V^/ 1  V(|)  |  =  \ 

The  equation  (|)(x,y,z)  =  0  represents  a  surface  in  space.  It  turns  out  that  the 
gradient  of  the  function  at  any  point  on  this  surface  is  normal  to  the  surface. 
Thus,  the  equation  of  a  plane  tangent  to  the  curve  at  that  point  can  be  found 
by  using  a  technique  presented  in  Chapter  9. 

The  simplest  way  to  obtain  the  gradient  is  by  using  function  DERIV,  available 
in  the  CALC  menu,  e.g.,  


A  program  to  calculate  the  gradient 

The  following  program,  which  you  can  store  into  variable  GRADIENT,  uses 
function  DERIV  to  calculate  the  gradient  of  a  scalar  function  of  X,Y,Z. 
Calculations  for  other  base  variables  will  not  work.  If  you  work  frequently  in 
the  (X,Y,Z)  system,  however,  this  function  will  facilitate  calculations: 

«  X  Y  Z  3  ^ARRY  DERIV  » 

Type  the  program  while  in  RPN  mode.  After  switching  to  ALG  mode,  you  can 
call  the  function  GRADIENT  as  in  the  following  example: 


GRflD I EHT ( X ^2+ Y A2+Z *2 
[2*X,2*Y?2*Z] 


Using  function  HESS  to  obtain  the  gradient 

The  function  HESS  can  be  used  to  obtain  the  gradient  of  a  function  as  shown 
next.  As  indicated  in  Chapter  14,  function  HESS  takes  as  input  a  function  of 


Page  1 5-2 


n  independent  variables  §(x]f  x2,  ...,xn)/  and  a  vector  of  the  functions  ['x/ 
/x2/---/xn/]-  Function  HESS  returns  the  Hessian  matrix  of  the  function  defined 
as  the  matrix  H  =  [hjj]  =  [d^/dXjdxj],  the  gradient  of  the  function  with  respect  to 
the  n-variables,  grad  f  =  [  d§/dx},  d$/dx2  ,  ...  d<\>/dxn],  and  the  list  of 
variables  ['x/  'x2'...'xn'].  Consider  as  an  example  the  function  §(X,Y,Z)  =  X2 
+  XY  +  XZ,  well  apply  function  HESS  to  this  scalar  field  in  the  following 


6: 

5: 

4: 

3: 

2: 

l: 

CX  Y  Z] 

3  = 

4: 

3: 

[2  1  11 

1  0  0 

Li  e  @J 

2: 

[2-X+Y+Z  X  X] 

l: 

[X  Y  Z] 

Thus,  the  gradient  is  [2X+Y+Z,  X,  X].  Alternatively,  one  can  use  function 
DERIV  as  follows:  DERIV(XA2+X*Y+X*Z,[X,Y,Z]),  to  obtain  the  same  result. 


Potential  of  a  gradient 

Given  the  vector  field,  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,  if  there  exists  a 
function  (|)(x,y,z),  such  that  f  =  d§/dx,  g  =  d§/dy,  and  h  =  d§/dz,  then  (|)(x,y,z) 
is  referred  to  as  the  potential  function  for  the  vector  field  F.  It  follows  that  F  = 
grad  §  =  V(|). 

The  calculator  provides  function  POTENTIAL,  available  through  the  command 
catalog  (L J  cat  ),  to  calculate  the  potential  function  of  a  vector  field,  if  it 
exists.  For  example,  if  F(x,y,z)  =  xi  +  yj  +  zk,  applying  function  POTENTIAL 
we  find: 

:  POTENT  I  RLCCx  y  zlCx  y  z> 
2        2  2 


Since  function  SQ(x)  represents  x2,  this  results  indicates  that  the  potential 
function  for  the  vector  field  F(x,y,z)  =  xi  +  yj  +  zk,  is  (|)(x,y,z)  =  (x2+y2+z2)/2. 

Notice  that  the  conditions  for  the  existence  of  (|)(x,y,z),  namely,  f  =  d§/dx,  g  = 
d^/dy,  and  h  =  d^/dz,  are  equivalent  to  the  conditions:  df/dy  =  dg/dx,  di/dz 
=  d\\/dx,  and  dg/dz  =  5h/5y.  These  conditions  provide  a  quick  way  to 
determine  if  the  vector  field  has  an  associated  potential  function.  If  one  of  the 
conditions  di/dy  =  dg/dx,  d\/dz  =  dh/dx,  dg/dz  =  dh/dy,  fails,  a  potential 


Page  1 5-3 


function  <\>(x,y,z)  does  not  exist.  In  such  case,  function  POTENTIAL  returns  an 
error  message.    For  example,  the  vector  field  F(x,y,z)  =  (x+y)i  +  (x-y+z)j  + 
xzk,  does  not  have  a  potential  function  associated  with  it,  since,  df/dz  ^ 
dh/dx.  The  calculator  response  in  this  case  is  shown  below: 


:  POTENT  I ALCCX+Y  X-Y+Z  XH 
50TEHTIflL<CX+Y,X-Y+Z, 
<*Z1, [X,Y,Z]> 


PC 


A  POTENTIAL 
Error: 

Bad  Argument 
Value 


I  X> 
7  X> 


"Bad  Argument  Value" 


Divergence 

The  divergence  of  a  vector  function,  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j+h(x,y,z)k, 
is  defined  by  taking  a  "dot-product"  of  the  del  operator  with  the  function,  i.e., 

divF  =  V.F=V  +  *  +  ® 
dx    dy  dz 

Function  DIV  can  be  used  to  calculate  the  divergence  of  a  vector  field.  For 
example,  for  F(X,Y,Z)  =  [XY,X2+Y2+Z2,YZ],  the  divergence  is  calculated,  in 
ALG  mode,  as  follows:   


Div([x-Y  X2+Y2+Z2  Y-z],[tf 
Y+2-Y+Y 


Laplacian 

The  divergence  of  the  gradient  of  a  scalar  function  produces  an  operator 
called  the  Laplacian  operator.  Thus,  the  Laplacian  of  a  scalar  function  (|)(x,y,z) 
is  given  by 

dx      dx  dx 

The  partial  differential  equation  V2([)  =  0  is  known  as  Laplace's  equation. 
Function  LAPL  can  be  used  to  calculate  the  Laplacian  of  a  scalar  function.  For 
example,  to  calculate  the  Laplacian  of  the  function  (|)(X,Y,Z)  =  (X2+Y2)cos(Z), 
use: 


Page  1  5-4 


LflPL((X^2+Y^2>*C0St:Z 

2*C0SCZ>  +  t:2*C0S(Z>  +  (X^ 
2+V^2>*-C0S(Z>> 

Curl 

The  curl  of  a  vector  field  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,  is  defined  by 
a  "cross-product"  of  the  del  operator  with  the  vector  field,  i.e., 


curlF  =  V  x  F  = 


f[]  fu  f[] 

ox  qy  & 

/(x,  y,  z)    g(x,  y,  z)    h(x,  y,  z) 


=  1 


dh    8g)    fdf  dh 


dy  dz 


+  J 


dz  dx 


+  k 


dh_dg} 
dy  dz 


The  curl  of  vector  field  can  be  calculated  with  function  CURL.  For  example, 
for  the  function  F(X,Y,Z)  =  [XY,X2+Y2+Z2,YZ],  the  curl  is  calculated  as  follows: 


:  CURl([x-Y  X2+Y2+Z2  Y-z]/ 
[Z-2-Z  0  2-X-X] 


Irrotational  fields  and  potential  function 

In  an  earlier  section  in  this  chapter  we  introduced  function  POTENTIAL  to 
calculate  the  potential  function  §(x,y,z)  for  a  vector  field,  F(x,y,z)  =  f(x,y,z)i+ 
g(x,y,z)j+  h(x,y,z)k,  such  that  F  =  grad  ^  =  V(|).    We  also  indicated  that  the 
conditions  for  the  existence  of     were:  di/dy  =  dg/dx,  d\/dz  =  dh/dx,  and 
dg/dz  =  dh/dy.  These  conditions  are  equivalent  to  the  vector  expression 

curl  F  =  VxF  =  0. 


A  vector  field  F(x,y,z),  with  zero  curl,  is  known  as  an  irrotational  field.  Thus, 
we  conclude  that  a  potential  function  (|)(x,y,z)  always  exists  for  an  irrotational 
field  F(x,y,z). 


Page  1 5-5 


As  an  example,  in  an  earlier  example  we  attempted  to  find  a  potential 
function  for  the  vector  field  F(x,y,z)  =  (x+y)i  +  (x-y+z)j  +  xzk,  and  got  an  error 
message  back  from  function  POTENTIAL.  To  verify  that  this  is  a  rotational 
field  (i.e.,  VxF  ^  0),  we  use  function  CURL  on  this  field: 


CURLCCX+Y  X-Y+Z  X-Z],[X 
[-1  -Z  0] 


On  the  other  hand,  the  vector  field  F(x,y,z)  =  xi  +  yj  +  zk,  is  indeed 
irrotational  as  shown  below: 


:  CURLCCX+Y  X-Y+Z  X-Z],[X 
'i  CURLCCX  Y  Z],CX  Y  Z]) 

C0  0  0] 


Vector  potential 

Given  a  vector  field  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j+h(x,y,z)k,  if  there  exists  a 
vector  function  0(x,y,z)  =  ^(x^zji+ii/jx^zjj+r^x^zjk,  such  that  F  =  curl  O  = 
Vx  O,  then  function  0(x,y,z)  is  referred  to  as  the  vector  potential  of  F(x,y,z). 

The  calculator  provides  function  VPOTENTIAL,  available  through  the 
command  catalog  (LxU_^z;  ),  to  calculate  the  vector  potential,  0(x,y,z),  given 
the  vector  field,  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j+h(x,y,z)k.  For  example,  given 
the  vector  field,  F(x,y,z)  =  -(yi+zj+xk),  function  VPOTENTIAL  produces 


:VP0TEHTinLC-[y  ; 

[•-M-fe 


z  x],Cx  *=* 
y  +z-x 


i.e.,  0(x,y,z)  =  -x2/2j  +  (-y2/2+zx)k. 


It  should  be  indicated  that  there  is  more  than  one  possible  vector  potential 
functions  O  for  a  given  vector  field  F.  For  example,  the  following  screen  shot 
shows  that  the  curl  of  the  vector  function  ®,  =  [X2+Y2+Z2,XYZ,X+Y+Z]  is  the 
vector  F  =  Vx  02  =  [1 -XY,2Z-1  ,ZY-2Y].  Application  of  function  VPOTENTIAL 


Page  1 5-6 


produces  the  vector  potential  function  02  =  [0,  ZYX-2YX,  Y-(2ZX-X)],  which  is 
different  from        The  last  command  in  the  screen  shot  shows  that  indeed  F  = 
Vx  02.    Thus,  a  vector  potential  function  is  not  uniquely  determined. 


:  CURliLx  +Y  +Z   X-Y-Z  X+? 

El -X-Y  2-Z-l  Z-Y-2-Y] 
:VPOTENTIflL(flNSmiX  Y  Z> 
[0  Z-Y-X-2-Y-X  Y-C2-^X-X)] 
:CURL(flHS(lUX  Y  Z]) 

Hl-Y-X  Z-2-1  Y-Z-Y-2] 


The  components  of  the  given  vector  field,  F(x,y,z)  =  f(x,y,z)i+g(x,y,z)j 
+h(x,y,z)k,  and  those  of  the  vector  potential  function,  ®(x,y,z)  = 
(|)(x,y,z)i+\|/(x,y,z)i+r|(x,y,z)k,  are  related  by  f  =  dr\/dy  -  dy/dx,  g  =  dfy/dz  - 
dr\/dx,  and  h  =  d\\r/dx  -  d<\>/dy. 

A  condition  for  function  ®(x,y,z)  to  exists  is  that  div  F  =  V*F  =  0,  i.e.,  d\/dx  + 
dg/dy  +  di/dz  =  0.  Thus,  if  this  condition  is  not  satisfied,  the  vector  potential 
function  0(x,y,z)  does  not  exist.  For  example,  given  F  =  [X+Y,X-Y,ZA2], 
function  VPOTENTIAL  returns  an  error  message,  since  function  F  does  not 
satisfy  the  condition  V*F  =  0: 


21 


VPOTENTIfll_([x+Y  X-Y  Z' 
4P0TEHT I  FlL  <  I X+Y  ?  X-Y  ?  Z 
'21, [X?Y?Z]> 


\SWI?M  +GEL   DEL-*  DEL  L  ins 


A  ^POTENTIAL 
Error: 

Bad  Argument 
Value 


VF 
VF 

"Bad  Rrgu ment  Value 


z2i 

,21 


The  condition  V*F  ^  0  is  verified  in  the  following  screen  shot: 


:  DI  v(Lx+Y  X-Y  Z2],CX  Y  Z]), 
1+-1+2-Z 


Page  1  5-7 


Chapter  1 6 
Differential  Equations 

In  this  Chapter  we  present  examples  of  solving  ordinary  differential  equations 
(ODE)  using  calculator  functions.  A  differential  equation  is  an  equation 
involving  derivatives  of  the  independent  variable.  In  most  cases,  we  seek  the 
dependent  function  that  satisfies  the  differential  equation. 

Basic  operations  with  differential  equations 

In  this  section  we  present  some  uses  of  the  calculator  for  entering,  checking 
and  visualizing  the  solution  of  ODEs. 

Entering  differential  equations 

The  key  to  using  differential  equations  in  the  calculator  is  typing  in  the 
derivatives  in  the  equation.  The  easiest  way  to  enter  a  differential  equation  is 
to  type  it  in  the  equation  writer.  For  example,  to  type  the  following  ODE: 
(x-1  )-(dy(x)/dx)2  +  2xy(x)  =  ex  sin  x,  use: 

CB^w  ED  QD(S  CEDCD  CDCDCD(X)CB^  S™)ED(3 

CD^lQDiSQD^  SQDSCDCD  CZDCTD  CDCDmtXJCX} 

(^)C5D  (3  QD  (^QD  0  QD^SEDO  CDCDCDCD 

The  derivative  dy/dx  is  represented  by  9x(y(x))  or  by  dly(x).  For 
solution  or  calculation  purposes,  you  need  to  specify  y(x)  in  the  expression, 
i.e.,  the  dependent  variable  must  include  its  independent  variable(s)  in  any 
derivative  in  the  equation. 

You  can  also  type  an  equation  directly  into  the  stack  by  using  the  symbol  d  in 
the  derivatives.  For  example,  to  type  the  following  ODE  involving  second- 
order  derivatives:  d2u(x)/dx2  +  3-u(x)-(du(x)/dx)  +  u(x)2  =  1/x,  directly  into 
the  screen,  use: 

C3!L_  (^QD(3CDCDCDGJ(X)(X)e™)  QD^CTl^ 

(^CT(3CDCxJC3  3(^C3D(3GI3  (l  (^(20(3      ^  _ 

(^CSO  CD  CD  CBt^CSOCSD  y_  (^GDOCD  CZDCZD 

C3         =   CDS  (AL^[^3(X}(ENTER) 


Page  16-1 


The  result  is  'dx  (3x  (u  (x)  )  )  +3*u  (x)  *dx  (u  (x) )  +uA2=i/x  '.  This  format 
shows  up  in  the  screen  when  the  _Textbook  option  in  the  display  setting 
([mode) lllli!)  is  not  selected.  Press  <\j?  to  see  the  equation  in  the  Equation 
Writer. 

An  alternative  notation  for  derivatives  typed  directly  in  the  stack  is  to  use  'dl' 
for  the  derivative  with  respect  to  the  first  independent  variable,  '62'  for  the 
derivative  with  respect  to  the  second  independent  variable,  etc.  A  second- 
order  derivative,  e.g.,  d2x/dt2,  where  x  =  x(t),  would  be  written  as  'd  1  d  1  x(t)// 
while  (dx/dt)2  would  be  written  'dlx(t)A2'.  Thus,  the  PDE  d2y/dt2  -  g(x,y)- 
(d2y/dx2)2  =  r(x,y),  would  be  written,  using  this  notation,  as  'd2d2y(x,t)- 
g(x,y)*dldly(x,t)A2=r(x,y)'. 

The  notation  using 'd'  and  the  order  of  the  independent  variable  is  the 
notation  preferred  by  the  calculator  when  derivatives  are  involved  in  a 
calculation.  For  example,  using  function  DERIV,  in  ALG  mode,  as  shown  next 
DERIV('x*f(x,t)+g(t,y)  =  h(x,y,t)/,t),  produces  the  following  expression: 
'x*d2f  (x,  t)  +dlg  (t,  y)  =d3h  (x,  y,  t) '.  Translated  to  paper,  this 
expression  represents  the  partial  differential  equation  x-(df/dt)  +  dg/d\  =  dh/dt. 

Because  the  order  of  the  variable  t  is  different  in  f(x,t),  g(t,y),  and  h(x,y,t), 
derivatives  with  respect  to  t  have  different  indices,  i.e.,  d2f(x,t),  dlg(t,y),  and 
d3h(x,y,t).  All  of  them,  however,  represent  derivatives  with  respect  to  the 
same  variable. 

Expressions  for  derivatives  using  the  order-of-variable  index  notation  do  not 
translate  into  derivative  notation  in  the  equation  writer,  as  you  can  check  by 
pressing  while  the  last  result  is  in  stack  level  1 .  However,  the  calculator 
understands  both  notations  and  operates  accordingly  regarding  of  the 
notation  used. 

Checking  solutions  in  the  calculator 

To  check  if  a  function  satisfy  a  certain  equation  using  the  calculator,  use 
function  SUBST  (see  Chapter  5)  to  replace  the  solution  in  the  form  'y  =  {(x)'  or 
'y  =  f(x,t)',  etc.,  into  the  differential  equation.  You  may  need  to  simplify  the 


Page  1 6-2 


result  by  using  function  EVAL  to  verify  the  solution.  For  example,  to  check  that 
u  =  A  sin  coDt  is  a  solution  of  the  equation  d2u/dt2  +  co02-u  =  0,  use  the 
following: 

In  ALG  mode: 

SUBST('dt(dt(u(t)))+  coOA2*u(t)  =  0','u(t)=A*SIN  (coO*t)')  [enter} 
EVAL(ANS(1))S 

In  RPN  mode: 

'dt(dt(u(t)))+  C00A2*u(t)  =  0'  (ENTER)  'u(t)=A*SIN  (a>0*t)'  {enter} 

SUBS'!"  EVflL 
The  result  is  '0=0'. 

For  this  example,  you  could  also  use:  'dt(dt(u(t))))+  coOA2*u(t)  =  0'  to  enter  the 
differential  equation. 

Slope  field  visualization  of  solutions 

Slope  field  plots,  introduced  in  Chapter  1  2,  are  used  to  visualize  the  solutions 
to  a  differential  equation  of  the  form  dy/dx  =  f(x,y).  A  slope  field  plot  shows 
a  number  of  segments  tangential  to  the  solution  curves,  y  =  f(x).  The  slope  of 
the  segments  at  any  point  (x,y)  is  given  by  dy/dx  =  f(x,y),  evaluated  at  any 
point  (x,y),  represents  the  slope  of  the  tangent  line  at  point  (x,y). 

Example  1  -  Trace  the  solution  to  the  differential  equation  y'  =  f(x,y)  =  sin  x 
cos  y,  using  a  slope  field  plot.  To  solve  this  problem,  follow  the  instructions  in 
Chapter  1  2  for  slopefield  plots. 

If  you  could  reproduce  the  slope  field  plot  in  paper,  you  can  trace  by  hand 
lines  that  are  tangent  to  the  line  segments  shown  in  the  plot.  This  lines 
constitute  lines  of  y(x,y)  =  constant,  for  the  solution  of  y'  =  f(x,y).  Thus,  slope 
fields  are  useful  tools  for  visualizing  particularly  difficult  equations  to  solve. 

In  summary,  slope  fields  are  graphical  aids  to  sketch  the  curves  y  =  g(x)  that 
correspond  to  solutions  of  the  differential  equation  dy/dx  =  f(x,y). 


Page  1 6-3 


The  CALC/DIFF  menu 

The  DIFFERENTIAL  EQNS..  sub-menu  within  the  CMC  (C5DS^_  )  menu 
provides  functions  for  the  solution  of  differential  equations.  The  menu  is  listed 
below  with  system  flag  1  1  7  set  to  CHOOSE  boxes: 


hhp:  HERU 


1.  DERIY.  ft  iriTEG . .. 

2.  LIHITS  SERIES.. 


3 .  DIFFERERTIAL  EGRS.. 


H .  GRAPH.. 

5.DERVK 

6.IRTYK 


DIFFERERTIAL  EGRS  HERU 

a.iLRp 

3 .  LAP 

H.LDEC 

5 .  CALCULUS.. 

These  functions  are  briefly  described  next.  They  will  be  described  in  more 
detail  in  later  parts  of  this  Chapter. 


DESOLVE:  Differential  Equation  SOLVEr,  provides  a  solution  if  possible 
ILAP:  Inverse  LAPlace  transform,  L1[F(s)]  =  f(t) 
LAP:  LAPlace  transform,  L[f(t)]=F(s) 

LDEC:  solves  Linear  Differential  Equations  with  Constant  coefficients,  including 
systems  of  differential  equations  with  constant  coefficients 

Solution  to  linear  and  non-linear  equations 

An  equation  in  which  the  dependent  variable  and  all  its  pertinent  derivatives 
are  of  the  first  degree  is  referred  to  as  a  linear  differential  equation. 
Otherwise,  the  equation  is  said  to  be  non-linear.    Examples  of  linear 
differential  equations  are:  d2x/dt2  +  p«(dx/dt)  +  co0-x  =  A  sin  cof  t,  and  dC/d\ 
+  u-(3C/3x)  =  D-(d2C/dx2). 

An  equation  whose  right-hand  side  (not  involving  the  function  or  its  derivatives) 
is  equal  to  zero  is  called  a  homogeneous  equation.  Otherwise,  it  is  called 
non-homogeneous.  The  solution  to  the  homogeneous  equation  is  known  as  a 
general  solution.  A  particular  solution  is  one  that  satisfies  the  non- 
homogeneous  equation. 


Page  1 6-4 


Function  LDEC 

The  calculator  provides  function  LDEC  (Linear  Differential  Equation  Command) 
to  find  the  general  solution  to  a  linear  ODE  of  any  order  with  constant 
coefficients,  whether  it  is  homogeneous  or  not.  This  function  requires  you  to 
provide  two  pieces  of  input: 

•  the  right-hand  side  of  the  ODE 

•  the  characteristic  equation  of  the  ODE 

Both  of  these  inputs  must  be  given  in  terms  of  the  default  independent  variable 
for  the  calculator's  CAS  (typically  'X').  The  output  from  the  function  is  the 
general  solution  of  the  ODE.    The  function  LDEC  is  available  through  in  the 
CALC/DIFF  menu.  The  examples  are  shown  in  the  RPN  mode,  however, 
translating  them  to  the  ALG  mode  is  straightforward. 

Example  1  -  To  solve  the  homogeneous  ODE:  d3y/dx3-4-(d2y/dx2)- 

1  1  -(dy/dx)+30-y  =  0,  enter:  0  [enter}   5  vr3  4*XA2- 1 1  *X+30  5  (enter) 

LDEC.   The  solution  is: 

frcCJ-<cCi»cCZ)    5-K  iP'CCJ-<7.cCi-cCZ)    -<3.K)  15.cCJ»Z.cCi-cCZ  ;-k 

 zh  *    +  hd  *       +  is  * 

where  cCO,  cCl,  and  cC2  are  constants  of  integration.  While  this  result 
seems  very  complicated,  it  can  be  simplified  if  we  take 

K1  =  (10*cC0-(7+cC1-cC2))/40,  K2  =  -(6*cC0-(cC1+cC2))/24, 

and 

K3  =  (15*cC0+(2*cC1-cC2))/15. 

Then,  the  solution  is 

y  =  Kre-3x+  K2-e5x+  K3-e2x. 

The  reason  why  the  result  provided  by  LDEC  shows  such  complicated 
combination  of  constants  is  because,  internally,  to  produce  the  solution,  LDEC 
utilizes  Laplace  transforms  (to  be  presented  later  in  this  chapter),  which 
transform  the  solution  of  an  ODE  into  an  algebraic  solution.  The  combination 


Page  1 6-5 


of  constants  result  from  factoring  out  the  exponential  terms  after  the  Laplace 
transform  solution  is  obtained. 


Example  2  -  Using  the  function  LDEC,  solve  the  non-homogeneous  ODE: 

" "  d3y/dx3-4.(d2y/dx2)-l  1  -(dy/dx)+30.y  =  x2. 

Enter: 

5     2  5  {enter)   5  X  3  4  *  X  2  ■■■■■  1  i  *  X + 3  0  s  (enter)  L  D  E  C 

The  solution,  shown  partially  here  in  the  Equation  Writer,  is: 

?50.cciKia5.ca+ia5.cca+a)  5-k  a70.cco-(i33.cCi-£ar.cca-a»     -O-h)  H50.Ka*EflK+aHi  t 

3000  10S0 

Replacing  the  combination  of  constants  accompanying  the  exponential  terms 
with  simpler  values,  the  expression  can  be  simplified  to  y  =  Kre"3x  +  K2-e5x  + 
K3-e2x+  (450-x2+330-x+241)/13500. 

We  recognize  the  first  three  terms  as  the  general  solution  of  the  homogeneous 
equation  (see  Example  1,  above).  If  yh  represents  the  solution  to  the 
homogeneous  equation,  i.e.,  yh  =  Kre"3x  +  K2-e5x  +  K3-e2x.  You  can  prove 
that  the  remaining  terms  in  the  solution  shown  above,  i.e.,  yp  = 
(450-x2+330-x+241  )/l  3500,  constitute  a  particular  solution  of  the  ODE. 


Note:  This  result  is  general  for  all  non-homogeneous  linear  ODEs,  i.e.,  given 
the  solution  of  the  homogeneous  equation,  yh(x),  the  solution  of  the 
corresponding  non-homogeneous  equation,  y(x),  can  be  written  as 

yM  =  yh(x)  +  yp(x), 

where  yp(x)  is  a  particular  solution  to  the  ODE. 

To  verify  that  yp  =  (450-x2+330-x+241  )/l  3500,  is  indeed  a  particular 
solution  of  the  ODE,  use  the  following: 

s  d 1 d 1 d 1 V  C X >  4*d 1 d 1 V < X > - 1 1 *d 1 Y <  X >  +30* Y < X >  =  X A2 s  (^) 

5  Y  C  X  > :::::  C  45B*X A2+330*X+24 1  >   1 3588  5  (enter) 
=:::=  ■::::  \         v  ni_ 


Page  1 6-6 


Allow  the  calculator  about  ten  seconds  to  produce  the  result:  'XA2  =  XA2\ 
Example  3  -  Solving  a  system  of  linear  differential  equations  with  constant 
coefficients. 

Consider  the  system  of  linear  differential  equations: 

x/(t)  +  2x2'(t)  =  a 

2x1'(t)  +  x2'(t)  =  0. 

[1  2\ 

In  algebraic  form,  this  is  written  as:  A-x'(t)  =  0,  where  A  =  .  The 

system  can  be  solved  by  using  function  LDEC  with  arguments  [0,0]  and  matrix 
A,  as  shown  in  the  following  screen  using  ALG  mode: 


RflD  KYI  HEM  f;= 
[HOME]- 


ALG 


LDEC([@  0],[^  f  ]] 

cVl+cV2   3-X,  cVl-cV2  : 

 ~,  '8       +  ~,  


The  solution  is  given  as  a  vector  containing  the  functions  [x^t),  x2(t)].  Pressing 
^▼7  will  trigger  the  Matrix  Writer  allowing  the  user  to  see  the  two 
components  of  the  vector.  To  see  all  the  details  of  each  component,  press  the 
illlilllili  soft  menu  key.  Verify  that  the  components  are: 


<  c  V 1 +c  V£  V£*EXP  <  3*X  >  + 
(cVl-cV2V2*EXP(-X> 


<  c  V 1 +c  V£  V£*EXP  <  3*X  > - 
(cVl-cV2V2*EXP(-X> 


H'MII'JVmilM'lllM'Hffl'HJEl  H'MII'JVmilM'lllM'Hffl'HJEl 


Function  DESOLVE 

The  calculator  provides  function  DESOLVE  (Differential  Equation  SOLVEr)  to 
solve  certain  types  of  differential  equations.  The  function  requires  as  input  the 
differential  equation  and  the  unknown  function,  and  returns  the  solution  to  the 
equation  if  available.  You  can  also  provide  a  vector  containing  the 
differential  equation  and  the  initial  conditions,  instead  of  only  a  differential 
equation,  as  input  to  DESOLVE.  The  function  DESOLVE  is  available  in  the 
CALC/DIFF  menu.  Examples  of  DESOLVE  applications  are  shown  below 
using  RPN  mode. 

Example  1  -  Solve  the  first-order  ODE: 


Page  1 6-7 


dy/dx  +  x2-y(x)  =  5. 

In  the  calculator  use: 

'dlytxJ+x^S^ytx^S'  [enter)  1  y(x)  5  [enter)  DESOLVE 

The  solution  provided  is  {'y 

(INT(5*EXP(xtA3/3)/xt/x)+cC0)*l/EXP(xA3/3)), },  i.e., 

y{x)  =  exp(-x3  /  3)  •  ( J5  •  exp(x3  /3)-dx  +  cC0 ) 
The  variable  ODETYPE 

You  will  notice  in  the  soft-menu  key  labels  a  new  variable  called  EESB 
(ODETYPE).  This  variable  is  produced  with  the  call  to  the  DESOL  function  and 
holds  a  string  showing  the  type  of  ODE  used  as  input  for  DESOLVE.  Press 
IMlXii  to  obtain  the  string  "lit  order  linear". 

Example  2  --  Solve  the  second-order  ODE: 

d2y/dx2  +  x  (dy/dx)  =  exp(x). 

In  the  calculator  use: 

'd  I  d  1  y  C  x )  +x*d  1  y 5-!  x )  :::::  EXP £  x ) '  (enter)  'y  (m  x  y '  [enter)  DESOLVE 
The  result  is  an  expression  having  two  implicit  integrations,  namely, 


l: 

y(x)=II)T 

in  t 

e  2 

mi 

For  this  particular  equation,  however,  we  realize  that  the  left-hand  side  of  the 
equation  represents  d/dx(x  dy/dx),  thus,  the  ODE  is  now  written: 

d/ dx(x  dy/ dx  )  =  exp  x, 

and 

x  dy/dx  =  exp  x  +  C. 


Page  1 6-8 


Next,  we  can  write 

dy/dx  =  (C  +  exp  x)/x  =  C/x  +  ex/x. 
In  the  calculator,  you  may  try  to  integrate: 

'd  1  y  (.  x  >  =====  ( C  +  EXP  C  x  > )  .■-  x '  (^D  'y  <  x  > 7  (enter)  DESOL VE 

The  result  is         {  'yM  =  INT((EXP(xt)+C)/xt,xt,x)+CO,  },  i.e., 

j(x)  =  f  •  -  +  —  dx  +  C0 
J  x 

Performing  the  integration  by  hand,  we  can  only  get  it  as  far  as: 

y(x)  <ix  +  C-lnx  +  C0 

J  x 

because  the  integral  of  exp(x)/x  is  not  available  in  closed  form. 
Example  3  -  Solving  an  equation  with  initial  conditions.  Solve 

d2y/dt2  +  5y  =  2  cos(t/2), 

with  initial  conditions 

y(0)  =  1 .2,  y'(0)  =  -0.5. 

In  the  calculator,  use: 

['dldly(t)+5*y(t)  =  2*COS(t/2)'  'y(O)  =  6/5'  /d]y(0)  =  -1/2']  [enter} 

'y(t)'  ^) 
DESOLVE 

Notice  that  the  initial  conditions  were  changed  to  their  Exact  expressions,  'y(O) 
=  6/5',  rather  than  /y(0)=1.2,/  and  'dly(O)  =  -1/2',  rather  than,  'dly(O)  =  - 
0.5'.  Changing  to  these  Exact  expressions  facilitates  the  solution. 


Note:  To  obtain  fractional  expressions  for  decimal  values  use  function  ->Q 
(See  Chapter  5). 


Page  1 6-9 


The  solution  is: 

4  'y<t>=cos<t*a^2>>* 

<B^I9>  +  <C0S<t*J"5>*<< 

95*<6^5>+-40V95>+-l^ 

2*<J"5*19V95*SIN<t*J"5 

>  ::■  1   

EHMHEZEgEgliaaiEig] 

Press  (W)(W)to  simplify  the  result  to 

'y(t)  =  -((19*V5*SIN(V5*tH148*COS(V5*t)+80*COS(t/2)))/190),. 

Press  C^D  EEOEQ  to  get  the  string  "Linear  w/  est  coef  f "  for  the  ODE  type 
in  this  case. 

Laplace  Transforms 

The  Laplace  transform  of  a  function  f(t)  produces  a  function  F(s)  in  the  image 
domain  that  can  be  utilized  to  find  the  solution  of  a  linear  differential  equation 
involving  f(t)  through  algebraic  methods.    The  steps  involved  in  this 
application  are  three: 

1 .  Use  of  the  Laplace  transform  converts  the  linear  ODE  involving  f(t)  into  an 
algebraic  equation. 

2.  The  unknown  F(s)  is  solved  for  in  the  image  domain  through  algebraic 
manipulation. 

3.  An  inverse  Laplace  transform  is  used  to  convert  the  image  function  found 
in  step  2  into  the  solution  to  the  differential  equation  f(t). 

Definitions 

The  Laplace  transform  for  function  f(t)  is  the  function  F(s)  defined  as 
L{f(t)}  =  F(s)=[f(t)-e-s'dt. 

The  image  variable  s  can  be,  and  it  generally  is,  a  complex  number. 

Many  practical  applications  of  Laplace  transforms  involve  an  original  function 
f(t)  where  t  represents  time,  e.g.,  control  systems  in  electric  or  hydraulic 


Page  16-10 


circuits.  In  most  cases  one  is  interested  in  the  system  response  after  time  t>0, 
thus,  the  definition  of  the  Laplace  transform,  given  above,  involves  an 
integration  for  values  of  t  larger  than  zero. 

The  inverse  Laplace  transform  maps  the  function  F(s)  onto  the  original  function 
f(t)  in  the  time  domain,  i.e.,  L  "^(s)}  =  f(t). 

The  convolution  integral  or  convolution  product  of  two  functions  f(t)  and  g(t), 
where  g  is  shifted  in  time,  is  defined  as 

(f*g)(t)  =  \j{u)-g{t-u)-du. 
Laplace  transform  and  inverses  in  the  calculator 

The  calculator  provides  the  functions  LAP  and  ILAP  to  calculate  the  Laplace 
transform  and  the  inverse  Laplace  transform,  respectively,  of  a  function  f(VX), 
where  VX  is  the  CAS  default  independent  variable,  which  you  should  set  to  'X'. 
Thus,  the  calculator  returns  the  transform  or  inverse  transform  as  a  function  of 
X.  The  functions  LAP  and  ILAP  are  available  under  the  CALC/DIFF  menu.  The 
examples  are  worked  out  in  the  RPN  mode,  but  translating  them  to  ALG  mode 
is  straightforward.  For  these  examples,  set  the  CAS  mode  to  Real  and  Exact. 

Example  1  -  You  can  get  the  definition  of  the  Laplace  transform  use  the 
following:  TOO'  (enter)  LAP  in  RPN  mode,  or  LflPCf  <X ))  in  ALG  mode. 
The  calculator  returns  the  result  (RPN,  left;  ALG,  right): 


fttUe  (U,X)dtt 

0 


LflPCf  (XH 


fltOe  CU,X3dU 


0 


Compare  these  expressions  with  the  one  given  earlier  in  the  definition  of  the 
Laplace  transform,  i.e., 


L{f(t)}  =  F(s)=[f(t)-e-stdt, 


and  you  will  notice  that  the  CAS  default  variable  X  in  the  equation  writer 
screen  replaces  the  variable  s  in  this  definition.     Therefore,  when  using  the 


Page  16-11 


function  LAP  you  get  back  a  function  of  X,  which  is  the  Laplace  transform  of 
f(X). 

Example  2  -  Determine  the  Laplace  transform  of  f(t)  =  e2t-sin(t).  Use: 
'EXP(2*X)*SIN(X)'  (enter)  LAP  The  calculator  returns  the  result:  l/(SQ(X-2)+l ). 
Press  (Ml)  to  obtain,  l/(X2-4X+5). 

When  you  translate  this  result  in  paper  you  would  write 

F(s)  =  L{e2t-smt}  =  ^^  

s2  -4-s  +  5 

Example  3  -  Determine  the  inverse  Laplace  transform  of  F(s)  =  sin(s).  Use: 
'SIN(X)'  (enter)  ILAP.    The  calculator  takes  a  few  seconds  to  return  the  result: 
/ILAP(SIN(X))//  meaning  that  there  is  no  closed-form  expression  f(t),  such  that  f(t) 
=  L-{sin(s)}. 

Example  4  -  Determine  the  inverse  Laplace  transform  of  F(s)  =  1/s3.  Use: 
'1/XA3'  (enter)  ILAP  (eval).  The  calculator  returns  the  result:  /XA2/2,/  which  is 
interpreted  as  L  ^{l/s3}  =  t2/2. 

Example  5  -  Determine  the  Laplace  transform  of  the  function  f(t)  =  cos  (a-t+b). 
Use:  'COS(a*X+b)'  (enter)  LAP  .  The  calculator  returns  the  result: 

Press  (Ml)  to  obtain  -(a  sin(b)  -  X  cos(b))/(X2+a2).  The  transform  is 
interpreted  as  follows:  L  {cos(a-t+b)}  =  (s-cos  b  -  a-sin  b)/(s2+a2). 

Laplace  transform  theorems 

To  help  you  determine  the  Laplace  transform  of  functions  you  can  use  a 
number  of  theorems,  some  of  which  are  listed  below.  A  few  examples  of  the 
theorem  applications  are  also  included. 

•     Differentiation  theorem  for  the  first  derivative.  Let  fG  be  the  initial  condition 
for  f(t),  i.e.,  f(0)  =  f0,  then 


Page  16-12 


L{df/dt}  =  s-F(s)  -  f0. 


Example  1  -  The  velocity  of  a  moving  particle  v(t)  is  defined  as  v(t)  =  dr/dt, 
where  r  =  r(t)  is  the  position  of  the  particle.  Let  rG  =  r(0),  and  R(s)  =L{r(t)},  then, 
the  transform  of  the  velocity  can  be  written  as  V(s)  =  L{v(t)}=L{dr/dt}=  s-R(s)-rQ. 


•     Differentiation  theorem  for  the  second  derivative.  Let  fQ  =  f(0),  and 
(df/dt)D  =  df/dt  1 1=0,  then  L{d2f/dt2}  =  s2-F(s)  -  s-f0  -  (df/dt)  c. 


Example  2  -  As  a  follow  up  to  Example  1 ,  the  acceleration  a(t)  is  defined  as 
a(t)  =  d2r/dt2.  If  the  initial  velocity  is  v0  =  v(0)  =  dr/dt  |t=0/  then  the  Laplace 
transform  of  the  acceleration  can  be  written  as: 

A(s)  =  L{a(t)}  =  L{d2r/dt2}=  s2-R(s)  -  s-r0  -  vc. 


•  Differentiation  theorem  for  the  n-th  derivative. 
Let  f  M,  =  dkf/dxk  1 1  =  o,  and  fG  =  f(0),  then 

L{dnf/dtn}  =  sn-F(s)  -  snl.f0        s.f(n2)0  -  f (nl)  0. 

•  Linearity  theorem.    L{af(t)+bg(t)}  =  a-L{f(t)}  +  b-L{g(t)}. 

•  Differentiation  theorem  for  the  image  function.  Let  F(s)  =  L{f(t)},  then 
dnF/dsn  =  L{(-t)n.f(t)}. 


Example  3  -  Let  f(t)  =  e~at,  using  the  calculator  with  'EXP(-a*X)'  [enter}  LAP,  you 
get  'l/(X+a)',  or  F(s)  =  l/(s+a).   The  third  derivative  of  this  expression  can 
be  calculated  by  using: 

X  (enter}  [^J  d    X^JCB  a  X  (enter}  [^J  d  (Ml) 

The  result  is 

'-6/(XA4+4*a*XA3+6*aA2*XA2+4*aA3*X+aA4)/,  or 
d3F/ds3  =  -6/(s4+4-a.s3+6.a2.s2+4.a3.s+a4). 


Page  16-13 


Now,  use  '(-X)A3*EXP(-a*X)'  gg)  LAP  The  result  is  exactly  the  same- 


Integration  theorem.  Let  F(s)  =  L{f(t)},  then 


\-\\'f(u)du 


■F(s). 


•     Convolution  theorem.  Let  F(s)  =  L{f(t)}  and  G(s)  =  L{g(t)},  then 


Example  4  -  Using  the  convolution  theorem,  find  the  Laplace  transform  of 


(f*g)(t),  if  f(t)  =  sin(t),  and  g(t)  =  exp(t).  To  find  F(s)  =  L{f(t)},  and  G(s)  =  L{g(t)}, 
use:  'SIN(X)'^)  LAP  (W).  Result,  '1/(XA2+1)',  i.e.,  F(s)  =  l/(s2+l). 
Also,  'EXP(X)'(^]  LAP.  Result,  'V(X-l)',  i.e.,  G(s)  =  l/(s-l).  Thus,  L{(f*g)(t)} 
=  F(s)G(s)  =  l/(s2+l)-l/(s-l)  =  l/((s-l)(s2+l))  =  l/(s3-s2+s-l). 


•     Shift  theorem  for  a  shift  to  the  right.  Let  F(s)  =  L{f(t)},  then 


•     Shift  theorem  for  a  shift  to  the  left.  Let  F(s)  =  L{f(t)},  and  a  >0,  then 


•  Similarity  theorem.  Let  F(s)  =  L{f(t)},  and  a>0,  then    L{f(a-t)}  = 
(l/a)-F(s/a). 

•  Damping  theorem.  Let  F(s)  =  L{f(t)},  then  L{e-bl-f(t)}  =  F(s+b). 

•  Division  theorem.  Let  F(s)  =  L{f(t)},  then 


L{f(t)}-L{g(t)}  =  F(s)-G(s) 


L{f(t-a)}=e-as-L{f(t)}  =  e^-Ffs). 


Page  16-14 


Laplace  transform  of  a  periodic  function  of  period  T: 


L{/(0>  = 


l-e 


sT 


■lf(t)-e-s>  -dt. 


Limit  theorem  for  the  initial  value:  Let  F(s)  =  L{f(t)},  then 


/0=lim/(0  =  lim[*-F(5)]. 


Limit  theorem  for  the  final  value:  Let  F(s)  =  L{f(t)},  then 


/00=lim/(0  =  lim[5-F(j)]. 

f-»00  S— »0 


Dirac's  delta  function  and  Heaviside's  step  function 

In  the  analysis  of  control  systems  it  is  customary  to  utilize  a  type  of  functions 
that  represent  certain  physical  occurrences  such  as  the  sudden  activation  of  a 
switch  (Heaviside's  step  function,  H(t))  or  a  sudden,  instantaneous,  peak  in  an 
input  to  the  system  (Dirac's  delta  function,  8(t)).  These  belong  to  a  class  of 
functions  known  as  generalized  or  symbolic  functions  [e.g.,  see  Friedman,  B., 
1956,  Principles  and  Techniques  of  Applied  Mathematics,  Dover  Publications 
Inc.,  New  York  (1 990  reprint)  ]. 

The  formal  definition  of  Dirac's  delta  function,  5(x),  is  8(x)  =  0,  for  x  ^0,  and 


Also,  if  f(x)  is  a  continuous  function,  then       f(x)S(x  —  x0)dx  =  f(x0). 


'-GO 


Page  16-15 


An  interpretation  for  the  integral  above,  paraphrased  from  Friedman  (1990), 
is  that  the  8-function  "picks  out"  the  value  of  the  function  f(x)  at  x  =  x0.  Dirac's 
delta  function  is  typically  represented  by  an  upward  arrow  at  the  point  x  =  xO, 
indicating  that  the  function  has  a  non-zero  value  only  at  that  particular  value 
of  x0. 


Heaviside's  step  function,  H(x),  is  defined  as 


H(x)  = 


1,  x>0 
0,  x<0 


Also,  for  a  continuous  function  f(x), 

/ {x)H(x  -  x0  )dx  =     f  (x)dx. 

J -co  Jx0 

Dirac's  delta  function  and  Heaviside's  step  function  are  related  by  dH/dx 
8(x).  The  two  functions  are  illustrated  in  the  figure  below. 


You  can  prove  that  L{H(t)}  =  1/s, 

from  which  it  follows  that      L{U0-H(t)}  =  U0/s, 

where  U0  is  a  constant.  Also,  L  ^{l/sJ^H^), 

and  L-MUo/s^lVHW. 

Also,  using  the  shift  theorem  for  a  shift  to  the  right,  L{f(t-a)}=e"°s-L{f(t)}  = 

e^-F(s),  we  can  write  L{H(t-k)}=e-b  L{H(t)}  =  e^-(l/s)  =  (l/s)-e"ks. 


Page  16-16 


Another  important  result,  known  as  the  second  shift  theorem  for  a  shift  to  the 
right  is  that  L         •F(s)}=f(t-a)-H(t-a),  with  F(s)  =  L{f(t)}. 


In  the  calculator  the  Heaviside  step  function  H(t)  is  simply  referred  to  as  '1 '. 
To  check  the  transform  in  the  calculator  use:  L  /  J  LENTER)  LAP.  The  result  is  '1/X', 
i.e.,  L{1}  =  1/s.     Similarly,  W  [enter]  LAP  ,  produces  the  result  /U0/X,/  i.e., 
L{U0}  =  U0/s. 

You  can  obtain  Dirac's  delta  function  in  the  calculator  by  using:  Ij^[enter)  ILAP 
The  result  is  'Delta  (X) '. 

This  result  is  simply  symbolic,  i.e.,  you  cannot  find  a  numerical  value  for,  say 

'Delta  (5) '. 


This  result  can  be  defined  the  Laplace  transform  for  Dirac's  delta  function, 
because  from  L      .0}=  5(t),  it  follows  that  L{5(t)}  =  1 .0 

Also,  using  the  shift  theorem  for  a  shift  to  the  right,  L{f(t-a)}=e"as-L{f(t)}  = 
e-as-F(s),  we  can  write  L{5(t-k)}=e-ks-L{5(t)}  =  e"ks-l  .0  =  e"ks. 

Applications  of  Laplace  transform  in  the  solution  of  linear  ODEs 

At  the  beginning  of  the  section  on  Laplace  transforms  we  indicated  that  you 
could  use  these  transforms  to  convert  a  linear  ODE  in  the  time  domain  into  an 
algebraic  equation  in  the  image  domain.  The  resulting  equation  is  then 
solved  for  a  function  F(s)  through  algebraic  methods,  and  the  solution  to  the 
ODE  is  found  by  using  the  inverse  Laplace  transform  on  F(s). 

The  theorems  on  derivatives  of  a  function,  i.e., 

L{df/dt}  =  s-F(s)  -  fo; 

L{d2f/dt2}  =  s2  F{s)  -  s-f0  -  (df/dt)  0, 

and,  in  general, 

L{dnf/dtn}  =  sn.F(s)  -  snl-f0        s.f(n2)0  -  f (nl)  G, 
are  particularly  useful  in  transforming  an  ODE  into  an  algebraic  equation. 


Page  16-17 


Example  1  -  To  solve  the  first  order  equation, 

dh/dt  +  k-h(t)  =  a-e"', 

by  using  Laplace  transforms,  we  can  write: 

L{dh/dt  +  k-h(t)}  =  Lja-e"'}, 

L{dh/dt}  +  k-L{h(t)}  =  a-Lje"1}. 

Note:  'EXP(-X)'  gg)  LAP  ,  produces  /1/(X+1)/,  i.e.,  L{e-f  }=1/(s+1).  

With  H(s)  =  L{h(t)},  and  L{dh/dt}  =  s-H(s)  -  hG,  where  hG  =  h(0),  the  transformed 
equation  is  s-H(s)-h0+k-H(s)  =  a/(s+l). 

Use  the  calculator  to  solve  for  H(s),  by  writing: 

/X*H-hO+k*H=a/(X+l),(^)  'H'  ISOL 

The  result  is         'H=((X+1  )*hO+a)/(XA2+(k+l )*X+k)'. 

To  find  the  solution  to  the  ODE,  h(t),  we  need  to  use  the  inverse  Laplace 
transform,  as  follows: 

OBJ->  CS  L^Jti^J  Isolates  right-hand  side  of  last  expression 

ILAP  Obtains  the  inverse  Laplace  transform 

a-ek'X+»k-D-ho-a)-eX 
X  k  'X 

The  result  is  (k-l)-e  -e  .  Replacing  X  with  t  in  this  expression 

and  simplifying,  results  in  h(t)  =  a/(k-l  )-e"f  +((k-l  )-h0-a)/(k-l  )-e"kt. 

Check  what  the  solution  to  the  ODE  would  be  if  you  use  the  function  LDEC: 

/a*EXP(-X)/  [enter}  'X+k'  [enter}  LDEC  (W) 


Page  16-18 


a-ek'X+»k-l)-cCB-a)-eX 
The  result  is:  Ck-13-eX-ek'X        /  i.e., 

h(t)  =  a/(k-l).e4  +((k-l).cC0-a)/(k-l).ekt. 

Thus,  cCO  in  the  results  from  LDEC  represents  the  initial  condition  h(0).  

Note:  When  using  the  function  LDEC  to  solve  a  linear  ODE  of  order  n  in  f(X), 
the  result  will  be  given  in  terms  of  n  constants  cCO,  cCl,  cC2,  cC(n-l), 
representing  the  initial  conditions  f(0),  f'(0),  {"(0),      f(n1)  (0).  

Example  2  -  Use  Laplace  transforms  to  solve  the  second-order  linear  equation, 

d2y/dt2+2y  =  sin  3t- 

Using  Laplace  transforms,  we  can  write: 

L{d2y/dt2+2y}  =  L{sin  3t}, 

L{d2y/dt2}  +  2  L{y(t)}  =  L{sin  3t}. 

Note:  'SIN(3*X)'(^)  LAP  (W)  produces  '3/(XA2+9)',  i.e., 

Ljsin  3t}^3/(s2+9). 


With  Y(s)  =  L{y(t)},  and  L{d2y/dt2}  =  s2-Y(s)  -  s-y0  -  yl7  where  yG  =  h(0)  and  Yl  = 
W(0),  the  transformed  equation  is 

s2-Y(s)-s-y0-y1  +  2-Y(s)  =  3/(s2+9). 

Use  the  calculator  to  solve  for  Y(s),  by  writing: 

/XA2*Y-X*y0-yl+2*Y=3/(XA2+9),(^)  'Y'  ISOL 

The  result  is 

fY=((XA2+9^y1+(y0*XA3+9*y0*X+3))/(XA4+1rXA2+18)^ 


Page  16-19 


To  find  the  solution  to  the  ODE,  y(t),  we  need  to  use  the  inverse  Laplace 
transform,  as  follows: 

OBJ->  1_4J  L*J  Isolates  right-hand  side  of  last  expression 

ILAPd^T)  Obtains  the  inverse  Laplace  transform 

The  result  is 

(7'Jg'yl+3'Jg)siH(^ 

14 

i.e., 

y(t)  =  -(1/7)  sin  3x  +  yQ  cos  V2x  +  (V2  (7Yl+3)/14)  sin  V2x. 
Check  what  the  solution  to  the  ODE  would  be  if  you  use  the  function  LDEC: 

'SIN(3*X)'(^)  'XA2+2'(£!3)  LDEC  d^T) 

The  result  is: 

(7'J?'cCl+3'J?)'SIN(^^ 

14 

i.e.,  the  same  as  before  with  cCO  =  yO  and  cCl  =  yl . 


Note:  Using  the  two  examples  shown  here,  we  can  confirm  what  we 
indicated  earlier,  i.e.,  that  function  ILAP  uses  Laplace  transforms  and  inverse 
transforms  to  solve  linear  ODEs  given  the  right-hand  side  of  the  equation  and 
the  characteristic  equation  of  the  corresponding  homogeneous  ODE. 


Example  3  -  Consider  the  equation 

d2y/dt2+y  =  8(t-3), 
where  8(t)  is  Dirac's  delta  function. 

Using  Laplace  transforms,  we  can  write: 

L{d2y/dt2+y}  =  L{S(t-3)}, 


Page  1 6-20 


L{d2y/dt2}  +  L{y(t)}  =  L{5(t-3)}. 

With  'Delta  (x-3)'  (enter)  LAP  ,  the  calculator  produces  EXP(-3*X),  i.e.,  L{8(t-3)} 
=  e"3s.  With  Y(s)  =  L{y(t)},  and  L{d2y/dt2}  =  s2-Y(s)  -  s-y0  -  yl7  where  yG  =  h(0) 
and  y}  =  h'(0),  the  transformed  equation  is  s2-Y(s)  -  s-y0  -  yi  +  Y(s)  =  e"3s.  Use 
the  calculator  to  solve  for  Y(s),  by  writing: 

'XA2*Y-X*yO-yl  +Y=EXP(-3*X)' 'Y'  ISOL 
The  result  is       'Y=(X*yO+(yl  +EXP(-(3*X))))/(XA2+1 )'. 

To  find  the  solution  to  the  ODE,  y(t),  we  need  to  use  the  inverse  Laplace 
transform,  as  follows: 

OBJ->        L*J  Isolates  right-hand  side  of  last  expression 

ILAP      (evalJ  Obtains  the  inverse  Laplace  transform 

The  result  is  'yl  *SIN(X)+yO*COS(X)+SIN(X-3)*Heaviside(X-3),. 


Notes: 

[1  ].  An  alternative  way  to  obtain  the  inverse  Laplace  transform  of  the 
expression  /(X*yO+(yl  +EXP(-(3*X))))/(XA2+1 )'  is  by  separating  the 
expression  into  partial  fractions,  i.e., 

'yO*X/(XA2+l)  +  yl/(XA2+l)  +  EXP(-3*X)/(XA2+1 )', 

and  use  the  linearity  theorem  of  the  inverse  Laplace  transform 

L  1{a.F(s)+b-G(s)}  =  a-L  1{F(s)}  +  b-L  ]{G(s)}, 

to  write, 

L  -i{yo.s/(s2+l  )+yi/(s2+l ))  +  e-V(s2+l ))  }  = 
y0-L  1  {s/(s2+ 1 )}+  y,  -L  1  { 1  /(s2+ 1 )}+  L  1  {e-3V(s2+ 1 ))}, 
Then,  we  use  the  calculator  to  obtain  the  following: 


Page  16-21 


'X/(XA2+1 )'         ILAP  Result,  'COS(X)',  i.e.,  L  VfAl  )}=  cos  t. 

'1/(XA2+1)'^)  ILAP  Result,  'SIN(X)',  i.e.,  L  ]{1 /(s2+l )}=  sin  t. 

'EXP(-3*X)/(XA2+1 )'  (mh)  ILAP    Result,  SIN(X-3)*Heaviside(X-3)'. 

[2].  The  very  last  result,  i.e.,  the  inverse  Laplace  transform  of  the  expression 
'(EXP(-3*X)/(XA2+1))',  can  also  be  calculated  by  using  the  second  shifting 
theorem  for  a  shift  to  the  right 

L-1{e-.F(s)}=Kt-a)-H(t-a)l 

if  we  can  find  an  inverse  Laplace  transform  for  l/(s2+l).  With  the  calculator, 
try  /1/(XA2+1)/  (enter)  ILAP.  The  result  is  'SIN(X)'.  Thus,  L  V^/V+l ))}  = 
sin(t-3)-H(t-3), 


Check  what  the  solution  to  the  ODE  would  be  if  you  use  the  function  LDEC: 
'Delta(X-3)'  (enter)  'Xa2+1'(^  LDEC  Qw) 

The  result  is: 

ySIN(X-3)*Heaviside(X-3)  +  cCl  *SIN(X)  +  cC0*COS(X)+'. 

Please  notice  that  the  variable  X  in  this  expression  actually  represents  the 
variable  t  in  the  original  ODE.  Thus,  the  translation  of  the  solution  in  paper 
may  be  written  as: 

y{f)  =  Co •  cosf  +  Q  •  sin/  +sin(7 - 3) •  H(t - 3) 

When  comparing  this  result  with  the  previous  result  for  y(t),  we  conclude  that 

cC0  =  y0,  cCt  =  y!. 


Page  16-22 


Defining  and  using  Heaviside's  step  function  in  the  calculator 

The  previous  example  provided  some  experience  with  the  use  of  Dirac's  delta 
function  as  input  to  a  system  (i.e.,  in  the  right-hand  side  of  the  ODE  describing 
the  system).  In  this  example,  we  want  to  use  Heaviside's  step  function,  H(t). 
In  the  calculator  we  can  define  this  function  as: 

'H(X)  =  IFTE(X>0,  1,  0)'  {enter)[^Jde^_ 

This  definition  will  create  the  variable         in  the  calculator's  soft  menu  key. 

Example  1  -  To  see  a  plot  of  H(t-2),  for  example,  use  a  FUNCTION  type  of 
plot  (see  Chapter  1  2): 

•  Press  C3D  J^, ,  simultaneously  in  RPN  mode,  to  access  to  the  PLOT  SETUP 
window. 

>  Change  type  to  function,  if  needed 

>  Change  EQ  to  'H(X-2)'. 

>  Make  sure  that  indep  is  set  to  'X'. 

>  Press  [nxt)  BQIHto  return  to  normal  calculator  display. 

•  Press  LjnJ ML  ,  simultaneously,  to  access  the  PLOT  window. 

>  Change  the  H-VIEW  range  to  0  to  20,  and  the  V-VIEW  range  to  -2  to  2. 

>  Press         D333  to  plot  the  function  . 

Use  of  the  function  H(X)  with  LDEC,  LAP,  or  ILAP,  is  not  allowed  in  the 
calculator.  You  have  to  use  the  main  results  provided  earlier  when  dealing 
with  the  Heaviside  step  function,  i.e.,    L{H(t)}  =  1/s,  L  ^{l /s}=H(t), 
L{H(t-k)}=e-ks  L{H(t)}  =  e-ks.(l/s)  =  -(l/sj-e^  and  L  V05  -F(s)}=f(t-a).H(t-a). 

Example  2  -  The  function  H(t-tG)  when  multiplied  to  a  function  f(t),  i.e.,  H(t-t0)f(t), 
has  the  effect  of  switching  on  the  function  f(t)  at  t  =  tQ.  For  example,  the 
solution  obtained  in  Example  3,  above,  was  y(t)  =  yG  cos  t  +  y}  sin  t  +  sin(t- 
3)-H(t-3).  Suppose  we  use  the  initial  conditions  yG  =  0.5,  and  y!  =  -0.25. 
Let's  plot  this  function  to  see  what  it  looks  like: 

•  Press  UnJi^ ,  simultaneously  if  in  RPN  mode,  to  access  to  the  PLOT 
SETUP  window. 


Page  16-23 


>  Change  type  to  function,  if  needed 

>  Change  EQ  to  /0.5*COS(X)-0.25*SIN(X)+SIN(X-3)*H(X-3)/. 

>  Make  sure  that  indep  is  set  to  'X'. 

>  Press  ylilLiii  Em  to  plot  the  function. 

>  Press  Mm  (W)  EMM  to  see  the  plot. 

The  resulting  graph  will  look  like  this: 


Notice  that  the  signal  starts  with  a  relatively  small  amplitude,  but  suddenly,  at 
t=3,  it  switches  to  an  oscillatory  signal  with  a  larger  amplitude.  The 
difference  between  the  behavior  of  the  signal  before  and  after  t  =  3  is  the 
"switching  on"  of  the  particular  solution  yp(t)  =  sin(t-3)-H(t-3).  The  behavior  of 
the  signal  before  t  =  3  represents  the  contribution  of  the  homogeneous 
solution,  yh(t)  =  yQ  cos  t  +  y}  sin  t. 

The  solution  of  an  equation  with  a  driving  signal  given  by  a  Heaviside  step 
function  is  shown  below. 

Example  3  -  Determine  the  solution  to  the  equation,  d2y/dt2+y  =  H(t-3), 
where  H(t)  is  Heaviside's  step  function.    Using  Laplace  transforms,  we  can 
write:  L{d2y/dt2+y}  =  L{H(t-3)},  L{d2y/dt2}  +  L{y(t)}  =  L{H(t-3)}.  The  last  term  in 
this  expression  is:  L{H(t-3)}  =  (l/s)-e-3s.  With  Y(s)  =  L{y(t)},  and  L{d2y/dt2}  = 
s2-Y(s)  -  s-yQ  -  y]f  where  yG  =  h(0)  and  y!  =  h'(0),  the  transformed  equation  is 
s2-Y(s)  -  s-yQ  -  yi  +  Y(s)  =  (l/s)-e"3s.  Change  CAS  mode  to  Exact,  if  necessary. 
Use  the  calculator  to  solve  for  Y(s),  by  writing: 


To  find  the  solution  to  the  ODE,  y(t),  we  need  to  use  the  inverse  Laplace 
transform,  as  follows: 


/XA2*Y-X*yO-yl+Y=(l/X)*EXP(-3*X),(^  'Y'  ISOL 


The  result  is 


'YHXA2>0+X>1+EXP(-3*X))/(XA3+X)'. 


Page  1 6-24 


OBJ^  3D  3D 
ILAP 


Isolates  right-hand  side  of  last  expression 
Obtains  the  inverse  Laplace  transform 


The  result  is      'yl  *SIN(X-1  )+yO*COS(X-l  )-(COS(X-3)-l  )*Heaviside(X-3)'. 
Thus,  we  write  as  the  solution:    y(t)  =  yQ  cos  t  +  yn  sin  t  +  H(t-3)*(  1  +sin(t-3)). 
Check  what  the  solution  to  the  ODE  would  be  if  you  use  the  function  LDEC: 
'H(X-3)'(^  [ENTER]  'XA2+T(^  LDEC 

The  result  is: 


sm<k> 


K.ttt 


Please  notice  that  the  variable  X  in  this  expression  actually  represents  the 
variable  t  in  the  original  ODE,  and  that  the  variable  ttt  in  this  expression  is  a 
dummy  variable.  Thus,  the  translation  of  the  solution  in  paper  may  be  written 
as: 

poo 

y{t)  =  Co-cost  +  Cl  -sin^  +  siiU-  ^  H(u  —  3)-e~ut  -du. 

Example  4  -  Plot  the  solution  to  Example  3  using  the  same  values  of  yQ  and  y] 
used  in  the  plot  of  Example  1 ,  above.  We  now  plot  the  function 

y(t)  =  0.5  cos  t  -0.25  sin  t  +  (1  +sin(t-3))-H(t-3). 

In  the  range  0  <  t  <  20,  and  changing  the  vertical  range  to  (-1,3),  the  graph 
should  look  like  this: 


Page  1 6-25 


Again,  there  is  a  new  component  to  the  motion  switched  at  t=3,  namely,  the 
particular  solution  yp(t)  =  [1 +sin(t-3)]-H(t-3),  which  changes  the  nature  of  the 
solution  for  t>3. 

The  Heaviside  step  function  can  be  combined  with  a  constant  function  and 
with  linear  functions  to  generate  square,  triangular,  and  saw  tooth  finite  pulses, 
as  follows: 

•  Square  pulse  of  size  UQ  in  the  interval  a  <  t  <  b: 

f(t)  =  Uo[H(t-a)-H(t-b)]. 

•  Triangular  pulse  with  a  maximum  value  Uo,  increasing  from  a  <  t  <  b, 
decreasing  from  b  <  t  <  c: 

f(t)  =  U0-  ((t-a)/(b-a)-[H(t-a)-H(t-b)]+(l-(t-b)/(b-c))[H(t-b)-H(t-c)]). 

•  Saw  tooth  pulse  increasing  to  a  maximum  value  Uo  for  a  <  t  <  b, 
dropping  suddenly  down  to  zero  at  t  =  b: 

f(t)  =  U„-  (t-a)/(b-a)  [H(t-a)-H(t-b)]. 

•  Saw  tooth  pulse  increasing  suddenly  to  a  maximum  of  Uo  at  t  =  a,  then 
decreasing  linearly  to  zero  for  a  <  t  <  b: 

f(t)  =  U0-[l-(t-a)/(b-l)]-[H(t-a)-H(t-b)]. 

Examples  of  the  plots  generated  by  these  functions,  for  Uo  =  1 ,  a  =  2,  b  =  3, 
c  =  4,  horizontal  range  =  (0,5),  and  vertical  range  =  (-1,  1 .5),  are  shown  in 
the  figures  below: 


lv 

1  1 

lv 

/\  . 

1  _i_ 

0.                                             5 .  1 
j. 

1  lv 

A 

f  N 

0.                                               F.  1 
f  j. 

0.                                             5 . 

Page  1 6-26 


Fourier  series 

Fourier  series  are  series  involving  sine  and  cosine  functions  typically  used  to 
expand  periodic  functions.  A  function  f(x)  is  said  to  be  periodic,  of  period  T, 
if  f(x+T)  =  f(t).  For  example,  because  sin(x+27i)  =  sin  x,  and  cos(x+27i)  =  cos 
x,  the  functions  sin  and  cos  are  27i-periodic  functions.  If  two  functions  f(x)  and 
g(x)  are  periodic  of  period  T,  then  their  linear  combination  h(x)  =  a-f(x)  + 
b-g(x),  is  also  periodic  of  period  T.  A  T-periodic  function  f(t)  can  be 
expanded  into  a  series  of  sine  and  cosine  functions  known  as  a  Fourier  series 
given  by 

Inn      7      .  2nn  \ 
fit)  =  a0  +  2^ \an  •  cos— —t  +  bn  •  sin— t 

n=\\  1  1  J 

where  the  coefficients  an  and  bn  are  given  by 

1  rT/2                     2  rT/2  2nn 
aCl=—\      f(t)'dt,    a=—\      fit)-  cos  t-dt, 

U       f  hll  J  V  7  n       J  J-T/2J  V  J  J 

Ti2  _  x    .  2nn 


h  =       fit)  -sin  t-dt. 


The  following  exercises  are  in  ALG  mode,  with  CAS  mode  set  to  Exact. 
(When  you  produce  a  graph,  the  CAS  mode  will  be  reset  to  Approx.  Make 
sure  to  set  it  back  to  Exact  after  producing  the  graph.)  Suppose,  for  example, 
that  the  function  f(t)  =  t2+t  is  periodic  with  period  T  =  2.  To  determine  the 
coefficients  a0/  a^  and  ^  for  the  corresponding  Fourier  series,  we  proceed  as 
follows:  First,  define  function  f(t)  =  t2+t  : 


:  DEF I  He(V  Ct  )=t  ) 

HOVRU 


Next,  we'll  use  the  Equation  Writer  to  calculate  the  coefficients: 


Page  1 6-27 


al 

m 

f(t) 

COS(>t)dt  I 

EDIT 

CURL"  | 

GIG  ■ 

EVflL  |FflCTO|  SIHP 

bl 

i 

fCt) 

SIN(TT't)dt  1 

EDIT 

CURS  | 

GIG  ■ 

EVflL  |FflCTO|  SIHP 

Thus,  the  first  three  terms  of  the  function  are: 

f(t)  *  1/3  -  (4/ti2).cos  M+(2/7i)-sin  (n-\). 


A  graphical  comparison  of  the  original  function  with  the  Fourier  expansion 
using  these  three  terms  shows  that  the  fitting  is  acceptable  for  t  <  1 ,  or 
thereabouts.  But,  then,  again,  we  stipulated  that  T/2  =  1 .  Therefore,  the 
fitting  is  valid  only  between  -1  <  t  <  1 . 


Function  FOURIER 

An  alternative  way  to  define  a  Fourier  series  is  by  using  complex  numbers  as 
follows: 


f(t)=Xcn-cxp(—^-), 


where 


Page  16-28 


cn=—  L/(0*exp(  t)-dt,    n  =  -oov..  -2 ,-1,0,1,2,- -00. 


Function  FOURIER  provides  the  coefficient  cn  of  the  complex-form  of  the 
Fourier  series  given  the  function  f(t)  and  the  value  of  n.  The  function  FOURIER 
requires  you  to  store  the  value  of  the  period  (T)  of  a  T-periodic  function  into 
the  CAS  variable  PERIOD  before  calling  the  function.  The  function  FOURIER  is 
available  in  the  DERIV  sub-menu  within  the  CALC  menu  ([j^jc^c_  ). 

Fourier  series  for  a  quadratic  function 

Determine  the  coefficients  c0,  q,  and  c2  for  the  function  f(t)  =  t2+t,  with  period 
1  =  2.  (Note:  Because  the  integral  used  by  function  FOURIER  is  calculated  in 
the  interval  [0,T],  while  the  one  defined  earlier  was  calculated  in  the  interval 
[-T/2,T/2],  we  need  to  shift  the  function  in  the  t-axis,  by  subtracting  T/2  from  t, 
i.e.,  we  will  use  g(t)  =  f(t-l )  =  (t-1  )2+(t-l ).) 

Using  the  calculator  in  ALG  mode,  first  we  define  functions  f(t)  and  g(t): 


:  DEF I  He(  f  Ct  )=t  +t ) 

HOVflL 
:  DEF  I  HEC'gCt  3=f  Ct  - 1 3'3 

Next,  we  move  to  the  CASDIR  sub-directory  under  HOME  to  change  the  value 
of  variable  PERIOD,  e.g.,        (hold)  updir_  (enter)[WJ  mMM  (l^(JJ  droT) 
lllj:[|!33  {enter} 


HOVRL 

:H0ME 

HOVflL 

iCflSDIR 

HOVflL 

^►PERIOD 

2 

F" F; I M 1 1 H L" I H | M ij [i U L | F; E H L H | F- E F; I ij |  VK 

Return  to  the  sub-directory  where  you  defined  functions  f  and  g,  and  calculate 
the  coefficients  (Accept  change  to  Complex  mode  when  requested): 


Page  1 6-29 


:FOURIER(gtX),0) 

2 
3 
2 

:FOURIER(g(XU) 

3 

2-i-TT+4 

2 

TT 

2 

:F0URIER(9(X),2) 

TT 

i-TT+1 

2 

TT 

2 

COLLECTtflNSdH 


TT 

2 

iCOLLECTtflNSClH 

i-TT+2 

2 

TT 

TT 

2 

iCOLLECTtflNSClH 

i-TT+1 

Thus,  c0  =  1/3,  c,  =  (n-\+2)/n2,  c2  =  (jc-i+l  )/(2tt2). 

The  Fourier  series  with  three  elements  will  be  written  as 

g(t)  ~  Re[(l/3)  +  (7i  i+2)A2  exp(i-7r  t)+  (n-i+1  )/(27r2)-exp(2i-7i-t)]. 
A  plot  of  the  shifted  function  g(t)  and  the  Fourier  series  fitting  follows: 


-i. 

The  fitting  is  somewhat  acceptable  for  0<t<2,  although  not  as  good  as  in  the 
previous  example. 


Page  1 6-30 


A  general  expression  for  cn 

The  function  FOURIER  can  provide  a  general  expression  for  the  coefficient  cn 
of  the  complex  Fourier  series  expansion.  For  example,  using  the  same 
function  g(t)  as  before,  the  general  term  cn  is  given  by  (figures  show  normal 
font  and  small  font  displays): 


2-tt" 

:F0URIERt9(X),n) 

Cn-TT+2-  i  )-e   1  ,|_|,TT+2-  i  ti2-tt2h 

i-n+i 

:F0URIER(3CK)jnJ 

(n.n+a.iJ.«a,x,n,n+a.i.na.na+3.n-n-a.i 

3  3  2-i-n-ir  * 
n  "TT  -e 

_3_3  a-i-n-n 

 2  

a 

The  general  expression  turns  out  to  be,  after  simplifying  the  previous  result, 

inn  +  2i)  ■  e2inM  +  2i2n2n2  +  3nx  -  2i 
c   =  —  

2n  n  -e 

We  can  simplify  this  expression  even  further  by  using  Euler's  formula  for 
complex  numbers,  namely,  e2inn  =  cos(2n7i)  +  i-sin(2n7i)  =  1  +  i-0  =  1,  since 
cos(2n7i)  =  1,  and  sin(2n7i)  =  0,  for  n  integer. 

Using  the  calculator  you  can  simplify  the  expression  in  the  equation  writer 
(L  r»  J  eqw  )  by  replacing  e2mTl  =  1.  The  figure  shows  the  expression  after 
simplification: 


EaaBnaBgnfflnMBiBEBini 

The  result  is  cn  =  (i-n-7i+2)/(n2-7i2). 

Putting  together  the  complex  Fourier  series 

Having  determined  the  general  expression  for  cn,  we  can  put  together  a  finite 
complex  Fourier  series  by  using  the  summation  function  (X)  in  the  calculator  as 
follows: 


Page  16-31 


•     First,  define  a  function  c(n)  representing  the  general  term  cn  in  the  complex 
Fourier  series. 


: DEFINE 

n  "TT 

HOVRL 

•  Next,  define  the  finite  complex  Fourier  series,  F(X,k),  where  X  is  the 
independent  variable  and  k  determines  the  number  of  terms  to  be  used. 
Ideally  we  would  like  to  write  this  finite  complex  Fourier  series  as 


F(X,k)=fjc(n)-txp(2'l^'n-X) 


n=-k 


However,  because  the  function  c(n)  is  not  defined  for  n  =  0,  we  will  be 
better  advised  to  re-write  the  expression  as 

F(X,k,c0)  =  c0  + 


2J.c{n)  •  exp(  X)  +  cirri)  •  exp(  X)l 

n=\  *  * 


Or,  in  the  calculator  entry  line  as: 


DEFINE('F(X,k,cO)  =  cO+Z(n=l  ,k,c(n)*EXP(2*i*7i*n*X/T)+ 
c(-n)*EXP(-(2*i*7i*n*X/T)),)/ 


where  T  is  the  period,  T  =  2.  The  following  screen  shots  show  the  definition  of 
function  F  and  the  storing  of  T  =  2: 


Page  16-32 


DEF I  HE  (  1 F  t  X  j  k  s_c0  >  =c0+A 

rr*n*X^PER  I OD  > +c  ( -n  >  * 
EXP  < -2*  i  *TT*n*X/PER  I OD 
>>  '  > 

a:tiaH:tiec.H«.»pi.]«iniMj 


:2H 

aajaaaffijaaBaHHaiBig] 


The  function  111!!!!!  can  be  used  to  generate  the  expression  for  the  complex 
Fourier  series  for  a  finite  value  of  k.  For  example,  for  k  =  2,  c0  =  l/3,and 
using  t  as  the  independent  variable,  we  can  evaluate  F(t,2,l/3)  to  get: 


=+41 

!+(0.20?0.32)-e 


(0.00,6.£ 


2.0£> 


This  result  shows  only  the  first  term  (cO)  and  part  of  the  first  exponential  term 
in  the  series.  The  decimal  display  format  was  changed  to  Fix  with  2  decimals 
to  be  able  to  show  some  of  the  coefficients  in  the  expansion  and  in  the 
exponent.  As  expected,  the  coefficients  are  complex  numbers. 

The  function  F,  thus  defined,  is  fine  for  obtaining  values  of  the  finite  Fourier 
series.  For  example,  a  single  value  of  the  series,  e.g.,  F(0.5,2,l/3),  can  be 
obtained  by  using  (CAS  modes  set  to  Exact,  step-by-step,  and  Complex): 

:f(.5,2±] 

737940956009,0.) 
:+NUM(flNS(l.H 

Accept  change  to  Approx  mode  if  requested.    The  result  is  the  value 
-0.40467....  The  actual  value  of  the  function  g(0.5)  is  g(0.5)  =  -0.25.  The 
following  calculations  show  how  well  the  Fourier  series  approximates  this 
value  as  the  number  of  components  in  the  series,  given  by  k,  increases: 

F  (0.5,  1,  1/3)  =  (-0.303286439037,0.) 

F  (0.5,  2,  1/3)  =  (-0.404607622676,0.) 

F(0.5,  3,  1/3)  =  (-0.192401031886,0.) 


Page  16-33 


F  (0.5,  4,  1/3)  =  (-0.167070735979,0.) 
F  (0.5,  5,  1/3)  =  (-0.294394690453,0.) 
F  (0.5,  6,  1/3)  =  (-0.305652599743,0.) 


To  compare  the  results  from  the  series  with  those  of  the  original  function,  load 
these  functions  into  the  PLOT  -  FUNCTION  input  form  (C5D^ , 
simultaneously  if  using  RPN  mode) 


Change  the  limits  of  the  Plot  Window  (GT3 ML  )  as  follows: 


^KPLOT  HiriDOH  - 

FuncTion^W 

3. 

V-Vi«H=-i. 

3. 

Indep  Loh:  I'-NJ-Um 

Hish: Default 

Step:  Default 

_ Pixels 

Enter  HiniHUH  indep 

uar  ualue 

Press  the  soft-menu  keys  IlilllS        to  produce  the  plot: 


-i. 

Notice  that  the  series,  with  5  terms,  "hugs"  the  graph  of  the  function  very 
closely  in  the  interval  0  to  2  (i.e.,  through  the  period  T  =  2).  You  can  also 
notice  a  periodicity  in  the  graph  of  the  series.  This  periodicity  is  easy  to 
visualize  by  expanding  the  horizontal  range  of  the  plot  to  (-0.5,4): 


Page  1 6-34 


Fourier  series  for  a  triangular  wave 

Consider  the  function 


x,    if  0  <  x  <  1 
2-x,  if\<x<2 


which  we  assume  to  be  periodic  with  period  T  =  2.  This  function  can  be 
defined  in  the  calculator,  in  ALG  mode,  by  the  expression 

DEFINE(/g(X)  =  IFTE(X<1  ,X,2-X)') 

If  you  started  this  example  after  finishing  example  1  you  already  have  a  value 
of  2  stored  in  CAS  variable  PERIOD.  If  you  are  not  sure,  check  the  value  of 
this  variable,  and  store  a  2  in  it  if  needed.   The  coefficient  c0  for  the  Fourier 
series  is  calculated  as  follows: 


:F0URIER(9(X),e) 

'  IFTE(Xt<l.,Xt,-(Xt-£h 

[  "  IFTE(Xt<l.,Xt,-(Xt-£h 

2 

:+NUM(flNS(l.H 

.5 

2 

The  calculator  will  request  a  change  to  Approx  mode  because  of  the 
integration  of  the  function  IFTE()  included  in  the  integrand.  Accepting,  the 
change  to  Approx  produces  c0  =  0.5.  If  we  now  want  to  obtain  a  generic 
expression  for  the  coefficient  cn  use: 


Page  16-35 


hUUKlLKI9|..::..|.riJ 

"2. 

IFTEtXt<l.JXtJ-tXt- 


Xt-n-te.  ?  1.3-3.  14159J 


0. 


The  calculator  returns  an  integral  that  cannot  be  evaluated  numerically 
because  it  depends  on  the  parameter  n.    The  coefficient  can  still  be 
calculated  by  typing  its  definition  in  the  calculator,  i.e., 


1  fi 

-•  \X-EXP 
?  Jo 


i  •  2  •  n  -  n  -  X 


•  dX  + 


^•fi(2-X)-EXP 


i  •  2  •  n  •  n  •  X 


dX 


where  T  =  2  is  the  period.  The  value  of  T  can  be  stored  using: 


:2H 

2 

Typing  the  first  integral  above  in  the  Equation  Writer,  selecting  the  entire 
expression,  and  using  iLlllliiEESil,  will  produce  the  following: 


l-n-TT  . 

e  -1TVTT-1 


.22  l-n-TT 
2-n  "TT  -e 


EDIT   lUF;l"   BIG  ■  E'.'HL  FACTO  l'IMF" 


Recall  the  emn  =  cos(n7i)  +  i-sin(nTi)  =  (-l)n  .  Performing  this  substitution  in  the 
result  above  we  have: 


Page  16-36 


(-1)  -i-n-iT-1 


Press  [enter] [enter]  to  copy  this  result  to  the  screen.  Then,  reactivate  the  Equation 
Writer  to  calculate  the  second  integra  defining  the  coefficient  cn/  namely, 


[2 


j-2TrTT-X 


(2-X>e 


1 


dX4 


(.-lTi-TT+lJ-e  -e 


EaaEnaoaiBEiMfliHEBini  BaaEnaBgaBEaHiHEBEa 

Once  again,  replacing  em7t  =  (-1)n,  and  using  e2m7t  =  1,  we  get: 


-i-n-TT+l-C-l) 


Press  ijNf^ijNTER]  to  copy  this  second  result  to  the  screen.  Now,  add  ANS(1 ) 
and  ANS(2)  to  get  the  full  expression  for  cn: 


n  n  


RHSCD+RHSC2) 

e1  n  TT+i-n-TT-l  (-(i-niTT)+1 

"22   i-n-ir       0  2  2 
2-n  *tt  -e  _2ti  -it  -i 

MoanimMaiaBBaaEiia 


Pressing  ^3?  will  place  this  result  in  the  Equation  Writer,  where  we  can 
simplify  (llllllillll)  it  to  read: 


i-n-ir 

-1 

e  + 

2  2  i 

■ri'TT 

n  "TT  -e 

EDIT   CURS    BIG    E'.'AL  FACTO  SIHF  H  EDIT   CURS   BIG  -  E'.'AL  FACTO  SIHF 


Page  16-37 


Once  again,  replacing  einn  =  (-l)n,  results  in 


fciffllllTI^iaPC^WIllMMKHild 

This  result  is  used  to  define  the  function  c(n)  as  follows: 

DEFINE('c(n)  =  -  (((-1  )An-l  )/(nA2*7iA2*(-l  )An)') 


i.e., 


DEFINE 


d  £.  .  . .  n 
n  "TT  -C —  1  i 
HOVRU 


Next,  we  define  function  F(X,k,cO)  to  calculate  the  Fourier  series  (if  you 
completed  example  1,  you  already  have  this  function  stored): 

DEFINE('F(X,k,cO)  =  cO+Z(n=l  ,k,c(n)*EXP(2*i*7i*n*X/T)+ 
c(-n)*EXP(-(2*i*7i*n*X/T)),)/ 

To  compare  the  original  function  and  the  Fourier  series  we  can  produce  the 
simultaneous  plot  of  both  functions.  The  details  are  similar  to  those  of 
example  1,  except  that  here  we  use  a  horizontal  range  of  0  to  2  and  a 
vertical  range  from  0  to  1,  and  adjust  the  equations  to  plot  as  shown  here: 


plot  -  Funaiong 


Page  16-38 


The  resulting  graph  is  shown  below  for  k  =  5  (the  number  of  elements  in  the 
series  is  2k+l,  i.e.,  1  1,  in  this  case): 


From  the  plot  it  is  very  difficult  to  distinguish  the  original  function  from  the 
Fourier  series  approximation.  Using  k  =  2,  or  5  terms  in  the  series,  shows  not 
so  good  a  fitting: 


plot  -  Funcnon§ 


The  Fourier  series  can  be  used  to  generate  a  periodic  triangular  wave  (or  saw 
tooth  wave)  by  changing  the  horizontal  axis  range,  for  example,  from  -2  to  4. 
The  graph  shown  below  uses  k  =  5: 


Fourier  series  for  a  square  wave 

A  square  wave  can  be  generated  by  using  the  function 


g(x)  = 


0,  ifO<x<\ 

1,  ifl<x<3 
0,   if  3<x<4 


Page  1 6-39 


In  this  case,  the  period  T,  is  4.  Make  sure  to  change  the  value  of  variable 
Ilillll  to  4  (use:  GD  (£roT)  ■BQwra)).    Function  g(X)  can  be  defined  in  the 
calculator  by  using 


DEFINE('g(X)  =  IFTE((X>1 )  AND  (X<3),  1 ,0)') 

The  function  plotted  as  follows  (horizontal  range:  0  to  4,  vertical  range:0  to 
1.2): 


V 

1 

+ 

■ . .  J 

Using  a  procedure  similar  to  that  of  the  triangular  shape  in  example  2,  above, 
you  can  find  that 


1  f  fl. 


and 


:Cn)=f 


i-2-n-TT-X 


3'iTVTT  i-n-TT 

c(n)= 

2       .  2 
-i-e  +i-e 

i-n-TT  S-i-n-TT 

*          2  2 

H4tiHII«l4{WI4Mfll*MM14J:iJ 

ein7t/2 

=  in  and  e3in7l/2  =  (-i)n 

Page  1 6-40 


((-1)cnti),1),<i-") 

ci.ri.j-- 

&rviK-l)n 

nmisTnaRmraniramsimTa 

DEFINE 


c(n)= 


((_15(n+D 


+  1, 


2-n-TT-C-l 
HOVRL 


The  simplification  of  the  right-hand  side  of  c(n),  above,  is  easier  done  on 
paper  (i.e.,  by  hand).  Then,  retype  the  expression  for  c(n)  as  shown  in  the 
figure  to  the  left  above,  to  define  function  c(n).  The  Fourier  series  is  calculated 
with  F(X,k,cO),  as  in  examples  l  and  2  above,  with  cO  =  0.5.    For  example, 
for  k  =  5,  i.e.,  with  I  l  components,  the  approximation  is  shown  below: 


A  better  approximation  is  obtained  by  using  k  =  I  0,  i.e., 


For  k  =  20,  the  fitting  is  even  better,  but  it  takes  longer  to  produce  the  graph: 


Page  1 6-4 1 


Fourier  series  applications  in  differential  equations 

Suppose  we  want  to  use  the  periodic  square  wave  defined  in  the  previous 
example  as  the  excitation  of  an  undamped  spring-mass  system  whose 
homogeneous  equation  is:  d2y/dX2  +  0.25y  =  0. 

We  can  generate  the  excitation  force  by  obtaining  an  approximation  with  k 
=  10  out  of  the  Fourier  series  by  using  SW(X)  =  F(X,  10,0.5): 


:  DEF I  NE('SU(X)=F(X,  1 0  ■ ,  ■  5* 
HOVRL 

We  can  use  this  result  as  the  first  input  to  the  function  LDEC  when  used  to 
obtain  a  solution  to  the  system  d2y/dx2  +  0-25Y  =  SW(X),  where  SW(X) 
stands  for  Square  Wave  function  of  X.  The  second  input  item  will  be  the 
characteristic  equation  corresponding  to  the  homogeneous  ODE  shown  above, 
i.e.,  'XA2+0.25'  . 

With  these  two  inputs,  function  LDEC  produces  the  following  result  (decimal 
format  changed  to  Fix  with  3  decimals). 


:  LDEctsNCX^X2"  000+0.  25& 
C4.  019E-9-CC0+C0.  000,-3! 


D El" 'J LI  ILHF-    LAP  LDEC 


Pressing        allows  you  to  see  the  entire  expression  in  the  Equation  writer. 
Exploring  the  equation  in  the  Equation  Writer  reveals  the  existence  of  two 
constants  of  integration,  cCO  and  cCl .  These  values  would  be  calculated 
using  initial  conditions.  Suppose  that  we  use  the  values  cCO  =  0.5  and  cCl 
=  -0.5,  we  can  replace  those  values  in  the  solution  above  by  using  function 
SUBST  (see  Chapter  5).  For  this  case,  use  SUBST(ANS(1  )/cC0=0.5)  (enter}, 
followed  by  SUBST(ANS(1  ),cCl  =-0.5)  [enter}.  Back  into  normal  calculator 
display  we  can  use: 


Page  1 6-42 


[4. 019E-9-CC0+C0.  000j  -3! 
:  SUBSTCRHSC1 .  0003,cC0=0  > 

C4.  019E-9-0.  500+C0.  000^ 
:  SUBSTCRHSC1 . 000),cCl=-i> 

:4.  019E-9-0.  500+C0.  000/ 


The  latter  result  can  be  defined  as  a  function,  FW(X),  as  follows  (cutting  and 
pasting  the  last  result  into  the  command): 


C4. 019E-9-0. 500+E0. 000/ 

:  DEFIHe('fNCX)=C4.  019E-9* 
HOVRL 


We  can  now  plot  the  real  part  of  this  function.  Change  the  decimal  mode  to 
Standard,  and  use  the  following: 


Eaimg-aig^tiifi-HMiTif-^i-mi 

The  solution  is  shown  below: 


^»FL0T  HiriDOH  - 

Funaionem 

60. 

H.5 

Indcp  Loh:  EHZEIE 

Hi?h: Default 

Step:  DflfOUU 

_PiX«  If 

Enter  HiniHUH  indep 

Fourier  Transforms 

Before  presenting  the  concept  of  Fourier  transforms,  we'll  discuss  the  general 
definition  of  an  integral  transform.  In  general,  an  integral  transform  is  a 
transformation  that  relates  a  function  f(t)  to  a  new  function  F(s)  by  an 


Page  1 6-43 


integration  of  the  form  F(s)=     k{sJ)-  f{f)-dt.    The  function  k(s,t)  is 

%:J  J  a 

known  as  the  kernel  of  the  transformation. 

The  use  of  an  integral  transform  allows  us  to  resolve  a  function  into  a  given 
spectrum  of  components.  To  understand  the  concept  of  a  spectrum,  consider 
the  Fourier  series 

00 

fit)  =  a0  +  ^  (an  •  cos  conx  +  bn  •  sin  conx\ 

n=\ 


representing  a  periodic  function  with  a  period  T.   This  Fourier  series  can  be 

00 

re-written  as  f(x)  =  a0  +  ^  An  •  cos(ft7wx  +  (j)n  ),  where 


n=\ 


A  =  J  a2  +  b2 ,    d)  = 


tan" 


for  n  =1 ,2,  ... 

The  amplitudes  An  will  be  referred  to  as  the  spectrum  of  the  function  and  will 
be  a  measure  of  the  magnitude  of  the  component  of  f(x)  with  frequency  fn  = 
n/T.  The  basic  or  fundamental  frequency  in  the  Fourier  series  is  f0  =  1/T,  thus, 
all  other  frequencies  are  multiples  of  this  basic  frequency,  i.e.,  fn  =  n-f0.  Also, 
we  can  define  an  angular  frequency,  con  =  2n7i/T  =  27i-fn  =       n-f0  =  n-co0, 
where  co0  is  the  basic  or  fundamental  angular  frequency  of  the  Fourier  series. 

Using  the  angular  frequency  notation,  the  Fourier  series  expansion  is  written 
as 

00 

f(x)  =  a0+YjAn'  cosOwx  +  (j)n ). 

n=\ 


GO 

=  a0  +  y£J(an  •  cos conx  +  bn  -sin^x) 

n=\ 


Page  1 6-44 


A  plot  of  the  values  An  vs.  con  is  the  typical  representation  of  a  discrete 
spectrum  for  a  function.  The  discrete  spectrum  will  show  that  the  function  has 
components  at  angular  frequencies  con  which  are  integer  multiples  of  the 
fundamental  angular  frequency  co0- 

Suppose  that  we  are  faced  with  the  need  to  expand  a  non-periodic  function 
into  sine  and  cosine  components.  A  non-periodic  function  can  be  thought  of 
as  having  an  infinitely  large  period.  Thus,  for  a  very  large  value  of  T,  the 
fundamental  angular  frequency,  co0  =  27i/T,  becomes  a  very  small  quantity, 
say  Aco.  Also,  the  angular  frequencies  corresponding  to  con  =  n-co0  =  n-Aco, 
(n  =  1,  2,  oo),  now  take  values  closer  and  closer  to  each  other,  suggesting 
the  need  for  a  continuous  spectrum  of  values. 

The  non-periodic  function  can  be  written,  therefore,  as 

/•OO 

f(x)=    [C(a))'COs(a>'x)  + S(a))'Sm(a>'x)]da>, 
Jo 


where 


anc 


1  r°° 

C(co)  =  / (x)  •  cos(&>  •  x)  •  dx, 

In  J-°° 

1     r  °° 

S(cd)  =  /  (x)  •  sin(&>  -x)-dx 

2n  J-°° 

The  continuous  spectrum  is  given  by 

A(<o)  =  ^[C(6))]2+[S(cd)]2 

The  functions  C(co),  S(co),  and  A(co)  are  continuous  functions  of  a  variable  co, 
which  becomes  the  transform  variable  for  the  Fourier  transforms  defined 
below. 

Example  1  -  Determine  the  coefficients  C(co),  S(co),  and  the  continuous 
spectrum  A(a>),  for  the  function  f(x)  =  exp(-x),  for  x  >  0,  and  f(x)  =  0,  x  <  0. 


Page  1 6-45 


In  the  calculator,  set  up  and  evaluate  the  following  integrals  to  calculate  C(co) 


and  S(co 


,  respectively.  CAS  modes  are  set  to  Exact  and  Real 


1 

2-ir" 


e  X-COS(>x)dx 


e  X-SIH(wx)dx-4 


Their  results  are,  respectively: 


The  continuous  spectrum,  A(co)  is  calculated  as: 


1 


Define  this  expression  as  a  function  by  using  function  DEFINE  ). 
Then,  plot  the  continuous  spectrum,  in  the  range  0  <  <d  <  10,  as: 


Definition  of  Fourier  transforms 

Different  types  of  Fourier  transforms  can  be  defined.  The  following  are  the  definitions 
of  the  sine,  cosine,  and  full  Fourier  transforms  and  their  inverses  used  in  this  Chapter: 


Fourier  sine  transform 


Page  1 6-46 


Fs  {fit)}  =  F{cd)  =  -  •  f  7(0  •  sin(^  •  t)  •  dt 
n  J0 

Inverse  sine  transform 

F;1  {F(co)}  =  f(t)  =  [f(cd)  •  sin(«  -t)-dt 
Fourier  cosine  transform 

Fc  {fit)}  =  F(a>)  =  -  •  f  7(0  •  cos(*>  •  0  •  dt 
n  Jo 

Inverse  cosine  transform 

F;1  {F(a>)}  =  f(t)  =  {>(«)  •  cos(«  -t)-dt 

Fourier  transform  (proper) 

F  {fit) }  =  F{a>)  =  -1-  •  f/(0 '  e4"  •  dt 
In  J-G0 

Inverse  Fourier  transform  (proper) 

F"1  {Fico)}  =  fit)  =  f°V(<»)  •  e~im  ■  dt 

J  -00 

Example  1  -  Determine  the  Fourier  transform  of  the  function  f(t)  =  exp(- 1),  for  t 
>0,  and  f(t)  =  0,  for  t<0. 

The  continuous  spectrum,  F(co),  is  calculated  with  the  integral: 

—  fV1+to)^  =  lim—  Ve^dt 
In  Jo  s^2n  Jo 

_L  1 

In  l  +  ico 

This  result  can  be  rationalized  by  multiplying  numerator  and  denominator  by 
the  conjugate  of  the  denominator,  namely,  1-ico.  The  result  is  now: 


lira— 
In 


1  -  exp(-(l  +  ico)s) 
\  +  ia> 


Page  1 6-47 


1  (  1 


CO 


InKX  +  co1 


\  +  co2  J 


which  is  a  complex  function. 

The  absolute  value  of  the  real  and  imaginary  parts  of  the  function  can  be 
plotted  as  shown  below 


Notes: 

The  magnitude,  or  absolute  value,  of  the  Fourier  transform,  |  F(co)  | ,  is  the 
frequency  spectrum  of  the  original  function  f(t).  For  the  example  shown  above, 
|F(co)|  =  1/[2ti(1 +co2)]1/2.  The  plot  of  |  F(co)  |  vs.  co  was  shown  earlier. 

Some  functions,  such  as  constant  values,  sin  x,  exp(x),  x2,  etc.,  do  not  have 
Fourier  transform.  Functions  that  go  to  zero  sufficiently  fast  as  x  goes  to 
infinity  do  have  Fourier  transforms. 


Properties  of  the  Fourier  transform 

Linearity:  If  a  and  b  are  constants,  and  f  and  g  functions,  then  F{a-f  +  b-g}  = 


Transformation  of  partial  derivatives.  Let  u  =  u(x,t).  If  the  Fourier  transform 
transforms  the  variable  x,  then 


aF{f}+bF{g}. 


F{5u/5x}  =  ico  F{u},  F{a2u/dx2}  =  -co2  F{u}, 
F{5u/3t}  =  3F{u}M  F{52u/3t2}  =  52F{u}/5t2 


Page  1 6-48 


convolution:  For  Fourier  transform  applications,  the  operation  of  convolution 
is  defined  as 


The  following  property  holds  for  convolution: 


F{f*g}  =  F{f}-F{g}. 


Fast  Fourier  Transform  (FFT) 


The  Fast  Fourier  Transform  is  a  computer  algorithm  by  which  one  can 
calculate  very  efficiently  a  discrete  Fourier  transform  (DFT).   This  algorithm 
has  applications  in  the  analysis  of  different  types  of  time-dependent  signals, 
from  turbulence  measurements  to  communication  signals. 

The  discrete  Fourier  transform  of  a  sequence  of  data  values  {xj},  j  =  0,  1, 
2,       n-l,  is  a  new  finite  sequence  {Xk},  defined  as 


The  direct  calculation  of  the  sequence  Xk  involves  n2  products,  which  would 
involve  enormous  amounts  of  computer  (or  calculator)  time  particularly  for 
large  values  of  n.  The  Fast  Fourier  Transform  reduces  the  number  of 
operations  to  the  order  of  n-log2n.     For  example,  for  n  =  1 00,  the  FFT 
requires  about  664  operations,  while  the  direct  calculation  would  require 
1  0,000  operations.  Thus,  the  number  of  operations  using  the  FFT  is  reduced 
by  a  factor  of  1 0000/664  «  1 5. 

The  FFT  operates  on  the  sequence  {xj}  by  partitioning  it  into  a  number  of 
shorter  sequences.  The  DFT's  of  the  shorter  sequences  are  calculated  and 
later  combined  together  in  a  highly  efficient  manner.    For  details  on  the 
algorithm  refer,  for  example,  to  Chapter  1  2  in  Newland,  D.E.,  1  993,  "An 


Page  1 6-49 


Introduction  to  Random  Vibrations,  Spectral  &  Wavelet  Analysis  -  Third 
Edition/'  Longman  Scientific  and  Technical,  New  York. 

The  only  requirement  for  the  application  of  the  FFT  is  that  the  number  n  be  a 
power  of  2,  i.e.,  select  your  data  so  that  it  contains  2,  4,  8,  1  6,  32,  62,  etc., 
points. 

Examples  of  FFT  applications 

FFT  applications  usually  involve  data  discretized  from  a  time-dependent  signal. 
The  calculator  can  be  fed  that  data,  say  from  a  computer  or  a  data  logger,  for 
processing.    Or,  you  can  generate  your  own  data  by  programming  a 
function  and  adding  a  few  random  numbers  to  it. 

Example  1  -  Define  the  function  f(x)  =  2  sin  (3x)  +  5  cos(5x)  +  0.5* RAND, 
where  RAND  is  the  uniform  random  number  generator  provided  by  the 
calculator.  Generate  1  28  data  points  by  using  values  of  x  in  the  interval 
(0,1  2.8).  Store  those  values  in  an  array,  and  perform  a  FFT  on  the  array. 

First,  we  define  the  function  f(x)  as  a  RPN  program: 

«  ^  x  '2*SIN(3*x)  +  5*COS(5*x)'  EVAL  RAND  5  *  +  ^NUM  » 

and  store  this  program  in  variable  I!I!II.    Next,  type  the  following  program  to 
generate  2m  data  values  between  a  and  b.  The  program  will  take  the  values 
of  m,  a,  and  b: 

«  ^  m  a  b  «  '2Am'  EVAL     n  «  '(b-a)/(n+l )'  EVAL  ^  Dx  «  1  n  FOR  j 
'a+(j-l)*Dx'  EVAL  f  NEXT  n  ^ARRY  »  »  »  » 

Store  this  program  under  the  name  GDATA  (Generate  DATA).  Then,  run  the 
program  for  the  values,  m  =  5,  a  =  0,  b  =  1  00.  In  RPN  mode,  use: 

CD  CD  QD  CD  CD  CD  CD  HnMI 

The  figure  below  is  a  box  plot  of  the  data  produced.  To  obtain  the  graph, 
first  copy  the  array  just  created,  then  transform  it  into  a  column  vector  by 
using:  OBJ^  CD  CD  ^ARRY  (Functions  OBJ^  and  -^ARRY  are  available 


Page  1 6-50 


in  the  command  catalog,  Lr>J  cat  ).  Store  the  array  into  variable  ZDAT  by 
using  function  STOZ  (also  available  through  L  r>  j  cat  ).    Select  Bar  in  the 
TYPE  for  graphs,  change  the  view  window  to  H-VIEW:  0  32,  V-VIEW:  -10  10, 
and  BarWidth  to  1 .    Press  fillilllL^J  to  return  to  normal  calculator  display. 


jilUjJJu 

1  n  | 

To  perform  the  FFT  on  the  array  in  stack  level  1  use  function  FFT  available  in 
the  MTH/FFT  menu  on  array  ZDAT:  Ellllii!  FFT.    The  FFT  returns  an  array  of 
complex  numbers  that  are  the  arrays  of  coefficients  Xk  of  the  DFT.  The 
magnitude  of  the  coefficients  Xk  represents  a  frequency  spectrum  of  the 
original  data.  To  obtain  the  magnitude  of  the  coefficients  you  could  transform 
the  array  into  a  list,  and  then  apply  function  ABS  to  the  list.  This  is 
accomplished  by  using:  OBJ^    (W)  ^LIST  C3D^ 

Finally,  you  can  convert  the  list  back  to  a  column  vector  to  be  stored  in  ZDAT, 
as  follows:  OBJ^  CD        CD  ^LIST  ^ARRY  STOZ 

To  plot  the  spectrum,  follow  the  instructions  for  producing  a  bar  plot  given 
earlier.  The  vertical  range  needs  to  be  changed  to  -1  to  80.  The  spectrum  of 
frequencies  is  the  following: 


1 

mIm.I.IIiI 

ii.ii 

The  spectrum  shows  two  large  components  for  two  frequencies  (these  are  the 
sinusoidal  components,  sin  (3x)  and  cos(5x)),  and  a  number  of  smaller 
components  for  other  frequencies. 


Page  16-51 


Example  2  -  To  produce  the  signal  given  the  spectrum,  we  modify  the 
program  GDATA  to  include  an  absolute  value,  so  that  it  reads: 

«  ^  m  a  b  «  '2Am'  EVAL  ^  n  «  '(b-a)/(n+l )'  EVAL  ^  Dx  «  1  n  FOR  j 
'a+(j-l)*Dx'  EVAL  f  ABS  NEXT  n  ^ARRY  »  »  »  » 

Store  this  version  of  the  program  under  GSPEC  (Generate  SPECtrum).  Run  the 
program  with  m  =  6,  a  =  0,  b  =  100.  In  RPN  mode,  use: 

□D  Cz3  QD  (jfD  CD  QD  QD  mSm 

Press  [enter]  when  done,  to  keep  an  additional  copy  of  the  spectrum  array. 
Convert  this  row  vector  into  a  column  vector  and  store  it  into  ZDAT.  Following 
the  procedure  for  generating  a  bar  plot,  the  spectrum  generated  for  this 
example  looks  as  shown  below.  The  horizontal  range  in  this  case  is  0  to  64, 
while  the  vertical  range  is  -1  to  1  0: 


.V 

,ll 

i,i 

li 

1 

.1 

1.1 

.1 

J 

ill 

n 

-L 

CM 

To  reproduce  the  signal  whose  spectrum  is  shown  ,  use  function  IFFT.  Since 
we  left  a  copy  of  the  spectrum  in  the  stack  (a  row  vector),  all  you  need  to  do  if 
find  function  IFFT  in  the  MTH/FFT  menu  or  through  the  command  catalog, 
L r>  J  cat  .  As  an  alternative,  you  could  simply  type  the  function  name,  i.e., 
type  [ai^{alpha}(I} (7}  (T) (t}[enter).  The  signal  is  shown  as  an  array  (row  vector) 
with  complex  numbers.  We  are  interested  only  in  the  real  part  of  the  elements. 
To  extract  the  real  part  of  the  complex  numbers,  use  function  RE  from  the 
CMPLX  menu  (see  Chapter  4),  e.g.,  type  {al^{alp^}(r](e}(e^r}  .  What  results  is 
another  row  vector.  Convert  it  into  a  column  vector,  store  it  into  SDAT,  and 
plot  a  bar  plot  to  show  the  signal.  The  signal  for  this  example  is  shown  below, 
using  a  horizontal  range  of  0  to  64,  and  a  vertical  range  of  -1  to  1 : 


Page  16-52 


V 

Jj-'4l.i..-,i'"J'i 

I,iI..i,i,.„i.  J.L.J.h. 

"   1  r  en. 

Except  for  a  large  peak  at  t  =  0,  the  signal  is  mostly  noise.  A  smaller  vertical 
scale  (-0.5  to  0.5)  shows  the  signal  as  follows:  


V 

It- , 

ll.Ul 

1, 

..J. 

Solution  to  specific  second-order  differential  equations 

In  this  section  we  present  and  solve  specific  types  of  ordinary  differential 
equations  whose  solutions  are  defined  in  terms  of  some  classical  functions, 
e.g.,  Bessel's  functions,  Hermite  polynomials,  etc.  Examples  are  presented  in 
RPN  mode. 

The  Cauchy  or  Euler  equation 

An  equation  of  the  form  x2-(d2y/dx2)  +  ax-  (dy/dx)  +  by  =  0,  where  a  and  b 
are  real  constants,  is  known  as  the  Cauchy  or  Euler  equation.  A  solution  to 
the  Cauchy  equation  can  be  found  by  assuming  that  y(x)  =  xn. 
Type  the  equation  as:  'xA2*dldly(x)+a*x*dly(x)+b*y(x)=0'  (enter} 
Then,  type  and  substitute  the  suggested  solution:  'y(x)  =  xAn'  {enter)  i!]3Iil 

The  result  is:  'xA2*(n*(xA(n-l -1  )*(n-l  )))+a*x*(n*xA(n-l  ))+b*xAn  =0,  which 
simplifies  to  'n*(n-l  )*xAn+a*n*xAn+b*xAn  =  0'.    Dividing  by  xAn,  results  in 
an  auxiliary  algebraic  equation:  'n*(n-l  )+a*n+b  =  0',  or. 

n2  +(a-\)-n  +  b  =  0  . 

•  If  the  equation  has  two  different  roots,  say     and  n2,  then  the  general 
solution  of  this  equation  is  y(x)  =  Krx  ^  +  K2-x  n2. 

•  If  b  =  (l-a)2/4,  then  the  equation  has  a  double  root  n}  =  n2  =  n  = 
(l-a)/2,  and  the  solution  turns  out  to  be   y(x)  =  (K}  +  K2-ln  x)xn. 


Page  16-53 


Legendre's  equation 

An  equation  of  the  form  (1 -x2)-(d2y/dx2)-2-x-  (dy/dx)+n-  (n+1)  -y  =  0,  where  n 
is  a  real  number,  is  known  as  the  Legendre's  differential  equation.  Any 
solution  for  this  equation  is  known  as  a  Legendre's  function.  When  n  is  a 
nonnegative  integer,  the  solutions  are  called  Legendre's  polynomials. 
Legendre's  polynomial  of  order  n  is  given  by 


M  (2n-2m)\ 


Pn(x)  =  Y(-l)m  K  }'  -xn~2m 

m=0  2n  -  m\-(n  -  m)\'{n  -  2m)l 


(2/i)!  (2/1-2)!  , 

X       \  ... 


2"-(n\y  2"-l!-(w-l)!(»-2)! 
where  M  =  n/2  or  (n-1  )/2,  whichever  is  an  integer. 

Legendre's  polynomials  are  pre-programmed  in  the  calculator  and  can  be 
recalled  by  using  the  function  LEGENDRE  given  the  order  of  the  polynomial,  n. 
The  function  LEGENDRE  can  be  obtained  from  the  command  catalog 
(CB-££  )  or  through  the  menu  ARITHMETIC/POLYNOMIAL  menu  (see 
Chapter  5).  In  RPN  mode,  the  first  six  Legendre  polynomials  are  obtained  as 
follows: 

0  LEGENDRE,  result:  1,  i.e.,      P0(x)  =  1.0. 

1  LEGENDRE,  result:  X,  i.e.,      P^x)  =  x. 

2  LEGENDRE,  result:  '(3*XA2-1  )/2',  i.e.,      P2(x)  =  (3x2-l)/2. 

3  LEGENDRE,  result:  '(5*XA3-3*X)/2',  i.e.,      P3(x)  =(5x3-3x)/2. 

4  LEGENDRE,  result:  '(35*XA4-30*XA2+3)/8',  i.e., 

P4(x)  =(35x4-30x2+3)/8. 

5  LEGENDRE,  result:  '(63*XA5-70*XA3+1 5*X)/8',  i.e., 

P5(x)  =(63x5-70x3+15x)/8. 

The  ODE   (l-x2)-(d2y/dx2)-2-x-  (dy/dx)+[n-  (n+l)-m2/(l-x2)]  y  =  0,  has  for 
solution  the  function  y(x)  =  Pnm(x)=  (1-x2)m/2-(dmPn/dxm).  This  function  is 
referred  to  as  an  associated  Legendre  function. 


Page  1 6-54 


Bessel's  equation 

The  ordinary  differential  equation  x2-(d2y/dx2)  +  x-  (dy/dx)+  (x2-v2)  -y  =  0, 
where  the  parameter  v  is  a  nonnegative  real  number,  is  known  as  Bessel's 
differential  equation.  Solutions  to  Bessel's  equation  are  given  in  terms  of 
Bessel  functions  of  the  first  kind  of  order  v: 


Jv(x)  =  xv 


y — 

r\  2m+v 
m=0  Z 


^  2m 


ht  

•  m\-T{v  +  m  + 1) 


where  v  is  not  an  integer,  and  the  function  Gamma  r(a)  is  defined  in  Chapter 
3. 


If  v  =  n,  an  integer,  the  Bessel  functions  of  the  first  kind  for  n  =  integer  are 
defined  by 


tl2m+n  -m\{n  +  m)\ 


Regardless  of  whether  we  use  v  (non-integer)  or  n  (integer)  in  the  calculator, 
we  can  define  the  Bessel  functions  of  the  first  kind  by  using  the  following  finite 
series: 


H_Ba  -H!-(n+H>! 


Thus,  we  have  control  over  the  function's  order,  n,  and  of  the  number  of 
elements  in  the  series,  k.  Once  you  have  typed  this  function,  you  can  use 
function  DEFINE  to  define  function  J(x,n,k).  This  will  create  the  variable  !!!!!!!!!!!!!!!! 
in  the  soft-menu  keys.  For  example,  to  evaluate  J3(0.1)  using  5  terms  in  the 
series,  calculate  J(0. 1 ,3,5),  i.e.,  in  RPN  mode:  CZDCTC^CBC^COi 
The  result  is  2.082031 57E-5. 


Page  1 6-55 


If  you  want  to  obtain  an  expression  for  J0(x)  with,  say,  5  terms  in  the  series, 
use  J(x,0,5).    The  result  is 

'l-0.25*xA3+0.015625*xA4-4.3403777E-4*xA6+6.782168E-6*xA8- 

6.78168*xA10'. 

For  non-integer  values  v,  the  solution  to  the  Bessel  equation  is  given  by 

y(x)  =  KrJv(x)+K2.J.v(x). 

For  integer  values,  the  functions  Jn(x)  and  J-n(x)  are  linearly  dependent,  since 

J„(x)  =  (-l)\l.„(x), 

therefore,  we  cannot  use  them  to  obtain  a  general  function  to  the  equation. 
Instead,  we  introduce  the  Bessel  functions  of  the  second  kind  defined  as 
Yv(x)  =  [Jv(x)  cos  V7i  -  J_v(x)]/sin  vn, 

for  non-integer  v,  and  for  n  integer,  with  n  >  0,  by 

Yn(x)  =  —  -Jn(x)-(ln-  +  y)  +  L^mTn  u   ~  'x 

7i  2  n  tTo  2      'm\\m  +  n)\ 


-n  n-\ 

-—■I 


(n-m-\)\  x2m 


n  22m-"-m\ 
where  y  is  the  Euler  constant,  defined  by 

y  =  lim[l  +  -  +  -  +  ...  +  --  In r]  «  0.57721 566490..., 

r^co      2    3  r 

and  hm  represents  the  harmonic  series 

hm  =1  +  -  +  -  +  ...  +  — 
2    3  m 

For  the  case  n  =  0,  the  Bessel  function  of  the  second  kind  is  defined  as 


Y0(x)  =  - 

71 


j  (x)  •  (In-  +  r)  +  Y  i-^  \  -x2m 


Page  1 6-56 


With  these  definitions,  a  general  solution  of  Bessel's  equation  for  all  values  of 
v  is  given  by  y(x)  =  KrJv(x)+K2-Yv(x). 


In  some  instances,  it  is  necessary  to  provide  complex  solutions  to  Bessel's 
equations  by  defining  the  Bessel  functions  of  the  third  kind  of  order  v  as 

Hn(1)(x)  =Jv(x)+i-Yv(x),  and  Hn<2'(x)  =  Jv(x)-i-Yv(x), 

These  functions  are  also  known  as  the  first  and  second  Hankel  functions  of 
order  v. 

In  some  applications  you  may  also  have  to  utilize  the  so-called  modified 
Bessel  functions  of  the  first  kind  of  order  v  defined  as  lv(x)=  i"v-Jv(i-x),  where  i  is 
the  unit  imaginary  number.  These  functions  are  solutions  to  the  differential 
equation  x2-(d2y/dx2)  +  x-  (dy/dx)-  (x2+v2)  y  =  0. 

The  modified  Bessel  functions  of  the  second  kind, 

Kv(x)  =  (tt/2)-[U  (xK  (x)]/sin  vtt, 

are  also  solutions  of  this  ODE. 

You  can  implement  functions  representing  Bessel's  functions  in  the  calculator 
in  a  similar  manner  to  that  used  to  define  Bessel's  functions  of  the  first  kind, 
but  keeping  in  mind  that  the  infinite  series  in  the  calculator  need  to  be 
translated  into  a  finite  series. 

Chebyshev  or  Tchebycheff  polynomials 

The  functions  Tn(x)  =  cos(n-cos"]  x),  and    Un(x)  =  sin[(n+l)  cos"1  x]/(l-x2)1/2,  n 
=  0,  1,  ...  are  called  Chebyshev  or  Tchebycheff  polynomials  of  the  first  and 
second  kind,  respectively.  The  polynomials  Tn(x)  are  solutions  of  the 
differential  equation  (l-x2)-(d2y/dx2)  -  x-  (dy/dx)  +  n2-y  =  0. 

In  the  calculator  the  function  TCHEBYCHEFF  generates  the  Chebyshev  or 
Tchebycheff  polynomial  of  the  first  kind  of  order  n,  given  a  value  of  n  >  0.  If 
the  integer  n  is  negative  (n  <  0),  the  function  TCHEBYCHEFF  generates  a 
Tchebycheff  polynomial  of  the  second  kind  of  order  n  whose  definition  is 


Page  1 6-57 


Un(x)  =  sin(n-arccos(x))/sin(arccos(x)). 

You  can  access  the  function  TCHEBYCHEFF  through  the  command  catalog 

(C3^ ). 

The  first  four  Chebyshev  or  Tchebycheff  polynomials  of  the  first  and  second 
kind  are  obtained  as  follows: 


0  TCHEBYCHEFF,  result:  1 , 

i.e., 

ToM 

=  1.0. 

-0  TCHEBYCHEFF,  result:  1, 

i.e., 

UoM 

=  1.0. 

1  TCHEBYCHEFF,  result:  X, 

i.e., 

T,(x) 

=  X. 

-1  TCHEBYCHEFF,  result:  1, 

i.e., 

U,(x) 

=  1.0. 

2  TCHEBYCHEFF,  result:  '2*XA2-1, 

i.e., 

T2(x) 

=2x2-l . 

-2  TCHEBYCHEFF,  result:  '2*X', 

i.e., 

U2(x) 

=2x. 

3  TCHEBYCHEFF,  result:  '4*XA3-3*X', 

i.e., 

T3(x) 

=  4x3-3x. 

-3  TCHEBYCHEFF,  result:  '4*XA2-T, 

i.e., 

U3(x) 

=  4x2-l . 

Laguerre's  equation 

Laguerre's  equation  is  the  second-order,  linear  ODE  of  the  form  x-(d2y/dx2) 
+(l-x)-  (dy/dx)  +  ny  =  0.  Laguerre  polynomials,  defined  as 

L0(x)  =  1,    Ln{x)  =  —  —  ,  n  =  1,2V.., 

n\  ax 

are  solutions  to  Laguerre's  equation.  Laguerre's  polynomials  can  also  be 
calculated  with: 


»=o  ml 


n(n-l)    2  (-I)"  „ 

=  l-n-x  +  —  --x  -...  +  ....  +  - — —-x 

4  n\ 


The  term 


=  C(njn) 


m\(n  —  m)\ 


Page  16-58 


is  the  m-th  coefficient  of  the  binomial  expansion  (x+y)n.  It  also  represents  the 
number  of  combinations  of  n  elements  taken  m  at  a  time.  This  function  is 
available  in  the  calculator  as  function  COMB  in  the  MTH/PROB  menu  (see 
also  Chapter  1  7). 

You  can  define  the  following  function  to  calculate  Laguerre's  polynomials: 

m=0 

When  done  typing  it  in  the  equation  writer  press  use  function  DEFINE  to 
create  the  function  L(x,n)  into  variable  IIII  . 

To  generate  the  first  four  Laguerre  polynomials  use,  L(x,0),  L(x,  1),  L(x,2),  L(x,3). 
The  results  are: 

Lo(x)=  • 

L,(x)=  1-x. 

L2(x)=  l-2x+  0.5x2 

L3(x)  =  l-3x+1.5x2-0.16666...x3. 

Weber's  equation  and  Hermite  polynomials 

Weber's  equation  is  defined  as  d2y/dx2+(n+l/2-x2/4)y  =  0,  for  n  =  0,  1, 
2,  ...  A  particular  solution  of  this  equation  is  given  by  the  function  ,  y(x)  = 
exp(-x2/4)H*(x/V2),  where  the  function  H*(x)  is  the  Hermite  polynomial: 

H0*  =  l,   Hn*(x)  =  (-l)"ex2^(e-x2),   n  =  l,2,.. 

ax 

In  the  calculator,  the  function  HERMITE,  available  through  the  menu 
ARITHMETIC/POLYNOMIAL.  Function  HERMITE  takes  as  argument  an  integer 
number,  n,  and  returns  the  Hermite  polynomial  of  n-th  degree.  For  example, 
the  first  four  Hermite  polynomials  are  obtained  by  using: 


Page  1 6-59 


0  HERMITE,  result:  1, 

1  HERMITE,  result:  ,2*X,/ 

2  HERMITE,  result:  ,4*XA2-2,/ 

3  HERMITE,  result:  '8*XA3-1  2*X', 


i.e.,  H0*  =  1. 
i.e.,  H]*  =  2x. 
i.e.,  H2*  =  4x2-2. 
i.e.,  H3*  =  8x3-12x. 


Numerical  and  graphical  solutions  to  ODEs 

Differential  equations  that  cannot  be  solved  analytically  can  be  solved 
numerically  or  graphically  as  illustrated  below. 

Numerical  solution  of  first-order  ODE 

Through  the  use  of  the  numerical  solver  (L  r>  Jnumslv  )f  you  can  access  an  input 
form  that  lets  you  solve  first-order,  linear  ordinary  differential  equations.  The 
use  of  this  feature  is  presented  using  the  following  example.  The  method  used 
in  the  solution  is  a  fourth-order  Runge-Kutta  algorithm  preprogrammed  in  the 
calculator. 

Example  1  --  Suppose  we  want  to  solve  the  differential  equation,  dv/dt  =  -1 .5 
v1/2,  with  v  =  4  at  t  =  0.  We  are  asked  to  find  v  for  t  =  2. 

First,  create  the  expression  defining  the  derivative  and  store  it  into  variable 
EQ.  The  figure  to  the  left  shows  the  ALG  mode  command,  while  the  right- 
hand  side  figure  shows  the  RPN  stack  before  pressing  Isto>j  . 


:-1.5-JI>EQ 

3: 

-1.5-Jy 

l: 

1  EQ  1 

Then,  enter  the  NUMERICAL  SOLVER  environment  and  select  the  differential 
equation  solver:  [j^Jnum.slv       jjjjjjilijTjjjjjjj  .  Enter  the  following  parameters: 


SOLVE  V'(T)=F(TJV) 

f-  '-1.5*-J"u' 
Ind«p=t    lnit:0        Final  2 
l'mI.ti:        Ini+:4  Final.| 
TOU,  000  1    Step  =  Df  1 1     _  Stiff 

Press  SOLVE  For  Final.  sol.n  ual.u« 


Page  1 6-60 


To  solve,  press:  BBEB  (wait)  EMM.  The  result  is  0.2499  «  0.25.  Press  H. 
Solution  presented  as  a  table  of  values 

Suppose  we  wanted  to  produce  a  table  of  values  of  v,  for  t  =  0.00,  0.25, 
2.00,  we  will  proceed  as  follows: 

First,  prepare  a  table  to  write  down  your  results.  Write  down  in  your  table  the 
step-by-step  results:  


t 

V 

0.00 

0.00 

0.25 

2.00 

Next,  within  the  SOLVE  environment,  change  the  final  value  of  the 
independent  variable  to  0.25,  use  : 

^.25IHaiCV)a)  MSB  (wait)  B3H 
(Solves  for  v  at  t  =  0.25,  v  =  3.285  ....  ) 

lini  lilllll  ^  .  5  lail  CTC^Iill(wait)!EiI 

(Changes  initial  value  of  t  to  0.25,  and  final  value  of  t  to  0.5,  solve  for  v(0.5) 
=  2.640...) 

tan  fssso^  .75naa5C& ana  (wait)  mm 

(Changes  initial  value  of  t  to  0.5,  and  final  value  of  t  to  0.75,  solve  for  v(0.75) 
=  2.066...) 

IIIElI:!!!!!  1  jjjjijijjjjjjj  O)  05  B3H33  (wait)  lEiEEiEii 

(Changes  initial  value  of  t  to  0.75,  and  final  value  of  t  to  1,  solve  for  v(l)  = 
1.562...) 

Repeat  for  t  =  1 .25,  1 .50,  1 .75,  2.00.  Press  ill!  after  viewing  the  last  result 
in  iililillii.  To  return  to  normal  calculator  display,  press  l^EJ  or  C^3Ilii]Ii.  The 
different  solutions  will  be  shown  in  the  stack,  with  the  latest  result  in  level  1 . 

The  final  results  look  as  follows  (rounded  to  the  third  decimal): 


Page  16-61 


t 

V 

0.00 

4.000 

0.25 

3.285 

0.50 

2.640 

0.75 

2.066 

1.00 

1.562 

1.25 

1.129 

1.50 

0.766 

1.75 

0.473 

2.00 

0.250 

Graphical  solution  of  first-order  ODE 

When  we  can  not  obtain  a  closed-form  solution  for  the  integral,  we  can 
always  plot  the  integral  by  selecting  Dif  f  Eq  in  the  TYPE  field  of  the  PLOT 
environment  as  follows:  suppose  that  we  want  to  plot  the  position  x(t)  for  a 
velocity  function  v(t)  =  exp(-t2),  with  x  =  0  at  t  =  0.  We  know  there  is  no 
closed-form  expression  for  the  integral,  however,  we  know  that  the  definition 
of  v(t)  is  dx/dt  =  exp(-t2). 

The  calculator  allows  for  the  plotting  of  the  solution  of  differential  equations  of 
the  form  Y'(T)  =  F(T,Y).  For  our  case,  we  let  Y  =  x  and  T  =  t,  therefore,  F(T,Y) 
=  f(t,  x)  =  exp(-t2).  Let's  plot  the  solution,  x(t),  for  t  =  0  to  5,  by  using  the 
following  keystroke  sequence: 

•  IJnJi^  (simultaneously,  if  in  RPN  mode)  to  enter  PLOT  environment 

•  Highlight  the  field  in  front  of  TYPE,  using  the       ^r?  keys.    Then,  press 
EiCEEH,  and  highlight  Diff  Eq,  using  the  ^^^7  keys.  Press  139. 

•  Change  field  F:  to 'EXP(-tA2)' 

•  Make  sure  that  the  following  parameters  are  set  to:  H-VAR:   0,  V-VAR: 

1 

•  Change  the  independent  variable  to  t  . 

•  Accept  changes  to  PLOT  SETUP:  [nxt)  MM 

•  EDJ^L  (simultaneously,  if  in  RPN  mode).    To  enter  PLOT  WINDOW 
environment 

•  Change  the  horizontal  and  vertical  view  window  to  the  following  settings: 

H-VIEW:    -1         5;     V-VIEW:    -1  1.5 


Page  1 6-62 


•  Also,  use  the  following  values  for  the  remaining  parameters:  Init:  0,  Final: 
5,  Step:  Default,  Tol:  0.0001,  Init-Soln:  0 

•  To  plot  the  graph  use:  EEMi 


+ 


n  1 1 II I  m  IMTl  I  HI  I  hi  H 


When  you  observe  the  graph  being  plotted,  you'll  notice  that  the  graph  is  not 
very  smooth.  That  is  because  the  plotter  is  using  a  time  step  that  may  be  a  bit 
large  for  a  smooth  graph.  To  refine  the  graph  and  make  it  smoother,  use  a 
step  of  0.1.  Press  IIIililiiE!  and  change  the  Step  :  value  to  0.1,  then  use 
EuTuI!  once  more  to  repeat  the  graph.  The  plot  will  take  longer  to  be 
completed,  but  the  shape  is  definitely  smoother  than  before.  Try  the  following: 
IIIiMll  ®  EHUD  to  see  axes  labels  and  range. 


Notice  that  the  labels  for  the  axes  are  shown  as  0  (horizontal,  for  t)  and  1 
(vertical,  for  x).  These  are  the  definitions  for  the  axes  as  given  in  the  PLOT 
SETUP  window  (C5Di^  )  i.e.,  H-VAR:  0,  and  V-VAR:  1 .  To  see  the  graphical 
solution  in  detail  use  the  following: 

Inxt j Inxt j [III]!    To  recover  menu  and  return  to  PICT  environment. 
!l(:l:li!lli!il)ll  To  determine  coordinates  of  any  point  on  the  graph. 

Use  the  GDCD  ^eys  *°  move  *ne  cursor  around  the  plot  area.  At  the  bottom 
of  the  screen  you  will  see  the  coordinates  of  the  cursor  as  (X,Y),  i.e.,  the 
calculator  uses  X  and  Y  as  the  default  names  for  the  horizontal  and  vertical 
axes,  respectively.  Press  [nxtJ H3D31  to  recover  the  menu  and  return  to  the 
PLOT  WINDOW  environment.  Finally,  press  Ion  j  to  return  to  normal  display. 


i 


-i. 


5. 


-i. 


Page  1 6-63 


Numerical  solution  of  second-order  ODE 

Integration  of  second-order  ODEs  can  be  accomplished  by  defining  the 
solution  as  a  vector.  As  an  example,  suppose  that  a  spring-mass  system  is 
subject  to  a  damping  force  proportional  to  its  speed,  so  that  the  resulting 

d2x  dx 
differential  equation  is:  — —  =  —1 8.75  •  x  —  1 .962  

dt2  dt 


or, 


x"  =  -  18.75  x-  1.962  x1, 


subject  to  the  initial  conditions,  v  =  x1  =  6,  x  =  0,  at  t  =  0.  We  want  to  find  x, 
x1  att  =  2. 


Re-write  the  ODE  as:  w1  =  Aw,  where  w  =  [  x  x1  ]T,  and  A  is  the  2  x  2 
matrix  shown  below. 


0  1 
-18.75  -1.962 


The  initial  conditions  are  now  written  as  w  =  [0  6]T,  for  t  =  0.  (Note:  The 
symbol  [  ]T  means  the  transpose  of  the  vector  or  matrix). 

To  solve  this  problem,  first,  create  and  store  the  matrix  A,  e.g.,  in  ALG  mode: 


re  ii 

 1-13,75  -1-962| 


Then,  activate  the  numerical  differential  equation  solver  by  using:  L  r>  J  numslv 
^▼7  Iliiilllllll  .    To  solve  the  differential  equation  with  starting  time  t  =  0  and 
final  time  t  =  2,  the  input  form  for  the  differential  equation  solver  should  look 
as  follows  (notice  that  the  Init:  value  for  the  Soln:  is  a  vector  [0,  6]): 


Page  1 6-64 


WBBBB  solve  rm:F(U)™ 
f :  fl*w 

Ind«p=t    lnit:0        Final  2 

i.n :  w    Init :  [  0 . ,,.  Fina  I.  ^^^^| 
ToU,  000  1    St*  p  =  Df  1 1     _  Stiff 

Ffgjj  SOLVE  for  final  join  ualm 

Press  Mm  (wait)  BSD  to  solve  for  w(t=2).  The  solution  reads  [.1  671  6...  - 
.6271...],  i.e.,  x(2)  =  0.16716,  and  x'(2)  =  v(2)  =  -0.6271.  Press  EM!  to 
return  to  SOLVE  environment. 

Solution  presented  as  a  table  of  values 

In  the  previous  example  we  were  interested  only  in  finding  the  values  of  the 
position  and  velocity  at  a  given  time  t.  If  we  wanted  to  produce  a  table  of 
values  of  x  and  x1,  for  t  =  0.00,  0.25,  2.00,  we  will  proceed  as  follows: 
First,  prepare  a  table  to  write  down  your  results: 


t 

X 

x' 

0.00 

0.00 

6.00 

0.25 

2.00 

Next,  within  the  SOLVE  environment,  change  the  final  value  of  the 
independent  variable  to  0.25,  use: 

^  .25 En  nr.  nr.  iEIIilii!  (wait) 

(Solves  for  w  at  t  =  0.25,  w  =  [0.968  1 .368].  ) 

::::v:::::::     I":"1::":":™:    /~\  C    ™v:::::::  ~r      ::!:::v:":""":::    /     ,     'i\  i::::"1:::::": 

iiii:!:!.-:!!!!!   LSijLSLSa  C±^>   .  D  ilfcLLLlli         LJ^  ii::;i:::!.::h::!.::i  (WOlt)  sjius 

(Changes  initial  value  of  t  to  0.25,  and  final  value  of  t  to  0.5,  solve  again  for 
w(0.5)  =  [0.748  -2.61 6]) 

SCIISI  iQOllli      .75 l!l!!i]]]|li(T)  QT>  S3!3!!!CEii!  (wait)  IHIHil 

(Changes  initial  value  of  t  to  0.5,  and  final  value  of  t  to  0.75,  solve  again  for 
w(0.75)  =  [0.0147  -2.859]) 

::::v:':':::::    :::":"!::"::"::!::  "I  ::::v:::::i:   /  r-— ^     /-r--.  V--------:"::   /.  4 ,     "iX  i:"::"":"::": 

iiii:!:!.-:!!!!!  !!!.!!.!.!!.!!!.!!■:■!  (_^j>  I  bb:!:™  (JO  (WOlt)  UJUJ 

(Changes  initial  value  of  t  to  0.75,  and  final  value  of  t  to  1,  solve  again  for 
w(l)  =  [-0.469  -0.607]) 


Page  1 6-65 


Repeat  for  t  =  1 .25,  1 .50,  1 .75,  2.00.  Press  liill!!!  after  viewing  the  last  result 
in  iilliillii.  To  return  to  normal  calculator  display,  press  L0A/ J  or  jNxrj The 
different  solutions  will  be  shown  in  the  stack,  with  the  latest  result  in  level  1 . 
The  final  results  look  as  follows: 


t 

X 

x" 

t 

X 

x" 

0.00 

0.000 

6.000 

1.25 

-0.354 

1.281 

0.25 

0.968 

1.368 

1.50 

0.141 

1.362 

!  0.50 

0.748 

-2.616 

1.75 

0.227 

0.268 

0.75 

-0.015 

-2.859 

2.00 

0.167 

-0.627 

1.00 

-0.469 

-0.607 

Graphical  solution  for  a  second-order  ODE 

Start  by  activating  the  differential  equation  numerical  solver,  (j^J  numslv 
^3?  IISHII  .  The  SOLVE  screen  should  look  like  this: 

^» SOLVE  r(T):F(U)« 

f  =  R*w 

Ind«p=t    lnit:0        Final  2  

i.n  =  y    Init  =  [  0 .  ,„  Fina  L  B^JMll 
TOU,  000  1    Step  =  Df  1 1     _  StiFf 

Press  SOLVE  For  Final.  sol.n  ual.u« 


Notice  that  the  initial  condition  for  the  solution  (Soln:  w  Init: [0.,  ...)  includes 
the  vector  [0,  6].  Press  [nxtJ  mlM. 

Next,  press  GT3J^  (simultaneously,  if  in  RPN  mode)  to  enter  the  PLOT 
environment.  Highlight  the  field  in  front  of  TYPE,  using  the  </^N><^7  keys. 
Then,  press  H33SB,  and  highlight  Diff  Eq,  using  the  keys.  Press 

DII.    Modify  the  rest  of  the  PLOT  SETUP  screen  to  look  like  this: 


woooooooooooooooooow  F  L '.'  T  _■  E  T  U  F  .; 

F:fl-H 

_StiFF 

Ind«p: t 

H-TiCh=iD.  Y-TiCh:10. 

^Pixels 

Choos«  typ«  oF  put 

Page  1 6-66 


Notice  that  the  option  V-Var:  is  set  to  1,  indicating  that  the  first  element  in  the 
vector  solution,  namely,  x',  is  to  be  plotted  against  the  independent  variable  t. 
Accept  changes  to  PLOT  SETUP  by  pressing  [nxtJ  mm. 

Press  C5D ML  (simultaneously,  if  in  RPN  mode)  to  enter  the  PLOT  WINDOW 
environment.  Modify  this  input  form  to  look  like  this: 


bbbbbplot 

HinDOH  -  DIFF 

2.5 

S. 

Init:  0. 

Final.: 

2.S 

Step: 

.1             To  L 

: .0001 

Init-Sol.n: 

[LL. .■■:.] 

Enter  HiniHUH  horizontal. 

To  plot  the  x  vs.  t  graph  use:  !L;iLl:!!;!!§L;i!  13333  .   The  plot  of  x'  vs.  t  looks  like  this: 


To  plot  the  second  curve  we  need  to  use  the  PLOT  SETUP  input  form  once, 
more.  To  reach  this  form  from  the  graph  above  use:  HlClHII 
®  EB  QD (simultaneously,  if  in  RPN  mode)  .  Change  the  value  of  the 
V-Var:  field  to  2,  and  press  III:!!;!;]  (do  not  press  11133:10  or  you  would  loose  the 
graph  produced  above).  Use:  13333  C^D  IB3333I  CHEEI  to  see  axes  labels  and 
range.  Notice  that  the  x  axis  label  is  the  number  0  (indicating  the 
independent  variable),  while  the  y-axis  label  is  the  number  2  (indicating  the 
second  variable,  i.e.,  the  last  variable  plotted).  The  combined  graph  looks 
like  this: 


5.' 

4 

-5., 

Press  [Nxf^[Nxf)  lllllil  B3BS33  Q^J  to  return  to  normal  calculator  display. 


Page  1 6-67 


Numerical  solution  for  stiff  first-order  ODE 

Consider  the  ODE:  dy/dt  =  -1 00y+l  00t+l  01 ,  subject  to  the  initial  condition 
y(0)=l. 

Exact  solution 

This  equation  can  be  written  as  dy/dt  +  1 00  y  =  1  00  t  +  101,  and  solved 
using  an  integrating  factor,  IF(t)  =  exp(100t),  as  follows  (RPN  mode,  with  CAS 
set  to  Exact  mode): 

'(100*t+101)*EXP(100*t)'(™^  '\' (enter)  RISCH 

The  result  is  '(t+1  )*EXP(1 00*t)'. 

Next,  we  add  an  integration  constant,  by  using:  'C  (^DC+D 

Then,  we  divide  by  Fl(x),  by  using:    /EXP(100*t)/  (^GED . 

The  result  is:  /((t+l)*EXP(100*t)+C)/EXP(100*t)/,  i.e.,  y(t)  =  1+  t  +Ge100t.  Use 
of  the  initial  condition  y(0)  =  1,  results  in  1  =  1  +  0  +  Ge°,  or  C  =  0,  the 
particular  solution  being  y(t)  =  1+t. 

Numerical  solution 

If  we  attempt  a  direct  numerical  solution  of  the  original  equation  dy/dt  =  - 
1  00y+l  00t+l  01 ,  using  the  calculator's  own  numerical  solver,  we  find  that 
the  calculator  takes  longer  to  produce  a  solution  that  in  the  previous  first-order 
example.     To  check  this  out,  set  your  differential  equation  numerical  solver 

((JfJ  NUM.SLV  ^r?  lllilMllll)  to: 


SOLVE  V'(T)=F(TJV) 
F=        -100.*Y+100.*t  +  l 
Ind«p=t    lnit:0        Final  2 
So  In:  V    Ini+:1         Final | 
To  I :  .  000  1    St*  p  =  Df  1 1     _  Stiff 

Fr«s  SOLVE  For  Final  join  ualu* 


Page  1 6-68 


Here  we  are  trying  to  obtain  the  value  of  y(2)  given  y(0)  =  1 .    With  the  Soin: 
Final  field  highlighted,  press  B3SQ.    You  can  check  that  a  solution  takes 
about  6  seconds,  while  in  the  previous  first-order  example  the  solution  was 
almost  instantaneous.  Press  jQN J  to  cancel  the  calculation. 

This  is  an  example  of  a  stiff  ordinary  differential  equation.  A  stiff  ODE  is 
one  whose  general  solution  contains  components  that  vary  at  widely  different 
rates  under  the  same  increment  in  the  independent  variable.  In  this  particular 
case,  the  general  solution,  y(t)  =  1+  t  +Ge100t,  contains  the  components  Y 
and  'Ge100t/,  which  vary  at  very  different  rates,  except  for  the  cases  C=0  or 
C*0  (e.g.,  for  C  =  1,  t  =0.1,  Ce100t  =22026). 

The  calculator's  ODE  numerical  solver  allows  for  the  solution  of  stiff  ODEs  by 
selecting  the  option  _stiff  in  the  solve  y'  (t)  =  f(t,y)  screen.  With  this 
option  selected  you  need  to  provide  the  values  of  df/dy  and  df/dt.  For  the 
case  under  consideration  df/dy  =  -1  00  and  di/d\  =  1  00. 

Enter  those  values  in  the  corresponding  fields  of  the  solve  y'  (t)  =  f(t,y) 
screen: 

jKHKB  SOLVE  V'(T)=F(T,V)^ 
F  =  '  -1...  *F*y:  '  -1...  3Fit:  100 

Ind«p=t    lnit=0        Final  2  

So  In:  V  Init:l  Final  3E1 
T^:,  000  1    Step  =  Df  1 1     ^  StiFf 

Pr«ff  SOLVE  For  Final  join  ual.u« 


|iniT+|S0LYE 


When  done,  move  the  cursor  to  the  soin: Final  field  and  press  BSSQ.  This 
time,  the  solution  in  produced  in  about  1  second.  Press  llllllil  to  see  the 
solution:  2.9999999999,  i.e.,  3.0. 

Note:  The  option  stiff  is  also  available  for  graphical  solutions  of  differential 
equations. 

Numerical  solution  to  ODEs  with  the  SOLVE/DIFF  menu 

The  SOLVE  soft  menu  is  activated  by  using  74  MENU  in  RPN  mode.  This 
menu  is  presented  in  detail  in  Chapter  6.    One  of  the  sub-menus,  DIFF, 


Page  1 6-69 


contains  functions  for  the  numerical  solution  of  ordinary  differential  equations 
for  use  in  programming.  These  functions  are  described  next  using  RPN  mode 
and  system  flag  1  1  7  set  to  SOFT  menus. 

The  functions  provided  by  the  SOLVE/DIFF  menu  are  the  following: 


U  


Function  RKF 

This  function  is  used  to  compute  the  solution  to  an  initial  value  problem  for  a 
first-order  differential  equation  using  the  Runge-Kutta-Fehlbert  4th  -5th  order 
solution  scheme.  Suppose  that  the  differential  equation  to  be  solved  is  given 
by  dy/dx  =  f(x,y),  with  y  =  0  at  x  =  0,  and  that  you  will  allow  a  convergence 
criteria  s  for  the  solution.  You  can  also  specify  an  increment  in  the 
independent  variable,  Ax,  to  be  used  by  the  function.    To  run  this  function  you 
will  prepare  your  stack  as  follows: 

3:     {V,  y, 'f(x,y)'} 

2:  {c  Ax} 

1  :  Xfinal 

The  value  in  the  first  stack  level  is  the  value  of  the  independent  variable  where 
you  want  to  find  your  solution,  i.e.,  you  want  to  find,  yfina|  =  fs(xfina|),  where  fs(x) 
represents  the  solution  to  the  differential  equation.  The  second  stack  level  may 
contain  only  the  value  of  8,  and  the  step  Ax  will  be  taken  as  a  small  default 
value.   After  running  function  illliil,  the  stack  will  show  the  lines: 

2:  {V,  y, 'f(x,y)'} 
1:  8 

The  value  of  the  solution,  yfina|,  will  be  available  in  variable  11111.  This 
function  is  appropriate  for  programming  since  it  leaves  the  differential 
equation  specifications  and  the  tolerance  in  the  stack  ready  for  a  new  solution. 
Notice  that  the  solution  uses  the  initial  conditions  x  =  0  at  y  =  0.  If,  your 
actual  initial  solutions  are  x  =  xinit  at  y  =  yinit,  you  can  always  add  these  values 
to  the  solution  provided  by  RKF,  keeping  in  mind  the  following  relationship: 


Page  1 6-70 


RKF  solution 

Actual  solution 

X 

y 

X 

y 

0 

0 

Xjnit 

Yinit 

Xfinal 

Yfinal 

Xjnit  +  Xfjnal 

Yinit  +  yfinal 

The  following  screens  show  the  RPN  stack  before  and  after  applying  function 

x+y,  e  =  0.001,  Ax  =  0.1. 


4: 

3: 

{XL 

2: 

mi 

l: 

2 

4: 
3 ' 

2:  (xy  x+y> 

l:  ,sgi 

baHMaa«iaH«iiaaHaiiaH«aiM:»d 


After  applying  function  RKF,  variable        contains  the  value  4.3880... 


Function  RRK 

This  function  is  similar  to  the  RKF  function,  except  that  RRK  (Rosenbrock  and 
Runge-Kutta  methods)  requires  as  the  list  in  stack  level  3  for  input  not  only  the 
names  of  the  independent  and  dependent  variables  and  the  function  defining 
the  differential  equation,  but  also  the  expressions  for  the  first  and  second 
derivatives  of  the  expression.  Thus,  the  input  stack  for  this  function  will  look  as 
follows: 

3:     {V,  y,  'f(x,y)'  'df/dx'  'd\/dy'  } 
2:  {c  Ax} 

"I  :  Xfinal 

The  value  in  the  first  stack  level  is  the  value  of  the  independent  variable  where 
you  want  to  find  your  solution,  i.e.,  you  want  to  find,  yfind  =  fs(xfina|),  where  fs(x) 
represents  the  solution  to  the  differential  equation.  The  second  stack  level  may 
contain  only  the  value  of  s,  and  the  step  Ax  will  be  taken  as  a  small  default 
value.  After  running  function  III:!!!!!,  the  stack  will  show  the  lines: 

2:    {V,  y,  'f(x,y)'  'tf/dx'  Wvy'  } 

1:  {s  Ax} 

The  value  of  the  solution,  yfina|,  will  be  available  in  variable  IE11II. 

This  function  can  be  used  to  solve  so-called  "stiff"  differential  equations. 


Page  16-71 


The  following  screen  shots  show  the  RPN  stack  before  and  after  application  of 
function  RRK: 


'-100*y+100*X 

-100'  y 


£  x  y 

+101T  100 

t  .001  .1  3 
l:  2 


3  5 

2:  {.  x  y  '-100*y+100*x 

+101 7  100    '-100'  > 

l:  .001 


The  value  stored  in  variable  y  is  3.00000000004. 


Function  RKFSTEP 

This  function  uses  an  input  list  similar  to  that  of  function  RKF,  as  well  as  the 
tolerance  for  the  solution,  and  a  possible  step  Ax,  and  returns  the  same  input 
list,  followed  by  the  tolerance,  and  an  estimate  of  the  next  step  in  the 
independent  variable.  The  function  returns  the  input  list,  the  tolerance,  and 
the  next  step  in  the  independent  variable  that  satisfies  that  tolerance.  Thus, 
the  input  stack  looks  as  follows: 

3:  {V,  y, 'f(x,y)'} 
2:  s 
1:  Ax 


After  running  this  function,  the  stack  will  show  the  lines: 

3:  fx',  y, 'f(x,y)'} 
2:  c 
1 :  (Ax)next 


Thus,  this  function  is  used  to  determine  the  appropriate  size  of  a  time  step  to 
satisfy  the  required  tolerance. 


The  following  screen  shots  show  the  RPN  stack  before  and  after  application  of 
function  RKFSTEP: 


4: 

3:  £  x  y  'x*y'  } 

2:  .001 
l:  .  1 

baHMaa«iaH«iiaaHaiiaH«aima^ 


4: 

3:  £  x  y  'x*y'  } 

2:  .001 

l:  .340493095001 


These  results  indicate  that  (Ax)next  =  0.34049... 


Page  1 6-72 


Function  RRKSTEP 

This  function  uses  an  input  list  similar  to  that  of  function  RRK,  as  well  as  the 
tolerance  for  the  solution,  a  possible  step  Ax,  and  a  number  (LAST)  specifying 
the  last  method  used  in  the  solution  (1,  if  RKF  was  used,  or  2,  if  RRK  was 
used).  Function  RRKSTEP  returns  the  same  input  list,  followed  by  the  tolerance, 
an  estimate  of  the  next  step  in  the  independent  variable,  and  the  current 
method  (CURRENT)  used  to  arrive  at  the  next  step.  Thus,  the  input  stack  looks 
as  follows: 

4:  {V,  y,  'f(x,y)'} 
3:  s 
2:  Ax 
1:  LAST 


After  running  this  function,  the  stack  will  show  the  lines: 

4:  {V,  y, 'f(x,y)'} 
3:  c 
2:  (Ax)next 
1:  CURRENT 


Thus,  this  function  is  used  to  determine  the  appropriate  size  of  a  time  step 
((Ax)next)  to  satisfy  the  required  tolerance,  and  the  method  used  to  arrive  at  that 
result  (CURRENT). 

The  following  screen  shots  show  the  RPN  stack  before  and  after  application  of 
function  RRKSTEP: 


£  x  y  '-100*y+100*x 
+101f  100   '-10'  > 
3:  .0001 
2:  .  1 

l:  1 


t  x  y  '-100*y+100*x 

+101 7    100    '-10'  > 

3:  .0001 
2:  5.58878551997E-3 

1. 

baH«aa«iaH«iiaaMiiaH«aiMa*a 


These  results  indicate  that  (Ax)next  =  0.00558...  and  that  the  RKF  method 
(CURRENT  =  1)  should  be  used. 


Page  1 6-73 


Function  RKFERR 

This  function  returns  the  absolute  error  estimate  for  a  given  step  when  solving 
a  problem  as  that  described  for  function  RKF.    The  input  stack  looks  as 
follows: 

2:  {V,  y, 'f(x,y)'} 
1:  Ax 


After  running  this  function,  the  stack  will  show  the  lines: 

4:     {V,  y, 'f(x,y)'} 

3:  c 

2:  Ay 

1 :  error 
Thus,  this  function  is  used  to  determine  the  increment  in  the  solution,  Ay,  as 
well  as  the  absolute  error  (error). 


The  following  screen  shots  show  the  RPN  stack  before  and  after  application  of 
function  RKFERR: 


6: 
5: 
4: 
3: 
2: 
l: 


£  x  y  'x*y'  } 


6: 
5: 
4: 
3 ' 

2:  .827201242271 

l:  -1.S95374S9701E-6 
baHMBaa«iaH«iiaa;«naHj^aiM=»s 


£  x  y  'x*y'  3 


These  result  show  that  Ay  =  0.827...  and  error  =  -1 .89...xl  0"6. 


Function  RSBERR 

This  function  performs  similarly  to  RKERR  but  with  the  input  elements  listed  for 
function  RRK.  Thus,  the  input  stack  for  this  function  will  look  as  follows: 

2:    {V,  y,  'f(x,y)'  'd\/dx'  'Sf/vy'  } 
1:  Ax 


After  running  the  function,  the  stack  will  show  the  lines: 
4:  {V,  y,  'f(x,y)'  'dt/dx*  Wvy'  }: 
3:  c 
2:  Ay 


error 


Page  1 6-74 


The  following  screen  shots  show  the  RPN  stack  before  and  after  application  of 
function  RSBERR: 


4: 
3: 

l: 


£  x  y 

+  101T 


'-100*y+100*x 

100    '-10'  > 

.  1 


4: 
3: 
l: 


t  x  y  '-100*y+100*x 
+1017  100   '-10'  > 

.  1 

4. 15144617136 
2.76211716614 


These  results  indicate  that  Ay  =  4.1  514...  and  error  =  2.762...,  for  Dx  =  0.1 . 
Check  that,  if  Dx  is  reduced  to  0.01,  Ay  =  -0.00307...  and  error  = 
0.000547. 


Note:  As  you  execute  the  commands  in  the  DIFF  menu  values  of  x  and  y  will 
be  produced  and  stored  as  variables  in  your  calculator.  The  results  provided 
by  the  functions  in  this  section  will  depend  on  the  current  values  of  x  and  y. 
Therefore,  some  of  the  results  illustrated  above  may  differ  from  what  you  get 
in  your  calculator. 


Page  1 6-75 


Chapter  1 7 

Probability  Applications 

In  this  Chapter  we  provide  examples  of  applications  of  calculator's  functions 
to  probability  distributions. 

The  MTH/PROBABILITY..  sub-menu  -  part  1 

The  MTH/PROBABILITY..  sub-menu  is  accessible  through  the  keystroke 
sequence  MhJmtw  .  With  system  flag  117  set  to  CHOOSE  boxes,  the 
following  list  of  MTH  options  is  provided  (see  left-hand  side  figure  below). 
We  have  selected  the  PROBABILITY.,  option  (option  7),  to  show  the  following 
functions  (see  right-hand  side  figure  below): 


HATH  HEMJ 

6. BASE..  1 

S .  FFT..  ! 
5 .  LUMF  LE!!.. 

10.  consTAnTS.. 

11.  special  FuncTions..  | 

PROBABILITY  HEMJ 

a™R™  %          ^  I 

3.! 

H . RAAD  I 

S .  RDZ 

e.uTPC  I 

In  this  section  we  discuss  functions  COMB,  PERM,  !  (factorial),  RAND,  and 
RDZ. 

Factorials,  combinations,  and  permutations 

The  factorial  of  an  integer  n  is  defined  as:  n!  =  n-  (n-1)  •  (n-2)... 3-2-1 .  By 
definition,  0!  =  1 .  Factorials  are  used  in  the  calculation  of  the  number  of 
permutations  and  combinations  of  objects.  For  example,  the  number  of 
permutations  of  r  objects  from  a  set  of  n  distinct  objects  is 

nPr  =  n(n  -\)(n  -\)...(n  -  r  +  1)  =  n\l{n-r)\ 
Also,  the  number  of  combinations  of  n  objects  taken  r  at  a  time  is 
n{n  -  X){n  -  2)...(n  -  r  + 1) 


\rj 


r\ 


r\{n  -  r)\ 


Page  17-1 


To  simplify  notation,  use  P(n,r)  for  permutations,  and  C(n,r)  for  combinations. 
We  can  calculate  combinations,  permutations,  and  factorials  with  functions 
COMB,  PERM,  and  !  from  the  MTH/PROBABILITY..  sub-menu.  The  operation 
of  those  functions  is  described  next: 

•  COMB(n,r):  Combinations  of  n  items  taken  r  at  a  time 

•  PERM(n,r):  Permutations  of  n  items  taken  r  at  a  time 

•  n!:  Factorial  of  a  positive  integer.  For  a  non-integer,  x!  returns 
r(x+l),  where  r(x)  is  the  Gamma  function  (see  Chapter  3).  The 
factorial  symbol  (!)  can  be  entered  also  as  the  keystroke  combination 

Example  of  applications  of  these  functions  are  shown  next: 


:COMB(10.,6.:i 

210. 

:PERM(10.,6.:i 

151200. 

:  12.! 

479001600. 

Random  numbers 

The  calculator  provides  a  random  number  generator  that  returns  a  uniformly 
distributed,  random  real  number  between  0  and  1 .  The  generator  is  able  to 
produce  sequences  of  random  numbers.  However,  after  a  certain  number  of 
times  (a  very  large  number  indeed),  the  sequence  tends  to  repeat  itself.  For 
that  reason,  the  random  number  generator  is  more  properly  referred  to  as  a 
pseudo-random  number  generator.  To  generate  a  random  number  with  your 
calculator  use  function  RAND  from  the  MTH/PROBABILITY  sub-menu.  The 
following  screen  shows  a  number  of  random  numbers  produced  using  RAND. 
The  numbers  in  the  left-hand  side  figure  are  produced  with  calling  function 
RAND  without  an  argument.  If  you  place  an  argument  list  in  function  RAND, 
you  get  back  the  list  of  numbers  plus  an  additional  random  number  attached 
to  it  as  illustrated  in  the  right-hand  side  figure. 


Page  1  7-2 


:RRHD 

.529199353633 

:RRHD 

4.35821314444E-2 

:RRHD 

. £94922982083 

.  2949229820881 

RRHDC5J 
{5-  4- 10396424443E-2) 
:RRHDC2-,5.) 

{2.  5.  .736370433305) 
RRND(l.,2.,3.:i 
kl.  2.  3.  4-07030793137^ 


Random  number  generators,  in  general,  operate  by  taking  a  value,  called  the 
"seed"  of  the  generator,  and  performing  some  mathematical  algorithm  on  that 
"seed"  that  generates  a  new  (pseudo)random  number.  If  you  want  to 
generate  a  sequence  of  number  and  be  able  to  repeat  the  same  sequence 
later,  you  can  change  the  "seed"  of  the  generator  by  using  function  RDZ(n), 
where  n  is  the  "seed,"  before  generating  the  sequence.  Random  number 
generators  operate  by  starting  with  a  "seed"  number  that  is  transformed  into 
the  first  random  number  of  the  series.  The  current  number  then  serves  as  the 
"seed"  for  the  next  number  and  so  on.  By  "re-seeding"  the  sequence  with  the 
same  number  you  can  reproduce  the  same  sequence  more  than  once.  For 
example,  try  the  following: 

Use  0.25  as  the  "seed." 
First  random  number  =  0.75285... 
Second  random  number  =  0.51  109... 
Third  random  number^  0.085429.... 


RDZ(0.25)  (ENTER) 

RAND()  (enter} 
RAND()  (enter} 
RAND()  (enter} 
Re-start  the  sequence: 
RDZ(0.25)  (enter} 
RAND()  (enter} 
RAND()  (enter} 
RANDQ  (enter} 


Use  0.25  as  the  "seed." 
First  random  number  =  0.75285... 
Second  random  number  =  0.51  1 09... 
Third  random  number^  0.085429.... 


To  generate  a  sequence  of  random  numbers  use  function  SEQ.  For  example, 
to  generate  a  list  of  5  random  numbers  you  can  use,  in  ALG  mode: 
SEQ<RflHD<  )  ::  j  ::  1 ::  5  j  1  >  .    In  RPN  mode,  use  the  following  program: 
■*:  ^  n  *  1  n  FOR  j  RND  NEXT  n  ^LIST  :*■ 

Store  it  into  variable  RLST  (Random  LiST),  and  use  L^LiJiE^  to  produce  a 
list  of  5  random  numbers. 


Function  RNDM(n,m)  can  be  used  to  generate  a  matrix  of  n  rows  and  m 
columns  whose  elements  are  random  integers  between  -1  and  l(see  Chapter 
10). 


Page  1 7-3 


Discrete  probability  distributions 

A  random  variable  is  said  to  be  discrete  when  it  can  only  take  a  finite  number 
of  values.  For  example,  the  number  of  rainy  days  in  a  given  location  can  be 
considered  a  discrete  random  variable  because  we  count  them  as  integer 
numbers  only.  Let  X  represent  a  discrete  random  variable,  its  probability  mass 
function  (pmf)  is  represented  by  f(x)  =  P[X=x],  i.e.,  the  probability  that  the 
random  variable  X  takes  the  value  x. 

The  mass  distribution  function  must  satisfy  the  conditions  that 

f(x)  >0,  for  all  x, 

and 

£/(*)  =  1.0 

allx 

A  cumulative  distribution  function  (cdf)  is  defined  as 
F(x)  =  P[X<x]  =  Zf(k) 

k<x 

Next,  we  will  define  a  number  of  functions  to  calculate  discrete  probability 
distributions.  We  suggest  that  you  create  a  sub-directory,  say, 
HOME\STATS\DFUN  (Discrete  FUNctions)  where  we  will  define  the 
probability  mass  function  and  cumulative  distribution  function  for  the  binomial 
and  Poisson  distributions. 


Binomial  distribution 

The  probability  mass  function  of  the  binomial  distribution  is  given  by 


f{n,p,x)  = 


•px-(i-prx,  *= 0,1,2,...,* 


where  (nx)  =  C(n,x)  is  the  combination  of  n  elements  taken  x  at  a  time.  The 
values  n  and  p  are  the  parameters  of  the  distribution.  The  value  n  represents 
the  number  of  repetitions  of  an  experiment  or  observation  that  can  have  one 
of  two  outcomes,  e.g.,  success  and  failure.  If  the  random  variable  X 
represents  the  number  of  successes  in  the  n  repetitions,  then  p  represents  the 


Page  1  7-4 


probability  of  getting  a  success  in  any  given  repetition.  The  cumulative 
distribution  function  for  the  binomial  distribution  is  given  by 


F(n,p,x)  =  ^f(n,p,x),    x  =  0,1,2,...,/? 

£=0 

Poisson  distribution 

The  probability  mass  function  of  the  Poisson  distribution  is  given  by 

f(A,x)  =  — ,    x  =  0,1,2,..., oo . 

x! 

In  this  expression,  if  the  random  variable  X  represents  the  number  of 
occurrences  of  an  event  or  observation  per  unit  time,  length,  area,  volume, 
etc.,  then  the  parameter  I  represents  the  average  number  of  occurrences  per 
unit  time,  length,  area,  volume,  etc.    The  cumulative  distribution  function  for 
the  Poisson  distribution  is  given  by 

X 

F(Z,x)  =  ^/(l,x),    x  =  0,1,2,...,  oo 

Next,  use  function  DEFINE  (UiJDiL_  )  to  define  the  following  probability  mass 
functions  (pmf)  and  cumulative  distribution  functions  (cdf): 

IY:  l"  !  ":  :°   l  "!  ■■  \'."S  ::  "x"    5      :::::     S~" ^  W  5"   """  •■  '  "S" ' 


!■■■    !  !■■.::■■■ 

  ■■  :     ■  ■  •    ■■■■  .■  ■  ■  ■ 

Z<  k  =0  j  X  j  pmfb  C  H  ,  P  V.  k  >  >  > 

^,x>  :::::  EXP<-X)*XAx^x  !  > 


■         :  :■■.[!■■■  :"- 


The  function  names  stand  for: 

•  pmfb:  probability  mass  function  for  the  binomial  distribution 

•  cdfb:  cumulative  distribution  function  for  the  binomial  distribution 

•  pmfp:  probability  mass  function  for  the  Poisson  distribution 

•  cdfp:  cumulative  distribution  function  for  the  Poisson  distribution 


Page  1  7-5 


Examples  of  calculations  using  these  functions  are  shown  next: 


:  pmfbCie, . 15i3> 

. 129833720754 
:  cdfbae,  .  15i3> 

.950030201121 

:  +HUM(pmfp(5?4>> 

. 175467369768 

:  ■*NUM<cdfp<5i4>> 

. 377336848837 

cdfp  I  pHfp  |  c4fb  \  pHfb  |        S  Wm 

■:dh  p  |  phFp  |  ojft- 1  pHfb  |  i 

Continuous  probability  distributions 

The  probability  distribution  for  a  continuous  random  variable,  X,  is 
characterized  by  a  function  f(x)  known  as  the  probability  density  function  (pdf). 
The  pdf  has  the  following  properties:  f(x)  >  0,  for  all  x,  and 

P[X<x]=  F(x)= 

p+co 

/  (x)dx  =  1 . 

J— 00 

Probabilities  are  calculated  using  the  cumulative  distribution  function  (cdf),  F(x), 
defined  by  P[X  <x]  =  F(x)  =  f  f{g)dl;  ,  where  P[X<x]  stands  for  "the 

J-00 

probability  that  the  random  variable  X  is  less  than  the  value  x". 

In  this  section  we  describe  several  continuous  probability  distributions 
including  the  gamma,  exponential,  beta,  and  Weibull  distributions.  These 
distributions  are  described  in  any  statistics  textbook.  Some  of  these 
distributions  make  use  of  a  the  Gamma  function  defined  earlier,  which  is 
calculated  in  the  calculator  by  using  the  factorial  function  as  r(x)  =  (x-1)!,  for 
any  real  number  x. 

The  gamma  distribution 

The  probability  distribution  function  (pdf)  for  the  gamma  distribution  is  given 
by 

1  X 

f  (x)  =  xa~l  •  exp(  ),  for    x  >  0?  a  >  0?  B  >  0; 

paT(a)  p 


Page  1 7-6 


The  corresponding  (cumulative)  distribution  function  (cdf)  would  be  given  by 
an  integral  that  has  no  closed-form  solution. 

The  exponential  distribution 

The  exponential  distribution  is  the  gamma  distribution  with  a  =  1 .  Its  pdf  is 
given  by 

1  x 
/(*)  =  — -exp(-—), /or  x>0,/?>0, 

while  its  cdf  is  given  by  F(x)  =  1  -  exp(-x/p),  for  x>0,  p  >0. 

The  beta  distribution 

The  pdf  for  the  gamma  distribution  is  given  by 

= +J*, \  - x*~l  •  0 - *y~i>fi>r  °  <  *  <  i>a  >  w  >  ° 

As  in  the  case  of  the  gamma  distribution,  the  corresponding  cdf  for  the  beta 
distribution  is  also  given  by  an  integral  with  no  closed-form  solution. 

The  Weibull  distribution 

The  pdf  for  the  Weibull  distribution  is  given  by 

f(x)  =  a-j3'  xp~x  •  exp(-a  •  xfi\    for  x  >  0,a  >  0,/?  >  0 
While  the  corresponding  cdf  is  given  by 

F(x)  =  1  -  exp(-a  •  xfi\    for  x>0,a  >0,j3  >  0 
Functions  for  continuous  distributions 

To  define  a  collection  of  functions  corresponding  to  the  gamma,  exponential, 
beta,  and  Weibull  distributions,  first  create  a  sub-directory  called  CFUN 


Page  1  7-7 


(Continuous  FUNctions)  and  define  the  following  functions  (change  to  Approx 
mode): 


Gamma  pdf:      'gpdf(x)  =  xA  (cx-l )  *exp  ( -  x/p)  /  (PAot*  gamma  (ot)  ) 1 
Gamma  cdf:  'gcdf(x)   =  J(0,  x,  gpdf  (t)  ,  t)  1 

Beta  pdf: 

1  Ppdf (x)=  GAMMA (a+P) *xA (a-1) * (1-x) A (P-l) / (GAMMA (a) *  GAMMA ( P )  )  ' 
Beta  cdf:  1  (Jcdf  (x)  =  1(0,  x,  Ppdf  (t)  ,t)  ' 

Exponential  pdf:  'epdf(x)  =  exp(-x/P)/P' 

Exponential  cdf:  'ecdf(x)  =  l  -  exp(-x/P)' 

Weibull  pdf:  'Wpdf(x)  =  a*p*xA  (p-l)  *exp  (-a*xAP)  ' 

Weibull  cdf:  'Wcdf(x)  =  1  -  exp(-oc*xaP)' 

Use  function  DEFINE  to  define  all  these  functions.  Next,  enter  the  values  of  a 
and  p,  e.g.,  Q^C^(^C3ffl(^  CZDC^D  ^w)CE)(3(^) 

Finally,  for  the  cdf  for  Gamma  and  Beta  cdf's,  you  need  to  edit  the  program 
definitions  to  add  ->NUM  to  the  programs  produced  by  function  DEFINE.  For 
example,  the  Gamma  cdf,  i.e.,  the  function  gcdf,  should  be  modified  to  read: 

->  x  '^NUM(  j  (0,x,gpdf  (t)  ,  t)  )  '  *  and  stored  back  into  EBB. 
Repeat  the  procedure  for  pcdf. 

Unlike  the  discrete  functions  defined  earlier,  the  continuous  functions  defined 
in  this  section  do  not  include  their  parameters  (a  and/or  p)  in  their  definitions. 
Therefore,  you  don't  need  to  enter  them  in  the  display  to  calculate  the 
functions.  However,  those  parameters  must  be  previously  defined  by  storing 
the  corresponding  values  in  the  variables  a  and  p.  Once  all  functions  and  the 
values  a  and  p  have  been  stored,  you  can  order  the  menu  labels  by  using 
function  ORDER.  The  call  to  the  function  will  be  the  following: 

ORDERa'a'/p'/gpdP/gcdP/PpdP/pcdP/epdP/ecdP/WpdP/WcdP}) 

Following  this  command  the  menu  labels  will  show  as  follows  (Press  jNxrj  to 
move  to  the  second  list.  Press  jNxrj  once  more  to  move  to  the  first  list): 


■zp-ji-  I  ccdf  |  Hpdf  |  Hojh 


Page  1 7-8 


Some  examples  of  application  of  these  functions,  for  values  of  a  =  2,  p  =  3, 
are  shown  below.  Notice  the  variable  IERR  that  shows  up  in  the  second 
screen  shot.  This  results  from  a  numerical  integration  for  function  gcdf. 


2>« 
3>p 


2. 
3. 


gpdf <l. 2) 

3.  93760061 33 1E-2I 


3c4f  |  Bpdf  I  Bcdf 


1.536 
■  1308 
. 154353006737 


ppdfC-2) 
pcdfC-2) 
epdf(2.3) 


GcdF  |  cpdf  |  i>z>if  |  Hpdf  |  H'i'JF 


571 

gpdf <1. 2) 

8.93760061381E-2 
gcdf <1.2> 

6. 1551 935550 1E-2 
ppdf <■£> 

1.536 


3cdf  |  ppdf 


530067S7 
ecdf <2.3> 

. 535440979639 
Npdf  < 1 . > 

.312011699422 
Ncdf  < 1 . > 

.364664716763 


GcdF  I  ip.jh  I  ecdf  I  Hp-jh  |  HcdF 


Continuous  distributions  for  statistical  inference 

In  this  section  we  discuss  four  continuous  probability  distributions  that  are 
commonly  used  for  problems  related  to  statistical  inference.  These 
distributions  are  the  normal  distribution,  the  Student's  t  distribution,  the  Chi- 
square  (x2)  distribution,  and  the  F-distribution.  The  functions  provided  by  the 
calculator  to  evaluate  probabilities  for  these  distributions  are  contained  in  the 
MTH/PROBABILITY  menu  introduced  earlier  in  this  chapter.  The  functions  are 
NDIST,  UTPN,  UTPT,  UTPC,  and  UTPF.  Their  application  is  described  in  the 
following  sections.  To  see  these  functions  activate  the  MTH  menu:  Cji  Jmth 
and  select  the  PROBABILITY  option: 


HATH  HEMJ 

c.base..  r 

S.FFT.. 

3.C0HFLEH..  li 

10.  consTAnTS.. 

11.  special  Funcnons..  | 

PROBABILITY  HEMJ 

6. UTPC 
F.UTPF 
S.UTPn 
3. UTPT 

iO.nDIST  I 

Normal  distribution  pdf 

The  expression  for  the  normal  distribution  pdf  is: 


Page  1 7-9 


G^ln  No- 
where ja  is  the  mean,  and  a2  is  the  variance  of  the  distribution.  To  calculate 
the  value  of  f(ju,a2,x)  for  the  normal  distribution,  use  function  NDIST  with  the 
following  arguments:  the  mean,  ju,  the  variance,  a2,  and,  the  value  x  ,  i.e., 
NDIST(|u,a2,x).  For  example,  check  that  for  a  normal  distribution, 
f(l. 0,0.5,2.0)  =  0.20755374. 

Normal  distribution  cdf 

The  calculator  has  a  function  UTPN  that  calculates  the  upper-tail  normal 
distribution,  i.e.,  UTPN(x)  =  P(X>x)  =  1  -  P(X<x).  To  obtain  the  value  of  the 
upper-tail  normal  distribution  UTPN  we  need  to  enter  the  following  values:  the 
mean,  \x;  the  variance,  a2;  and,  the  value  x,  e.g.,  UTPN((|u,a2,x) 

For  example,  check  that  for  a  normal  distribution,  with  ja  =  1 .0,  a2  =  0.5, 
UTPN(0.75)  =  0.638163.  Use  UTPN(1 .0,0.5,0.75)  =  0.638163. 

Different  probability  calculations  for  normal  distributions  [X  is  N(ja,a2)]  can  be 
defined  using  the  function  UTPN,  as  follows: 

•  P(X<a)  =  1  -  UTPN(m  a2,a) 

•  P(a<X<b)  =  P(X<b)  -  P(X<a)  =  1  -  UTPN(n,  a2,b)  -  (1  -  UTPN(n,  a2,a)) 
=  UTPNfri,  a2,a)  -  UTPN(ju,  a2,b) 

•  P(X>c)  =  UTPN(m  a2,c) 

Examples:  Using  ju  =  1 .5,  and  a2  =  0.5,  find: 

P(X<1 .0)  =  1  -  P(X>1 .0)  =  1  -  UTPN(1 .5,  0.5,  1 .0)  =  0.239750. 
P(X>2.0)  =  UTPN(1 .5,  0.5,  2.0)  =  0.239750. 

P(l  .0<X<2.0)  =  F(l  .0)  -  F(2.0)  =  UTPN(1 .5,0.5,1 .0)  -  UTPN(1 .5,0.5,2.0) 
=  0.7602499  -  0.2397500  =  0.524998. 


Page  17-10 


The  Student-t  distribution 

The  Student-t,  or  simply,  the  t-,  distribution  has  one  parameter  v,  known  as  the 
degrees  of  freedom  of  the  distribution.  The  probability  distribution  function 
(pdf)  is  given  by 

f(t)  =  2  (l  +  _)    2  >_00<^<00 

r(^)-V^ 

where  r(a)  =  (a-1)!  is  the  GAMMA  function  defined  in  Chapter  3. 

The  calculator  provides  for  values  of  the  upper-tail  (cumulative)  distribution 
function  for  the  t-distribution,  function  UTPT,  given  the  parameter  v  and  the 
value  of  t,  i.e.,  UTPT(v,t).  The  definition  of  this  function  is,  therefore, 

f(t)dt  =  1  -  f  f{t)dt  =  \-P(T<t) 

t  J-oo 

For  example,  UTPT(5,2.5)  =  2.7245. ..E-2.  Other  probability  calculations  for 
the  t-distribution  can  be  defined  using  the  function  UTPT,  as  follows: 

•  P(T<a)  =  1  -  UTPT(v,a) 

•  P(a<T<b)  =  P(T<b)  -  P(T<a)  =  1  -  UTPT(v,b)  -    (1  -  UTPT(v,a))  = 
UTPT(v,a)  -  UTPT(v,b) 

•  P(T>c)  =  UTPT(v,c) 

Examples:  Given  v  =  1  2,  determine: 

P(T<0.5)  =  1-UTPT(1 2,0.5)  =  0.68694.. 

P(-0.5<T<0.5)  =  UTPT(12,-0.5)-UTPT(l  2,0.5)  =  0.3738... 

P(T>  -1 .2)  =  UTPT(1  2,-1 .2)  =  0.8733... 

The  Chi-square  distribution 

The  Chi-square  (x2)  distribution  has  one  parameter  v,  known  as  the  degrees  of 
freedom.  The  probability  distribution  function  (pdf)  is  given  by 


Page  17-11 


1         --1  -- 
/(jc)  =  —  x2    -e  2,v>0,x>0 

The  calculator  provides  for  values  of  the  upper-tail  (cumulative)  distribution 
function  for  the  x2-distribution  using  [UTPC]  given  the  value  of  x  and  the 
parameter  v.  The  definition  of  this  function  is,  therefore, 

UTPC(v,  x)  =  f  C°f(x)dx  =  1-1"  f{x)dx  =  1  -  P(X  <  x) 

J  t  J  -00 

To  use  this  function,  we  need  the  degrees  of  freedom,  v,  and  the  value  of  the 
chi-square  variable,  x,  i.e.,  UTPC(v,x).  For  example,  UTPQ5,  2.5)  = 
0.776495... 

Different  probability  calculations  for  the  Chi-squared  distribution  can  be 
defined  using  the  function  UTPC,  as  follows: 

•  P(X<a)  =  1  -  UTPC(v,a) 

•  P(a<X<b)  =  P(X<b)  -  P(X<a)  =  1  -  UTPC(v,b)  -    (1  -  UTPC(v,a))  = 
UTPC(v,a)  -  UTPC(v,b) 

•  P(X>c)  =  UTPC(v,c) 

Examples:  Given  v  =  6,  determine: 

P(X<5.32)  =  1-UTPC(6,5.32)  =  0.4965.. 
P(1.2<X<10.5)  =  UTPC(6,1.2)-UTPC(6,10.5)  =  0.8717... 
P(X>  20)  =  UTPC(6,20)  =  2.769.. E-3 

The  F  distribution 

The  F  distribution  has  two  parameters  vN  =  numerator  degrees  of  freedom, 
and  vD  =  denominator  degrees  of  freedom.  The  probability  distribution 
function  (pdf)  is  given  by 


Page  17-12 


The  calculator  provides  for  values  of  the  upper-tail  (cumulative)  distribution 
function  for  the  F  distribution,  function  UTPF,  given  the  parameters  vN  and  vD, 
and  the  value  of  F.  The  definition  of  this  function  is,  therefore, 


UTPF(vN,vD,F)  =  \  f{F)dF  =  \-\    f(F)dF  =  \-P(Z<F) 

J  t  J  -oo 

For  example,  to  calculate  UTPF(10,5,  2.5)  =  0.161834... 

Different  probability  calculations  for  the  F  distribution  can  be  defined  using  the 
function  UTPF,  as  follows: 

•  P(F<a)  =  1  -  UTPF(vN,  vD,a) 

•  P(a<F<b)  =  P(F<b)  -  P(F<a)  =  1  -UTPF(vN,  vD,b)-  (1  -  UTPF(vN,  vD,a)) 

=  UTPF(vN,  vD,a)  -  UTPF(vN,  vD,b) 

•  P(F>c)  =  UTPF(vN,  vD,a) 

Example:  Given  vN  =  10,  vD  =  5,  find: 

P(F<2)  =  1-UTPF(1 0,5,2)  =  0.7700... 

P(5<F<10)  =  UTPF(1 0,5,5) -UTPF(1 0,5, 10)  =  3.4693. .E-2 

P(F>5)  =  UTPF(1 0,5,5)  =  4.4808. .E-2 

Inverse  cumulative  distribution  functions 

For  a  continuous  random  variable  X  with  cumulative  density  function  (cdf)  F(x) 
=  P(X<x)  =  p,  to  calculate  the  inverse  cumulative  distribution  function  we  need 
to  find  the  value  of  x,  such  that  x  =  F_1(p).    This  value  is  relatively  simple  to 
find  for  the  cases  of  the  exponential  and  Weibull  distributions  since  their  cdf's 
have  a  closed  form  expression: 


Page  17-13 


•  Exponential,  F(x)  =  1  -exp(-x/p) 

•  Weibull,  F(x)=  l-exp(-axp) 

(Before  continuing,  make  sure  to  purge  variables  a  and  p).  To  find  the  inverse 
cdPs  for  these  two  distributions  we  need  just  solve  for  x  from  these  expressions, 
i.e., 


Exponential: 


: SOLVE 


-x 

6 

P=l-e  ,'x' 

x=-(p-LH(-(F-l»J| 


Weibull: 


SOLVElp=l-e  /x1 


x=e 


LH[  LH(-(p-1h1 
P 


For  the  Gamma  and  Beta  distributions  the  expressions  to  solve  will  be  more 
complicated  due  to  the  presence  of  integrals,  i.e., 

^  f X         1  a-\  ,      Z  x  r 

•     Gamma,     p  =  z     -exp(  )dz 

ioJ3ar(a)  P 

.     Beta,  f-r(a  +  l)    ^      _  ^ 

A  numerical  solution  with  the  numerical  solver  will  not  be  feasible  because  of 
the  integral  sign  involved  in  the  expression.    However,  a  graphical  solution  is 
possible.  Details  on  how  to  find  the  root  of  a  graph  are  presented  in  Chapter 
1  2.  To  ensure  numerical  results,  change  the  CAS  setting  to  Approx.  The 
function  to  plot  for  the  Gamma  distribution  is 

Y(X)  =  j(0,X,zA(a-l)*exp(-z/p)/(pAa*GAMMA(a)),z)-p 


For  the  Beta  distribution,  the  function  to  plot  is 


Y(X)  = 

I(0,X,  z  A  (a- 1 )  *  ( 1  -z) A  (p- 1 )  *  GAMMA(a+p)/(GAMMA(a)  *  GAMMA(p)),z)-p 

To  produce  the  plot,  it  is  necessary  to  store  values  of  a,  p,  and  p,  before 
attempting  the  plot.    For  example,  for  a  =  2,  p  =  3,  and  p  =  0.3,  the  plot  of 
Y(X)  for  the  Gamma  distribution  is  shown  below.  (Please  notice  that,  because 


Page  17-14 


of  the  complicated  nature  of  function  Y(X),  it  will  take  some  time  before  the 
graph  is  produced.  Be  patient.) 


a: 

V 

-6.5 

-3.1 

e.s 

There  are  two  roots  of  this  function  found  by  using  function  USED  within  the 
plot  environment.  Because  of  the  integral  in  the  equation,  the  root  is 
approximated  and  will  not  be  shown  in  the  plot  screen.  You  will  only  get  the 
message  Constant?  Shown  in  the  screen.  However,  if  you  press  [enter]  at  this 
point,  the  approximate  root  will  be  listed  in  the  display.  Two  roots  are  shown 
in  the  right-hand  figure  below. 


V 

-6.5 

Constant? 

6.5 

(-1.9,0.) 


Alternatively,  you  can  use  function  lililE;!!  1(1193)1  to  estimate  the  roots  by 
tracing  the  curve  near  its  intercepts  with  the  x-axis.  Two  estimates  are  shown 
below: 


V 

V 

—  6.5 

r=5.aHE-3 

V 

-I  1  1  1  ^^_L — 

V 

-6.5 

f:5.71E-H 

These  estimates  suggest  solutions  x  =  -1 .9  and  x  =  3.3.  You  can  verify  these 
"solutions"  by  evaluating  function  Yl  (X)  for  X  =  -1 .9  and  X  =  3.3,  i.e., 


:Y1(-1.9) 
:Y1(3.3) 


.009244101213 
, 000970724234 


Page  17-15 


For  the  normal,  Student's  t,  Chi-square  (y2),  and  F  distributions,  which  are 
represented  by  functions  UTPN,  UTPT,  UPTC,  and  UTPF  in  the  calculator,  the 
inverse  cuff  can  be  found  by  solving  one  of  the  following  equations: 


•  Normal, 

•  Student's  t, 

•  Chi-square, 

•  F  distribution: 


p  =  1  -  UTPN(li,g2,x) 

p  =  1  -  UTPT(v,t) 

p  =  1  -  UTPC(v,x) 

p  =  1  -  UTPF(vN,vD,F) 


Notice  that  the  second  parameter  in  the  UTPN  function  is  a2,  not  a2, 
representing  the  variance  of  the  distribution.  Also,  the  symbol  v  (the  lower- 
case Greek  letter  no)  is  not  available  in  the  calculator.  You  can  use,  for 
example,  y  (gamma)  instead  of  v.  The  letter  y  is  available  thought  the 
character  set  (CB^s)- 


For  example,  to  obtain  the  value  of  x  for  a  normal  distribution,  with  ju  =  10, 
a2  =  2,  with  p  =  0.25,  store  the  equation  /p=l-UTPH<!  \x».  <j2*.  x)'  into 
variable  EQ  (figure  in  the  left-hand  side  below).  Then,  launch  the  numerical 
solver,  to  get  the  input  form  in  the  right-hand  side  figure: 


'P=l .  -UTPHCu,cr2,xi>EQ 
P=l.-UTPNCtJL,(T2,x> 


^  EtUflTIOl"!  §§§§§§§§§§§§§§§§§ 

Eq=p=l,-UTPH<UL,g2,X> 
p:  p^j^rn^t^  M.: 

Enter  ual.u«  or  press  SOLVE 


The  next  step  is  to  enter  the  values  of  li,  a2,  and  p,  and  solve  for  x: 


SOLVE  EGUATIOM 
Eq:p=l.-UTPhKtJL3<T£?X> 
P=    -25  K=  10  

&Z:     2  X= 


9.04612.. 


Enter  ugUjg  or  pfjJJ  SOLVE 
1 1  I  H^M^Tl  1 1  II  lllllllHII 


This  input  form  can  be  used  to  solve  for  any  of  the  four  variables  involved  in 
the  equation  for  the  normal  distribution. 


Page  17-16 


To  facilitate  solution  of  equations  involving  functions  UTPN,  UTPT,  UTPC,  and 
UTPF,  you  may  want  to  create  a  sub-directory  UTPEQ  were  you  will  store  the 
equations  listed  above: 


'P=l .  -UTPHCu,cr2,xi>EQH 
P=1.-UTPNCui(t2,x 
'p=l--UTPTKt3*EQT 

P=l. -UTPT t>?U 


'p=l--UTPCKxi>EQC 

P=l.-UTPC<^jX> 
'P=l  -  -UTPFt-*N^D,F)»EQF 
P=1.-UTPF^HJ^D,F> 


Thus,  at  this  point,  you  will  have  the  four  equations  available  for  solution.  You 
needs  just  load  one  of  the  equations  into  the  EQ  field  in  the  numerical  solver 
and  proceed  with  solving  for  one  of  the  variables.  Examples  of  the  UTPT, 
UTPC,  and  UPTF  are  shown  below: 


SOLVE  EGUfiTIOn 
Eq=p=l.-UTPT<^t> 
P=  .25 

r-  15 


-.691196948958 


Enter  ualm  or  pregj  SOLVE 


SOLVE  EGUATIOn 

Eg:p=l.-UTPC(-v?x> 
p=  .68 
r-  10 


11.498778181: 


Enter  uaUje  or  pfjjj  SOLVE 


inFO  |S0LVE 


SOLVE  EGUfiTIOn 

Eq:p=l.-UTPF(-vH?-vD?F> 


.25 

i& :  14 


vn=  18 


Enter  uqlue  or  press  SOLVE 


Notice  that  in  all  the  examples  shown  above,  we  are  working  with  p  =  P(X<x). 
In  many  statistical  inference  problems  we  will  actually  try  to  find  the  value  of  x 
for  which  P(X>x)  =  a.  Furthermore,  for  the  normal  distribution,  we  most  likely 
will  be  working  with  the  standard  normal  distribution  in  which  li  =0,  and  a2  = 
1 .  The  standard  normal  variable  is  typically  referred  to  as  Z,  so  that  the 
problem  to  solve  will  be  P(Z>z)  =  a.  For  these  cases  of  statistical  inference 
problems,  we  could  store  the  following  equations: 


Page  17-17 


,ffl=UTPN(0.,l.Jz)*EQNfl 

o:=UTPH(0.  , 1. ?z> 
'ffl=UTPTKt)>EQTfl 

oc=UTPT<^Pt> 


'*=UTPCK>d'KQCfl 

«=UTPC(-Vjx> 
:  ,ffl=UTPF(^H/yDiF)*EQFfi 

ffl=UTPF<^HP^DPF> 


With  these  four  equations,  whenever  you  launch  the  numerical  solver  you 
have  the  following  choices: 


HeHory: 

Select: 

PROG 

sECnfl 

PROG 

r-Li| 

PROG 

sECF 

PROG 

Sil 

»E4C 

PROG 

SECT 

PROG 

Examples  of  solution  of  equations  EQNA,  EQTA,  EQCA,  and  EQFA  are 
shown  below: 


SOLVE  EGURTIOn 
Eq=ffl=UTPN<8.  ,11PZ) 

«:  .05 


1 .64435362695 


Enter  uql.ue  or  press  SOLVE 


SOLVE  ECURTIOn 
Eq:ffl=UTPT<"fPt> 


.05 

15 


Enter  uqlue  or  press  SOLVE 


SOLVE  EGURTIOn 
Eq=«=UTPC<^X> 


05 
25 


37.6524341335 


Enter  uaUe  or  press  SOLVE 

fcaaa^M^Mffl  1 1  n  hn  mm 


SOLVE  EGURTIOn 

Eq=«=UTPF<^H,^D,F> 

«=    .05  5^  

=     3  F:  aEggBi 


Enter  ualue  or  press  SOLVE 

ehi^m^ti  1 1  ii  hn  n  inn 


Page  17-18 


Chapter  1 8 
Statistical  Applications 

In  this  Chapter  we  introduce  statistical  applications  of  the  calculator  including 
statistics  of  a  sample,  frequency  distribution  of  data,  simple  regression, 
confidence  intervals,  and  hypothesis  testing. 

Pre-programmed  statistical  features 

The  calculator  provides  pre-programmed  statistical  features  that  are  accessible 
through  the  keystroke  combination  Lr^J_^(  (same  key  as  the  number  L5J 
key).  The  statistical  applications  available  in  the  calculator  are: 


3 

Frequencies.. 

3 

Fit  data.. 

H 

L'UHH'iry  ftatf.. 

5 

Hypoth.  tests.. 

6 

ConF.  interuaL. 

These  applications  are  presented  in  detail  in  this  Chapter.  First,  however,  we 
demonstrate  how  to  enter  data  for  statistical  analysis. 

Entering  data 

For  the  analysis  of  a  single  set  of  data  (a  sample)  we  can  use  applications 
number  1,  2,  and  4  from  the  list  above.  All  of  these  applications  require  that 
the  data  be  available  as  columns  of  the  matrix  SDAT.  This  can  be 
accomplished  by  entering  the  data  in  columns  using  the  matrix  writer, 

This  operation  may  become  tedious  for  large  number  of  data  points.  Instead, 
you  may  want  to  enter  the  data  as  a  list  (see  Chapter  8)  and  convert  the  list 
into  a  column  vector  by  using  program  CRMC  (see  Chapter  10). 
Alternatively,  you  can  enter  the  following  program  to  convert  a  list  into  a 
column  vector.  Type  the  program  in  RPN  mode: 

*  OBJ^  1  2  ^LIST  HARRYS- 


Page  18-1 


Store  the  program  in  a  variable  called  LXC.  After  storing  this  program  in  RPN 
mode  you  can  also  use  it  in  ALG  mode. 

To  store  a  column  vector  into  variable  SDAT  use  function  STOZ,  available 
through  the  catalog  (CB-^  ),  e.g.,  STOZ  (ANS(l))  in  ALG  mode. 

Example  1  -  Using  the  program  LXC,  defined  above,  create  a  column  vector 
using  the  following  data:  2.1  1.2  3.1  4.5  2.3  1.1  2.3  1.5  1.6 
2.2    1.2  2.5. 

In  RPG  mode,  type  in  the  data  in  a  list: 

{2.1   1.2   3.1  4.5   2.3    1.1  2.3    1.5    1.6   2.2    1.2   2.5  }(^!!I!H 
Use  function  STOS  to  store  the  data  into  SDAT. 

Calculating  single-variable  statistics 

Assuming  that  the  single  data  set  was  stored  as  a  column  vector  in  variable 
ZDAT.  To  access  the  different  STAT  programs,  press  Lr?J_^  .  Press  ECU  to 
select  1.  Single-var..  There  will  be  available  to  you  an  input  form  labeled 
SINGLE-VARIABLE  STATISTICS,  with  the  data  currently  in  your  SDAT  variable 
listed  in  the  form  as  a  vector.  Since  you  only  have  one  column,  the  field  Col: 
should  have  the  value  l  in  front  of  it.  The  Type  field  determines  whether  you 
are  working  with  a  sample  or  a  population,  the  default  setting  is  Sample. 
Move  the  cursor  to  the  horizontal  line  preceding  the  fields  Mean,  Std  Dev, 
Variance,  Total,  Maximum,  Minimum,  pressing  the  I^IDl!  soft  menu  key  to 
select  those  measures  that  you  want  as  output  of  this  program.  When  ready, 
press  Sill!:!!!!!.  The  selected  values  will  be  listed,  appropriately  labeled,  in  the 
screen  of  your  calculator. 

Example  1  --  For  the  data  stored  in  the  previous  example,  the  single-variable 
statistics  results  are  the  following: 

Mean:  2.133,  Std  Dev:  0.964,  Variance:  0.929 
Total:  25.6,  Maximum:  4.5,  Minimum:  1.1 


Page  1  8-2 


Definitions 

The  definitions  used  for  these  quantities  are  the  following: 

Suppose  that  you  have  a  number  data  points  xl7  x2,  x3/  representing 
different  measurements  of  the  same  discrete  or  continuous  variable  x.  The  set 
of  all  possible  values  of  the  quantity  x  is  referred  to  as  the  population  of  x.  A 
finite  population  will  have  only  a  fixed  number  of  elements  Xj.  If  the  quantity  x 
represents  the  measurement  of  a  continuous  quantity,  and  since,  in  theory, 
such  a  quantity  can  take  an  infinite  number  of  values,  the  population  of  x  in 
this  case  is  infinite.  If  you  select  a  sub-set  of  a  population,  represented  by 
the  n  data  values  {xl7  x2,  xn},  we  say  you  have  selected  a  sample  of  values 
of  x. 


Samples  are  characterized  by  a  number  of  measures  or  statistics.  There  are 
measures  of  central  tendency,  such  as  the  mean,  median,  and  mode,  and 
measures  of  spreading,  such  as  the  range,  variance,  and  standard  deviation. 

Measures  of  central  tendency 

The  mean  (or  arithmetic  mean)  of  the  sample,  x,  is  defined  as  the  average 
value  of  the  sample  elements, 

x=  —  ^xr 

n  i=x 

The  value  labeled  Total  obtained  above  represents  the  summation  of  the 
values  of  x,  or  ZXj  =  n-  x.  This  is  the  value  provided  by  the  calculator  under 
the  heading  Mean.  Other  mean  values  used  in  certain  applications  are  the 
geometric  mean,  xg,  or  the  harmonic  mean,  xh,  defined  as: 


-ni  —  _L-V_L 

Examples  of  calculation  of  these  measures,  using  lists,  are  available  in 
Chapter  8. 


The  median  is  the  value  that  splits  the  data  set  in  the  middle  when  the 
elements  are  placed  in  increasing  order.  If  you  have  an  odd  number,  n,  of 
ordered  elements,  the  median  of  this  sample  is  the  value  located  in  position 


Page  1  8-3 


(n+l)/2.  If  you  have  an  even  number,  n,  of  elements,  the  median  is  the 
average  of  the  elements  located  in  positions  n/2  and  (n+l)/2.  Although  the 
pre-programmed  statistical  features  of  the  calculator  do  not  include  the 
calculation  of  the  median,  it  is  very  easily  to  write  a  program  to  calculate  such 
quantity  by  working  with  lists.  For  example,  if  you  want  to  use  the  data  in 
ZDAT  to  find  the  median,  type  the  following  program  in  RPN  mode  (see 
Chapter  21  for  more  information  on  programming  in  User  RPL  language).: 

■*:  ^  nC  *  RCLZ  DUP  SIZE  2  GET  IF  1  >  THEN  nC  COL-  SWAP  DROP 
OBJ^  1  +  ->ARRY  END  OBJ^  OBJ^  DROP  DROP  DUP     n  ■*:  ^LIST  SORT 
IF  'n  MOD  2  ==  0'  THEN  DUP  'n/27  EVAL  GET  SWAP  '(n+1  )/2'  EVAL  GET  + 
2  /  ELSE  '(n+l)/2'  EVAL  GET  END  "Median"  ^TAG  * 

Store  this  program  under  the  name  MED.  An  example  of  application  of  this 
program  is  shown  next. 

Example  2  -  To  run  the  program,  first  you  need  to  prepare  your  SDAT  matrix. 
Then,  enter  the  number  of  the  column  in  ZDAT  whose  median  you  want  to  find, 
and  press  lllulll!.     For  the  data  currently  in  IDAT  (entered  in  an  earlier 
example),  use  program  MED  to  show  that  Median:  2.15. 

The  mode  of  a  sample  is  better  determined  from  histograms,  therefore,  we 
leave  its  definition  for  a  later  section. 

Measures  of  spread 

1  n 

The  variance  (Var)  of  the  sample  is  defined  as  sx  =  /  t  (xf  —  x)  . 

n-\  M 

The  standard  deviation  (St  Dev)  of  the  sample  is  just  the  square  root  of  the 
variance,  i.e.,  sx. 

The  range  of  the  sample  is  the  difference  between  the  maximum  and  minimum 
values  of  the  sample.  Since  the  calculator,  through  the  pre-programmed 
statistical  functions  provides  the  maximum  and  minimum  values  of  the  sample, 
you  can  easily  calculate  the  range. 


Page  1  8-4 


Coefficient  of  variation 

The  coefficient  of  variation  of  a  sample  combines  the  mean,  a  measure  of 
central  tendency,  with  the  standard  deviation,  a  measure  of  spreading,  and  is 
defined,  as  a  percentage,  by:  Vx  =  (sx/  x)l  00. 

Sample  vs.  population 

The  pre-programmed  functions  for  single-variable  statistics  used  above  can  be 
applied  to  a  finite  population  by  selecting  the  Type:  Population  in  the 
single-variable  statistics  screen.  The  main  difference  is  in  the  values 
of  the  variance  and  standard  deviation  which  are  calculated  using  n  in  the 
denominator  of  the  variance,  rather  than  (n-1 ). 


Example  3  -  If  you  were  to  repeat  the  exercise  in  Example  1  of  this  section, 
using  Population  rather  than  Sample  as  the  Type,  you  will  get  the  same 
values  for  the  mean,  total,  maximum,  and  minimum.  The  variance  and 
standard  deviation,  however,  will  be  given  by:  Variance:  0.852,  Std  Dev: 
0.923. 


Obtaining  frequency  distributions 

The  application  2.  Frequencies.,  in  the  STAT  menu  can  be  used  to  obtain 
frequency  distributions  for  a  set  of  data.  Again,  the  data  must  be  present  in 
the  form  of  a  column  vector  stored  in  variable  ZDAT.  To  get  started,  press 
(j-H  stat<^v?  iiiiii=!=ili:iiiilii .  The  resulting  input  form  contains  the  following  fields: 


EDAT:  the  matrix  containing  the  data  of  interest. 

Col:  the  column  of  ZDAT  that  is  under  scrutiny. 

X-Min:  the  minimum  class  boundary  (default  =  -6.5). 

Bin  Count:  the  number  of  classes(default  =  1  3). 

Bin  Width:  the  uniform  width  of  each  class  (default  =  1 ). 


Definitions 

To  understand  the  meaning  of  these  parameters  we  present  the  following 
definitions:  Given  a  set  of  n  data  values:  {xl7  x2,  xn}  listed  in  no  particular 
order,  it  is  often  required  to  group  these  data  into  a  series  of  classes  by 
counting  the  frequency  or  number  of  values  corresponding  to  each  class. 
(Note:  the  calculators  refers  to  classes  as  bins). 


Page  1  8-5 


Suppose  that  the  classes,  or  bins,  will  be  selected  by  dividing  the  interval  (xbot, 
xtop),  into  k  =  Bin  Count  classes  by  selecting  a  number  of  class  boundaries, 
i.e.,  {xB],  xB2,  xBk+1},  so  that  class  number  1  is  limited  by  xBrxB2,  class 
number  2  by  xB2-  xB3,  and  so  on.  The  last  class,  class  number  k,  will  be 
limited  by  xBk-  xBk+1. 

The  value  of  x  corresponding  to  the  middle  of  each  class  is  known  as  the  class 
mark,  and  is  defined  as  xMj  =  (xBj  +  xB  i+1)/2,  for  i  =  1 ,  2,  k. 

If  the  classes  are  chosen  such  that  the  class  size  is  the  same,  then  we  can 
define  the  class  size  as  the  value  Bin  Width  =  Ax  =  (xmax  -  xmin)  /  k, 

and  the  class  boundaries  can  be  calculated  as  xBj  =  xbot  +  (i  -  1 )  *  Ax. 

Any  data  point,  Xj,  j  =  1 ,  2,       n,  belongs  to  the  i-th  class,  if  xB;<  Xj  <  xB  i+1 

The  application  2.  Frequencies.,  in  the  STAT  menu  will  perform  this  frequency 
count,  and  will  keep  track  of  those  values  that  may  be  below  the  minimum 
and  above  the  maximum  class  boundaries  (i.e.,  the  outliers). 

Example  1  -  In  order  to  better  illustrate  obtaining  frequency  distributions,  we 
want  to  generate  a  relatively  large  data  set,  say  200  points,  by  using  the 
following: 

•  First,  seed  the  random  number  generator  using:  RDZt 25)  in  ALG  mode, 
or  25  [enter)  RDZ  in  RPN  mode  (see  Chapter  17). 

•  Type  in  the  following  program  in  RPN  mode: 

■*:  ^  n  *  1  n  FOR  j  RAND  100  *  2  RND  NEXT  n  ^LIST  :*■ 
and  save  it  under  the  name  RDLIST  (RanDom  number  LIST  generator). 

•  Generate  the  list  of  200  number  by  using  RDLIST(200)  in  ALG  mode,  or 
298  [enter)  in  RPN  mode. 

•  Use  program  LXC  (see  above)  to  convert  the  list  thus  generated  into  a 
column  vector. 

•  Store  the  column  vector  into  SDAT,  by  using  function  STOZ. 


Page  1  8-6 


•  Obtain  single-variable  information  using:  LrD  stat  BIS.  Use  Sample  for 
the  Type  of  data  set,  and  select  all  options  as  results.  The  results  for  this 
example  were: 

Mean:  51.0406,  Std  Dev:  .29.5893...,  Variance:  875.529... 
Total:  10208.12,  Maximum:  99.35,  Minimum:  0.13 

This  information  indicates  that  our  data  ranges  from  values  close  to  zero  to 
values  close  to  1  00.  Working  with  whole  numbers,  we  can  select  the  range 
of  variation  of  the  data  as  (0,1  00).    To  produce  a  frequency  distribution  we 
will  use  the  interval  (1 0,90)  dividing  it  into  8  bins  of  width  1  0  each. 

•  Select  the  program  2.  Frequencies.,  by  using  LHJ_^^3?  BIS.  The 
data  is  already  loaded  in  ZDAT,  and  the  option  Col  should  hold  the  value 
1  since  we  have  only  one  column  in  ZDAT. 

•  Change  X-Min  to  1 0,  Bin  Count  to  8,  and  Bin  Width  to  1  0,  then  press 

:::::::::P!::::::: 

Using  the  RPN  mode,  the  results  are  shown  in  the  stack  as  a  column  vector  in 
stack  level  2,  and  a  row  vector  of  two  components  in  stack  level  1 .  The 
vector  in  stack  level  1  is  the  number  of  outliers  outside  of  the  interval  where 
the  frequency  count  was  performed.  For  this  case,  I  get  the  values  [  25.  22.] 
indicating  that  there  are,  in  my  ZDAT  vector,  25  values  smaller  than  1 0  and 
22  larger  than  90. 

•  Press      J  to  drop  the  vector  of  outliers  from  the  stack.  The  remaining 
result  is  the  frequency  count  of  data.  This  can  be  translated  into  a  table 
as  shown  below. 

This  table  was  prepared  from  the  information  we  provided  to  generate  the 
frequency  distribution,  although  the  only  column  returned  by  the  calculator  is 
the  Frequency  column  (fj.  The  class  numbers,  and  class  boundaries  are  easy 
to  calculate  for  uniform-size  classes  (or  bins),  and  the  class  mark  is  just  the 
average  of  the  class  boundaries  for  each  class.  Finally,  the  cumulative 
frequency  is  obtained  by  adding  to  each  value  in  the  last  column,  except  the 
first,  the  frequency  in  the  next  row,  and  replacing  the  result  in  the  last  column 


Page  1  8-7 


of  the  next  row.  Thus,  for  the  second  class,  the  cumulative  frequency  is 
1  8+1  5  =  33,  while  for  class  number  3,  the  cumulative  frequency  is  33  +  16  = 
49,  and  so  on.  The  cumulative  frequency  represents  the  frequency  of  those 
numbers  that  are  smaller  than  or  equal  to  the  upper  boundary  of  any  given 
class. 


Class 
No. 
i 

Class 

Bound. 

Class 
mark. 
Xm, 

Frequency 
f; 

Cumulative 
frequency 

XB; 

XB  i+1 

<XBt 

outlier 

below 
range 

25 

1 

10 

!  20 

15 

18 

18 

2 

20 

30 

25 

14 

32 

3 

30 

40 

35 

17 

49 

4 

40 

50 

45 

17 

66 

5 

50 

60 

55 

22 

88 

6 

60 

70 

65 

22 

110 

7 

70 

80 

75 

24 

134 

k  =  8 

80 

90 

85 

19 

153 

>XBk 

outliers 

above 
range 

22 

Given  the  (column)  vector  of  frequencies  generated  by  the  calculator,  you  can 
obtain  a  cumulative  frequency  vector  by  using  the  following  program  in  RPN 
mode: 

■*:  DUP  SIZE  1  GET  ^  freq  k  *  {k  1}  0  CON  ^  cfreq  *  'freq(l,l)'  EVAL 
'cfreq(l,l)'  STO  2  k  FOR  j  'cfreq(j-l  ,1 )  +freq(j,l)'  EVAL  'cfreq  (j,l)'  STO 
NEXT  cfreq  * 

Save  it  under  the  name  CFREQ.  Use  this  program  to  generate  the  list  of 
cumulative  frequencies  (press  MSSM  with  the  column  vector  of  frequencies  in 
the  stack).  The  result,  for  this  example,  is  a  column  vector  representing  the  last 
column  of  the  table  above. 


Page  1  8-8 


Histograms 

A  histogram  is  a  bar  plot  showing  the  frequency  count  as  the  height  of  the 
bars  while  the  class  boundaries  shown  the  base  of  the  bars.  If  you  have  your 
raw  data  (i.e.,  the  original  data  before  the  frequency  count  is  made)  in  the 
variable  ZDAT,  you  can  select  Histogram  as  your  graph  type  and  provide 
information  regarding  the  initial  value  of  x,  the  number  of  bins,  and  the  bin 
width,  to  generate  the  histogram.  Alternatively,  you  can  generate  the  column 
vector  containing  the  frequency  count,  as  performed  in  the  example  above, 
store  this  vector  into  SDAT,  and  select  Barpiot  as  your  graph  type.  In  the 
next  example,  we  show  you  how  to  use  the  first  method  to  generate  a 
histogram. 

Example  1  -  Using  the  200  data  points  generated  in  the  example  above 
(stored  as  a  column  vector  in  SDAT),  generate  a  histogram  plot  of  the  data 
using  X-Min  =  10,  Bin  Count  =  16,  and  Bin  Width  =  5. 

•  First,  press  CSDi^  (simultaneously,  if  in  RPN  mode)  to  enter  the  PLOT 
SETUP  screen.  Within  this  screen,  change  Type:  to  Histogram,  and  check 
that  the  option  Col:  1  is  selected.  Then,  press  (nxt)WEEM. 

•  Next,  press  LjnJ ML  (simultaneously,  if  in  RPN  mode)  to  enter  the  PLOT 
WINDOW  -  HISTOGRAM  screen.  Within  that  screen  modify  the 
information  to  H-View:  1  0   90,  V-View:  0     1  5,  Bar  Width:  5. 

•  Press  MBMi  llillillll  to  generate  the  following  histogram: 


•  Press  H3EEHS  to  return  to  the  previous  screen.  Change  the  V-view  and  Bar 
Width  once  more,  now  to  read  V-View:  0  30,  Bar  Width:  10.  The  new 
histogram,  based  on  the  same  data  set,  now  looks  like  this: 


Page  1  8-9 


A  plot  of  frequency  count,  fj,  vs.  class  marks,  xMj,  is  known  as  a  frequency 
polygon.  A  plot  of  the  cumulative  frequency  vs.  the  upper  boundaries  is 
known  as  a  cumulative  frequency  ogive.  You  can  produce  scatterplots  that 
simulate  these  two  plots  by  entering  the  proper  data  in  columns  1  and  2  of  a 
new  ZDAT  matrix  and  changing  the  Type:  to  scatter  in  the  PLOT  SETUP 
window. 


Fitting  data  to  a  function  y  =  f(x) 

The  program  3,  Fit  data..,  available  as  option  number  3  in  the  STAT  menu, 
can  be  used  to  fit  linear,  logarithmic,  exponential,  and  power  functions  to 
data  sets  (x,y),  stored  in  columns  of  the  SDAT  matrix.  In  order  for  this 
program  to  be  effective,  you  need  to  have  at  least  two  columns  in  your  SDAT 
variable. 


Example  1  -  Fit  a  linear  relationship  to  the  data  shown  in  the  table  below: 


X 

0 

1 

2 

3 

4 

5 

y 

0.5 

2.3 

3.6 

6.7 

7.2 

11 

First,  enter  the  two  rows  of  data  into  column  in  the  variable  SDAT  by 
using  the  matrix  writer,  and  function  STOS. 

To  access  the  program  3.  Fit  data..,  use  the  following  keystrokes: 
iJ+J_sm<^?<^?WBM  The  input  form  will  show  the  current  SDAT, 
already  loaded.  If  needed,  change  your  set  up  screen  to  the  following 
parameters  for  a  linear  fitting: 


_fit  data;! 

EDAT :  _ 

Hod«u  Linear  Fit 


[[  0.   .5  ]  [  1. 


Enter  statistic  l_dgtg  

1 1 1  HI  1 1 1 1  \^mfi  1 1 II I  III  HB  I  ■ 


Page  18-10 


•  To  obtain  the  data  fitting  press  EC9.  The  output  from  this  program, 
shown  below  for  our  particular  data  set,  consists  of  the  following  three 
lines  in  RPN  mode: 

3:  '0.195238095238  +  2.00857142857*X' 
2:  Correlation:  0.983781424465 
1 :  Covariance:  7.03 

Level  3  shows  the  form  of  the  equation.  In  this  case,  y  =  0.06924  +  0.00383 
x.  Level  2  shows  the  sample  correlation  coefficient,  and  level  1  shows  the 
covariance  of  x-y. 

Definitions 

For  a  sample  of  data  points  (x,y),  we  define  the  sample  covariance  as 

1  n 

sxy  = — -y) 

n  —  v  i=x 

The  sample  correlation  coefficient  for  x,y  is  defined  as 


r  = 

xy 


xy 


Where  sx,  sy  are  the  standard  deviations  of  x  and  y,  respectively,  i.e. 

1        n  1  n 


si 


n-\  i=l  n-\  i=l 


The  values  sxy  and  rxy  are  the  "Covariance"  and  "Correlation,"  respectively, 
obtained  by  using  the  "Fit  data"  feature  of  the  calculator. 

Linearized  relationships 

Many  curvilinear  relationships  "straighten  out"  to  a  linear  form.  For  example, 
the  different  models  for  data  fitting  provided  by  the  calculator  can  be 
linearized  as  described  in  the  table  below. 


Page  18-11 


Type  of 
Fitting 

Actual 
Model 

Linearized 
Model 

Indep. 
variable 

Depend. 
Variable 

Covar. 

Linear 

y  =  a  +  bx 

[same] 

X 

y 

Sxy 

Log. 

y  =  a  +  b  ln(x) 

[same] 

ln(x) 

y 

S|n(x),y 

Exp. 

y  =  a  eDX 

ln(y)  =  ln(a)  +  bx 

X 

ln(y) 

Sx,ln(y) 

Power 

y  =  axD 

ln(y)  =  ln(a)  +  b  ln(x) 

ln(x) 

ln(y) 

S|n(x),ln(y) 

The  sample  covariance  of  ^,r|  is  given  by  s,  =  ^ (<^.  -^){rji  -rj) 

n-l 

Also,  we  define  the  sample  variances  of  E,  and  r|,  respectively,  as 

n    i  i=i                             n    i  i=x 
The  sample  correlation  coefficient  r^  is  r,  =  

The  general  form  of  the  regression  equation  is   r|  =  A  +  B^. 
Best  data  fitting 

The  calculator  can  determine  which  one  of  its  linear  or  linearized  relationship 
offers  the  best  fitting  for  a  set  of  (x,y)  data  points.  We  will  illustrate  the  use  of 
this  feature  with  an  example.  Suppose  you  want  to  find  which  one  of  the  data 
fitting  functions  provides  the  best  fit  for  the  following  data: 


X 

0.2 

0.5 

1 

1.5 

2 

4 

5 

10 

y 

3.16 

2.73 

2.12 

1.65 

1.29 

0.47 

0.29 

0.01 

First,  enter  the  data  as  a  matrix,  either  by  using  the  Matrix  Writer  and 
entering  the  data,  or  by  entering  two  lists  of  data  corresponding  to  x  and  y 
and  using 


Page  18-12 


the  program  CRMC  developed  in  Chapter  10.  Next,  save  this  matrix  into  the 
statistical  matrix  2DAT,  by  using  function  STOZ. 

Finally,  launch  the  data  fit  application  by  using:  Lr^J  _JBT  ^r?  <^r?  ECEII1  .  The 
display  shows  the  current  ZDAT,  already  loaded.  Change  your  set  up  screen 
to  the  following  parameters  if  needed: 


EDflT: 

EC 

■  2  3-16  ]  C  .... 

1 

Y-C0l: 2 

Mods:  i. : 

Press  IE!!!!,  to  get: 

1 :  '3.99504833324*EXP(-.579206831 203*X)' 
2:  Correlation:  -0.996624999526 
3:  Covariance:  -6.23350666124 

The  best  fit  for  the  data  is,  therefore,  y  =  3.995  e058x. 

Obtaining  additional  summary  statistics 

The  application  4.  Summary  stats.,  in  the  STAT  menu  can  be  useful  in  some 
calculations  for  sample  statistics.  To  get  started,  press  L  r>  j  STAl  once  more, 
move  to  the  fourth  option  using  the  down-arrow  key  ^3? ,  and  press  !!!!!E!I!!!!!!!. 
The  resulting  input  form  contains  the  following  fields: 

SDAT:  the  matrix  containing  the  data  of  interest. 

X-Col,  Y-Col:      these  options  apply  only  when  you  have  more  than  two 

columns  in  the  matrix  ZDAT.    By  default,  the  x  column  is 

column  1 ,  and  the  y  column  is  column  2. 
_zx_  ZY...:       summary  statistics  that  you  can  choose  as  results  of  this 

program  by  checking  the  appropriate  field  using  [VCHK] 

when  that  field  is  selected. 

Many  of  these  summary  statistics  are  used  to  calculate  statistics  of  two 
variables  (x,y)  that  may  be  related  by  a  function  y  =  f(x).  Therefore,  this 
program  can  be  thought  off  as  a  companion  to  program  3.  Fit  data.. 

Example  1  -  For  the  x-y  data  currently  in  SDAT,  obtain  all  the  summary 
statistics. 


Page  18-13 


•  To  access  the  summary  stats...  option,  use:  OJ^^?^?^iIl 

•  Select  the  column  numbers  corresponding  to  the  x-  and  y-data,  i.e.,  X-Col: 
1,  and  Y-Col:  2. 

•  Using  the  IKiilllEi  key  select  all  the  options  for  outputs,  i.e.,  _ZX,  _IY,  etc. 

•  Press  B™1  to  obtain  the  following  results: 

XX:  24.2,  ZY:  1 1 .72,  ZX2:  148.54,  ZY2:  26.6246,  ZXY:  12.602,  NZ:8 


Note:  There  are  two  other  applications  under  the  STAT  menu,  namely,  5. 
Hypth.  tests.,  and  6,  Conf.  Interval..  These  two  applications  will  be  discussed 
later  in  the  chapter. 


Calculation  of  percentiles 

Percentiles  are  measures  that  divide  a  data  set  into  100  parts.  The  basic 
procedure  to  calculate  the  1 00  p-th  Percentile  (0  <  p  <  1)  in  a  sample  of  size 
n  is  as  follows: 

1 .  Order  the  n  observations  from  smallest  to  largest. 

2.  Determine  the  product  n-p 

A.  If  n-p  is  not  an  integer,  round  it  up  to  the  next  integer  and  find  the 
corresponding  ordered  value. 

B.  If  n-p  is  an  integer,  say  k,  calculate  the  mean  of  the  k-th  and  (k-1)  th 
ordered  observations. 


Note:  Integer  rounding  rule,  for  a  non-integer  x.yz...,  if  y  >  5,  round  up  to 
x+1 ;  if  y  <  5,  round  up  to  x. 


This  algorithm  can  be  implemented  in  the  following  program  typed  in  RPN 
mode  (See  Chapter  21  for  programming  information): 

■*:  SORT  DUP  SIZE  ^  p  X  n  *:  n  p  *  ^  k  *  IF  k  CEIL  k  FLOOR  -  NOT  THEN  X 
k  GET  X  k  1  +  GET  +  2  /  ELSE  k  0  RND  X  SWAP  GET  END  :*■  * 


Page  18-14 


which  well  store  in  variable  %TILE  (percent-tile).  This  program  requires  as 
input  a  value  p  within  0  and  1,  representing  the  lOOp  percentile,  and  a  list 
of  values.  The  program  returns  the  1  OOp  percentile  of  the  list. 

Example  1  -  Determine  the  37%  percentile  of  the  list  {21013512367 
9}.  In  RPN  mode,  enter  0.27  (^{21013512367  9}  [enter}  |%fflJO. 
In  ALG  mode,  enter  %TILE(0.27,{2, 1 ,0, 1 ,3,5, 1 ,2,3,6,7,9}.  The  result  is  1 . 

The  STAT  soft  menu 

All  the  pre-programmed  statistical  functions  described  above  are  accessible 
through  a  STAT  soft  menu.  The  STAT  soft  menu  can  be  accessed  by  using,  in 
RPN  mode,  the  command:  96  MENU 

You  can  create  your  own  program,  say  !!=!:!i!i!!i!ii!i!!l!i,  to  activate  the  STAT  soft  menu 
directly.  The  contents  of  this  program  are  simply:  £  96  MENU  3\ 

The  STAT  soft  menu  contains  the  following  functions: 


Pressing  the  key  corresponding  to  any  of  these  menus  provides  access  to 
different  functions  as  described  below. 

The  DATA  sub-menu 

The  DATA  sub-menu  contains  functions  used  to  manipulate  the  statistics  matrix 
ZDATA: 


The  operation  of  these  functions  is  as  follows: 

2+  :  add  row  in  level  1  to  bottom  of  ZDATA  matrix. 

Z-  :  removes  last  row  in  SDATA  matrix  and  places  it  in  level  of  1  of  the  stack. 

The  modified  ZDATA  matrix  remains  in  memory. 
CLE  :  erases  current  ZDATA  matrix. 


Page  18-15 


ZDAT:  places  contents  of  current  ZDATA  matrix  in  level  1  of  the  stack. 
C5DZDAT:  stores  matrix  in  level  1  of  stack  into  ZDATA  matrix. 

The  ZPAR  sub-menu 

The  ZPAR  sub-menu  contains  functions  used  to  modify  statistical  parameters. 
The  parameters  shown  correspond  to  the  last  example  of  data  fitting. 


Intercept:  3.995048333 
Slope:  -.579206331203 
hlodel:  EXPFIT 

The  parameters  shown  in  the  display  are: 

Xcol:  indicates  column  of  ZDATA  representing  x  (Default:  1) 
Ycol:  indicates  column  of  SDATA  representing  y  (Default:  2) 
Intercept:  shows  intercept  of  most  recent  data  fitting  ((Default:  0) 
Slope:  shows  slope  of  most  recent  data  fitting  (Default:  0) 
Model:  shows  current  data  fit  model  (Default:  UNFIT) 

The  functions  listed  in  the  soft  menu  keys  operate  as  follows: 

XCOL:  entered  as  n  EI5EH,  changes  Xcol  to  n. 

YCOL:  entered  as  n  EH31,  changes  Ycol  to  n. 

ZPAR:    shows  statistical  parameters. 

RESET:  reset  parameters  to  default  values 

INFO:  shows  statistical  parameters 

The  MODL  sub-menu  within  ZPAR 

This  sub-menu  contains  functions  that  let  you  change  the  data-fitting  model  to 
UNFIT,  LOGFIT,  EXPFIT,  PWRFIT  or  BESTFIT  by  pressing  the  appropriate  button. 

The  1 VAR  sub  menu 

The  1VAR  sub  menu  contains  functions  that  are  used  to  calculate  statistics  of 
columns  in  the  SDATA  matrix. 


Page  18-16 


The  functions  available  are  the  following: 

TOT:  show  sum  of  each  column  in  ZDATA  matrix. 

MEAN:  shows  average  of  each  column  in  ZDATA  matrix. 

SDEV:  shows  standard  deviation  of  each  column  in  ZDATA  matrix. 

MAXZ:  shows  maximum  value  of  each  column  in  ZDATA  matrix. 

MINZ:  shows  average  of  each  column  in  ZDATA  matrix. 

BINS:  used  as  xs/  Ax,  n  [BINS],  provides  frequency  distribution  for  data  in 

Xcol  column  in  ZDATA  matrix  with  the  frequency  bins  defined  as 

[xs,xs+Ax],  [xs,xs+2Ax],...,  [xs,xs+nAx]. 
VAR:  shows  variance  of  each  column  in  ZDATA  matrix. 

PSDEV:  shows  population  standard  deviation  (based  on  n  rather  than  on  (n-1)) 

of  each  column  in  ZDATA  matrix. 
PVAR:  shows  population  variance  of  each  column  in  SDATA  matrix. 
MINZ:  shows  average  of  each  column  in  ZDATA  matrix. 

The  PLOT  sub-menu 

The  PLOT  sub-menu  contains  functions  that  are  used  to  produce  plots  with  the 
data  in  the  ZDATA  matrix. 


The  functions  included  are: 

BARPL:  produces  a  bar  plot  with  data  in  Xcol  column  of  the  ZDATA  matrix. 
HISTP:  produces  histogram  of  the  data  in  Xcol  column  in  the  SDATA  matrix, 
using  the  default  width  corresponding  to  1  3  bins  unless  the  bin  size  is 
modified  using  function  BINS  in  the  1  VAR  sub-menu  (see  above). 
SCATR:  produces  a  scatterplot  of  the  data  in  Ycol  column  of  the  ZDATA 
matrix  vs.  the  data  in  Xcol  column  of  the  ZDATA  matrix.  Equation 
fitted  will  be  stored  in  the  variable  EQ. 


Page  18-17 


The  FIT  sub-menu 

The  FIT  sub-menu  contains  functions  used  to  fit  equations  to  the  data  in 
columns  Xcol  and  Ycol  of  the  ZDATA  matrix. 


l:  ^| 

H^iHHI;aiJ;H^IJ;H.VjE4];l;l»l>l 

The  functions  available  in  this  sub-menu  are: 

ZLINE:  provides  the  equation  corresponding  to  the  most  recent  fitting. 
LR:  provides  intercept  and  slope  of  most  recent  fitting. 
PREDX:  used  as  y  lllillli!!:!:!,  given  y  find  x  for  the  fitting  y  =  f(x). 
PREDY:  used  as  x  111:111!,  given  x  find  y  for  the  fitting  y  =  f(x). 
CORR:  provides  the  correlation  coefficient  for  the  most  recent  fitting. 
COV:  provides  sample  co-variance  for  the  most  recent  fitting 
PCOV:  shows  population  co-variance  for  the  most  recent  fitting. 

The  SUMS  sub-menu 

The  SUMS  sub-menu  contains  functions  used  to  obtain  summary  statistics  of 
the  data  in  columns  Xcol  and  Ycol  of  the  ZDATA  matrix. 


IX  :  provides  the  sum  of  values  in  Xcol  column. 
ZY  :  provides  the  sum  of  values  in  Ycol  column. 
ZXA2  :  provides  the  sum  of  squares  of  values  in  Xcol  column. 
SYA2  :  provides  the  sum  of  squares  of  values  in  Ycol  column. 
ZX*Y:  provides  the  sum  of  x-y,  i.e.,  the  products  of  data  in  columns  Xcol  and 
Ycol. 

NE  :  provides  the  number  of  columns  in  the  SDATA  matrix. 

Example  of  STAT  menu  operations 

Let  ZDATA  be  the  matrix  shown  in  next  page. 

•  Type  the  matrix  in  level  1  of  the  stack  by  using  the  Matrix  Writer. 

•  To  store  the  matrix  into  SDATA,  use:  illliEi!!  C3D  BED 


Page  18-18 


Calculate  statistics  of  each  column:  EEED  EH39: 


lil'iiYil'ii 


IffiTH 


produces  [38.5  87.5  82799.8] 
produces  [5.5.  12.5  11828.54...] 
produces  [3.39...  6.78...  21097.01 ...] 
produces  [10  21.5  55066] 
produces  [1.1  3.7  7.8] 
produces  [1 1.52  46.08  445084146.33] 
produces  [3.142...  6.284...  19532.04...] 
produces  [9.87...   39.49...  381500696.85. 


Data: 


1.1 

3.7 

7.8 

3.7 

8.9 

101 

2.2 

5.9 

25 

5.5 

12.5 

612 

6.8 

15.1 

2245 

9.2 

19.9 

24743 

10.0 

21.5 

55066 

Generate  a  scatterplot  of  the  data  in  columns  1  and  2  and  fit  a  straight 
line  to  it: 


■■!:!r:-vr:i  :i:!lll!:«v":« 


resets  statistical  parameters 


6: 

Kcol:  1- 
Vcol:  2. 
Intercept:  0. 
Slope:  0. 
Model:  L INF IT 
ll^llHJlfEJJll^llllti^HiliJ 


[nxtJ  mm  MSm  ESSE 


produces  scatterplot 

draws  data  fit  as  a  straight  line 


Page  18-19 


nr*m'^:T:Tjjgjj«n,uir*i,Tji 

SliluSa  returns  to  main  display 

Determine  the  fitting  equation  and  some  of  its  statistics: 


siiiii  fill  iiiiiH 


3 

1  IIBII! 


jjjCHHj,j.j 


produces  ' i.5+2*xf 

produces  Intercept:    1.5,    Slope:  2 

produces  0.75 

produces  3.  50 

produces  1 .0 

produces  23.04 

produces  1 9.74... 


Obtain  summary  statistics  for  data  in  columns  1  and  2:  ilillii!  i!]3I!:!l!: 


produces  38.5 
produces  87.5 
produces  280.87 
produces  1  370.23 
produces  61 9.49 
produces  7 


Fit  data  using  columns  1  (x)  and  3  (y)  using  a  logarithmic  fitting: 

®  BED  Em  3  mSM       select  Ycol  =  3,  and 
lllll  Emm  select  Model  =  Logfit 


II 


Hcol:  1. 
Vcol:  3. 
Intercept:  1.5 
Slope:  2. 
Model:  LOGFIT 
nijTrjjTrriT^jiiii^iynui 


Page  1  8-20 


f  ,  ,.,-r  *\  !:sii!ls:i"!i:":  :"":'::ivi:": 


produce  scattergram  of  y  vs.  x 
show  line  for  log  fitting 


Obviously,  the  log-fit  is  not  a  good  choice. 

ili!]!!!!!!!!!!  returns  to  normal  display. 

•     Select  the  best  fitting  by  using: 

BIB  mm  mm  iiliiiilii!         shows  EXPFIT  as  the  best  fit  for  these  data 


P1 
6: 

Kcol:  1. 

Vcol:  3. 
Intercept:  2. 6545321 32 

Slope:  .992727735591 

Model:  EXPFIT 


{  nxt  )  IliYil  Hp |::1"111|]T| 

am 

2300  mm 


5 0 


(jvxT)  lllijElj!  113333  I1I3I1 


produces  '2.6545*EXP(0.9927*X)' 

produces  0.99995...  (good  correlation) 

produces  6.81  39 

produces  463.37 

produce  scattergram  of  y  vs.  x 

show  line  for  log  fitting 


To  return  to  STAT  menu  use:  [nxtJ mEM 
To  get  your  variable  menu  back  use 


Page  18-21 


Confidence  intervals 

Statistical  inference  is  the  process  of  making  conclusions  about  a  population 
based  on  information  from  sample  data.    In  order  for  the  sample  data  to  be 
meaningful,  the  sample  must  be  random,  i.e.,  the  selection  of  a  particular 
sample  must  have  the  same  probability  as  that  of  any  other  possible  sample 
out  of  a  given  population.  The  following  are  some  terms  relevant  to  the 
concept  of  random  sampling: 

•  Population:  collection  of  all  conceivable  observations  of  a  process  or 
attribute  of  a  component. 

•  Sample:  sub-set  of  a  population. 

•  Random  sample:  a  sample  representative  of  the  population. 

•  Random  variable:  real-valued  function  defined  on  a  sample  space.  Could 
be  discrete  or  continuous. 

If  the  population  follows  a  certain  probability  distribution  that  depends  on 
a  parameter  9,  a  random  sample  of  observations  (X1,X2,X3,...  ,  Xn),  of  size 
n,  can  be  used  to  estimate  0. 

•  Sampling  distribution:  the  joint  probability  distribution  of  X1,X2,X3,...  ,  Xn. 

•  A  statistic:  any  function  of  the  observations  that  is  quantifiable  and  does 
not  contain  any  unknown  parameters.  A  statistic  is  a  random  variable 
that  provides  a  means  of  estimation. 

•  Point  estimation:  when  a  single  value  of  the  parameter  9  is  provided. 

•  Confidence  interval:  a  numerical  interval  that  contains  the  parameter  9  at 
a  given  level  of  probability. 

•  Estimator:  rule  or  method  of  estimation  of  the  parameter  9. 

•  Estimate:  value  that  the  estimator  yields  in  a  particular  application. 

Example  1  -  Let  X  represent  the  time  (hours)  required  by  a  specific 
manufacturing  process  to  be  completed.  Given  the  following  sample  of  values 
of  X:  2.2    2.5    2.1    2.3    2.2.    The  population  from  where  this  sample  is 
taken  is  the  collection  of  all  possible  values  of  the  process  time,  therefore,  it  is 
an  infinite  population.  Suppose  that  the  population  parameter  we  are  trying 


Page  1  8-22 


to  estimate  is  its  mean  value,  u..  We  will  use  as  an  estimator  the  mean  value 

_      |  n 

of  the  sample,  X,  defined  by  (a  rule):  X  =  —  •  /  Xx . 

n  i=i 

For  the  sample  under  consideration,  the  estimate  of  ja  is  the  sample  statistic  x 
=  (2.2+2.5+2. 1+2. 3+2. 2)/5  =  2.36.  This  single  value  of  X,  namely  x  = 
2.36,  constitutes  a  point  estimation  of  the  population  parameter  ju. 

Estimation  of  Confidence  Intervals 

The  next  level  of  inference  from  point  estimation  is  interval  estimation,  i.e., 
instead  of  obtaining  a  single  value  of  an  estimator  we  provide  two  statistics,  a 
and  b,  which  define  an  interval  containing  the  parameter  0  with  a  certain 
level  of  probability.  The  end  points  of  the  interval  are  known  as  confidence 
limits,  and  the  interval  (a,b)  is  known  as  the  confidence  interval. 

Definitions 

Let  (C\,CU)  be  a  confidence  interval  containing  an  unknown  parameter  9. 

•  Confidence  level  or  confidence  coefficient  is  the  quantity  (1-a),  where  0  < 
a  <  1 ,  such  that  P[Q  <  6  <  CJ  =  1  -  a,  where  P[  ]  represents  a 
probability  (see  Chapter  1  7).  The  previous  expression  defines  the  so- 
called  two-sided  confidence  limits. 

•  A  lower  one-sided  confidence  interval  is  defined  by  Pr[Q  <  0]  =  1  -  a. 

•  An  upper  one-sided  confidence  interval  is  defined  by  Pr[0  <  CJ  =  1  -  a. 

•  The  parameter  a  is  known  as  the  significance  level.  Typical  values  of  a 
are  0.01,  0.05,  0.1,  corresponding  to  confidence  levels  of  0.99,  0.95, 
and  0.90,  respectively. 

Confidence  intervals  for  the  population  mean  when  the 
population  variance  is  known 

Let  X  be  the  mean  of  a  random  sample  of  size  n,  drawn  from  an  infinite 
population  with  known  standard  deviation  a.  The  100(1 -a)  %  [i.e.,  99%, 
95%,  90%,  etc.],  central,  two-sided  confidence  interval  for  the  population 
mean  ju  is  (  X-za/2-a/Vn  ,  X+za/2-a/Vn  ),  where  za/2  is  a  standard  normal 
variate  that  is  exceeded  with  a  probability  of  a  /2.  The  standard  error  of 
the  sample  mean,  X,  is  -a/Vn. 


Page  1  8-23 


The  one-sided  upper  and  lower  100(1 -a)  %  confidence  limits  for  the 
population  mean  u.  are,  respectively,  X+za-a/Vn  ,  and  X-za-a/Vn  .  Thus,  a 
lower,  one-sided,  confidence  interval  is  defined  as  (-00  ,  X+za-a/Vn),  and  an 
upper,  one-sided,  confidence  interval  as  (X-za-a/Vn,+oo).    Notice  that  in 
these  last  two  intervals  we  use  the  value  za,  rather  than  za/2. 

In  general,  the  value  zk  in  the  standard  normal  distribution  is  defined  as  that 
value  of  z  whose  probability  of  exceedence  is  k,  i.e.,  Pr[Z>zk]  =  k,  or  Pr[Z<zk] 
=  1  -  k.    The  normal  distribution  was  described  in  Chapter  17. 

Confidence  intervals  for  the  population  mean  when  the 
population  variance  is  unknown 

Let  X  and  S,  respectively,  be  the  mean  and  standard  deviation  of  a  random 
sample  of  size  n,  drawn  from  an  infinite  population  that  follows  the  normal 
distribution  with  unknown  standard  deviation  a.  The  100-(l-a)  %  [i.e.,  99%, 
95%,  90%,  etc.]  central  two-sided  confidence  interval  for  the  population  mean 
ju,  is  (  X-  tn.l  a/2  -S  /Vn  ,  X+  tn.l  a/2  -S/Vn  ),  where  tn.l  a/2  is  Student's  t  variate 
with  v  =  n-1  degrees  of  freedom  and  probability  a/2  of  exceedence. 

The  one-sided  upper  and  lower  100-  (1-a)  %  confidence  limits  for  the 
population  mean  \i  are,  respectively, 

X  +  tn_W2  -S/Vn  ,  and  X-  tn_W2  S  /Vn. 

Small  samples  and  large  samples 

The  behavior  of  the  Student's  t  distribution  is  such  that  for  n>30,  the 
distribution  is  indistinguishable  from  the  standard  normal  distribution.  Thus, 
for  samples  larger  than  30  elements  when  the  population  variance  is  unknown, 
you  can  use  the  same  confidence  interval  as  when  the  population  variance  is 
known,  but  replacing  a  with  S.  Samples  for  which  n>30  are  typically  referred 
to  as  large  samples,  otherwise  they  are  small  samples. 

Confidence  interval  for  a  proportion 

A  discrete  random  variable  X  follows  a  Bernoulli  distribution  if  X  can  take  only 
two  values,  X  =  0  (failure),  and  X  =  1  (success).    Let  X  ~  Bernoulli(p),  where  p 


Page  1  8-24 


is  the  probability  of  success,  then  the  mean  value,  or  expectation,  of  X  is  E[X] 
=  p,  and  its  variance  is  Var[X]  =  p(l-p). 

If  an  experiment  involving  X  is  repeated  n  times,  and  k  successful  outcomes 
are  recorded,  then  an  estimate  of  p  is  given  by  p'=  k/n,  while  the  standard 
error  of  p'  is  ap>=  V(p-(l-p)/n)  .  In  practice,  the  sample  estimate  for  p,  i.e.,  p' 
replaces  p  in  the  standard  error  formula. 


For  a  large  sample  size,  n>30,  and  n-p  >  5  and  n-(l-p)>5,  the  sampling 
distribution  is  very  nearly  normal.  Therefore,  the  lOO(l-a)  %  central  two-sided 
confidence  interval  for  the  population  mean  p  is  (p'+za/2-ap<,  p'+za/2-cy  ). 
For  a  small  sample  (n<30),  the  interval  can  be  estimated  as  (p'-tn.-|  a/2-ap',p'+tn. 

l,a/2,c>p')- 

Sampling  distribution  of  differences  and  sums  of  statistics 

Let  S]  and  S2  be  independent  statistics  from  two  populations  based  on 
samples  of  sizes  n]  and  n2,  respectively.  Also,  let  the  respective  means  and 
standard  errors  of  the  sampling  distributions  of  those  statistics  be  jus1  and  u.S2, 
and  as1  and  aS2,  respectively.  The  differences  between  the  statistics  from  the 
two  populati  ons,  S]-S2,  have  a  sampling  distribution  with  mean  u.  $i-s2  —  mi  ~ 
juS2,  and  standard  error  a  S1_S2  =  (as12+  aS22)1/2.  Also,  the  sum  of  the  statistics 
Tt+T2  has  a  mean  u.  S1+S2  =  u.s1  +u.S2,  and  standard  error  asl+S2  =  (as12  + 

Estimators  for  the  mean  and  standard  deviation  of  the  difference  and  sum  of 
the  statistics  S]  and  S2  are  given  by: 


_2  2 
°  S\    ,    °  S2 


In  these  expressions,  and  X2  are  the  values  of  the  statistics  and  S2  from 
samples  taken  from  the  two  populations,  and  as12  and  aS22  are  the  variances 
of  the  populations  of  the  statistics  S]  and  S2  from  which  the  samples  were 
taken. 


Page  1  8-25 


Confidence  intervals  for  sums  and  differences  of  mean  values 

If  the  population  variances  a^and  a22  are  known,  the  confidence  intervals  for 
the  difference  and  sum  of  the  mean  values  of  the  populations,  i.e.,  jut+u^,  are 
given  by: 


+  ^,(X1±X2)  +  za/2 


For  large  samples,  i.e.,  n}  >  30  and  n2  >  30,  and  unknown,  but  equal, 
population  variances  c]2=  a22,  the  confidence  intervals  for  the  difference  and 
sum  of  the  mean  values  of  the  populations,  i.e.,  u^iu^,  are  given  by: 


{Xx±X2)-zal2 


.  +  ^^Xx±X2)  +  zal2 


■  +  ■ 


If  one  of  the  samples  is  small,  i.e.,  <  30  or  n2  <  30,  and  with  unknown,  but 
equal,  population  variances  o2  =  a22,  we  can  obtain  a  "pooled"  estimate  of 
the  variance  of  |a1±jLi2,  as  sp2  =  [(nrl  )-s12+(n2-l  )-s22]/(  n1+n2-2). 


In  this  case,  the  centered  confidence  intervals  for  the  sum  and  difference  of 
the  mean  values  of  the  populations,  i.e.,  (ii+(i2,  are  given  by: 

((X,  ±X2)-tVta/2  -s2p,  (Xi  ±X2)  +  tv>a/2  -s]) 


where  v  =  n!+n2-2  is  the  number  of  degrees  of  freedom  in  the  Student's  t 
distribution. 


In  the  last  two  options  we  specify  that  the  population  variances,  although 
unknown,  must  be  equal.  This  will  be  the  case  in  which  the  two  samples  are 
taken  from  the  same  population,  or  from  two  populations  about  which  we 
suspect  that  they  have  the  same  population  variance.   However,  if  we  have 


Page  1  8-26 


reason  to  believe  that  the  two  unknown  population  variances  are  different,  we 
can  use  the  following  confidence  interval 

((Jf,  ±  x2 )  -  tVra/2  ■  4_±j2 ,  (xx  ±  x2 )  +  tv,a/2  ■  4_±j2 ) 

where  the  estimated  standard  deviation  for  the  sum  or  difference  is 


and  n,  the  degrees  of  freedom  of  the  t  variate,  are  calculated  using  the  integer 
value  closest  to 

[(512/n1)/(n1-l)]  +  [(522/n2)/(/72-l)] 

Determining  confidence  intervals 

The  application  6.  Conf  Interval  can  be  accessed  by  using  Lr> J_^gi^ 
SECIIi.  The  application  offers  the  following  options: 


Confidence  intervals 

a.I-inT:  MI-M2.. 

3.2-inT:  i  P.. 

H.s-inT:  Pi-pa.. 

5.T-inT:  i  H_. 

e.T-inT:  m-wa.. 

These  options  are  to  be  interpreted  as  follows: 

1.  Z-INT:  1  ju.:  Single  sample  confidence  interval  for  the  population  mean, 
ju,  with  known  population  variance,  or  for  large  samples  with  unknown 
population  variance. 

2.  Z-INT:  jli1-|li2.:  Confidence  interval  for  the  difference  of  the  population 
means,  jar  \i2/  with  either  known  population  variances,  or  for  large 
samples  with  unknown  population  variances. 


Page  1  8-27 


3.  Z-INT:  1  p.:   Single  sample  confidence  interval  for  the  proportion,  p,  for 
large  samples  with  unknown  population  variance. 

4.  Z-INT:  pi-  p2.:  Confidence  interval  for  the  difference  of  two  proportions, 
prp2/  for  large  samples  with  unknown  population  variances. 

5.  T-INT:  1  ju.:   Single  sample  confidence  interval  for  the  population  mean, 
ju,  for  small  samples  with  unknown  population  variance. 

6.  T-INT:  |ul-ja2.:    Confidence  interval  for  the  difference  of  the  population 
means,  u.r  ju2,  for  small  samples  with  unknown  population  variances. 

Example  1  -  Determine  the  centered  confidence  interval  for  the  mean  of  a 
population  if  a  sample  of  60  elements  indicate  that  the  mean  value  of  the 
sample  is  x  =  23.2,  and  its  standard  deviation  is  s  =  5.2.  Use  a  =  0.05. 
The  confidence  level  is  C  =  1-a  =  0.95. 


Select  case  1  from  the  menu  shown  above  by  pressing  IEIIIIIL  Enter  the  values 
required  in  the  input  form  as  shown: 


conF.  im.  =  i      KnoHn  t 
23.3 
5.2 
60. 


Press  IllSIill  to  obtain  a  screen  explaining  the  meaning  of  the  confidence 
interval  in  terms  of  random  numbers  generated  by  a  calculator.  To  scroll 
down  the  resulting  screen  use  the  down-arrow  key  ^3? .  Press  1132!!  when 
done  with  the  help  screen.  This  will  return  you  to  the  screen  shown  above. 


To  calculate  the  confidence  interval,  press  The  result  shown  in  the 

calculator  is:   

mmiS.Z  ConFid«n«  inttrva L mm 

critics  z=±l.  959964 
n  Hin  =£1.93424 
u  hox  =£4.61576 


^■^■IlMiiFiffniiiwnnafliM 


Page  1  8-28 


The  result  indicates  that  a  95%  confidence  interval  has  been  calculated.  The 
Critical  z  value  shown  in  the  screen  above  corresponds  to  the  values  ±za/2  in 
the  confidence  interval  formula  (  X-za/2-a/Vn  ,  X+za/2-a/Vn  ).  The  values  u. 
Min  and  u.  Max  are  the  lower  and  upper  limits  of  t