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UNIVERSITY OF ILLINOIS 
DIGITAL COMPUTER 

LIBBARY ROUTINE 11 - S^J 

Interpolation (DOI or SADOl) 
Closed 

51 

Parameters must be read into locations 5 and k before the 

program is read into the store. A number of program 

parameters also need to be seto 

0^ 1, 2^ 5^ h, 5, 6y 7 plus d locations starting at the 

address entered in S before read-in. 

A maximum rounding =-off error of (2 - l) x 2°^^ where 

d = depth of interpolation o 

(6 + 5.5d + 2,'^d ) ms where d = depth of interpolation o 

The routine interpolates to arbitrary depths ^s ^^ o^^ o^ 
more tables which may be stored in the machine » It is assumed that the table 
consists of a sequential set of function values, f (x^) for increasing arguments^ 
X., equally spaced by h = x^^ - x^ > 0. f (x^)^ls to be stored in location 
(a + [l/6] X.] where h = l/n, n is an integer « f(0) is stored in location a. 
(Location a itself need not be filled, e.go^ with a = 500, h = oOl, the entire 
table might comprise the 11 entries for x^ = e85, -84, 085, -oo, o95 which would 
be stored in locations 583-593 )• The arguments of the entries can be trans- 
lated by an arbitrary amount provided that the argument of any fimction 
desired is similarly translated o 

The routine will be atitomatically left ' whenever two successive 
interpolates differ by less than e where e < 2" ^ is entered in Sk before read- in « 
If it is desired to go to fixed depth, S^ should be set to before read-ino 
Enter with x in R. and the following words in the main program 



TITLE 

TYPE 

NUMBER OF WORDS 

PARAMETERS 



TEMPORARY STORAGE 

ACCURACY 

DURATION 
DESCRIPTION 



P 



p+1 



50 dF 
50 pF 
26 qF 
00 aF 



p+2 00 F 
00 hJ 
whee q is the location of this routine, ( the routine may be entered with x in 



- 2 - 



if control is then transferred to the right hand side of q). The routine will 

be left with f (x) in both A and Qo 

RESTRICnoiilS 

(1) The depth of interpolation d^ and the argiment step h^ 
must satisfy |hd| < lo 

(2) If x^ and X be the arguments at the beginning and end 
of the table respectively, the routine should not be used to interpolate for an 

X that lies within; 

[(d-^1) h/2] C^ x^ or [ (d-l) h/2] of x if d;. is oddo 



METHOD 
used? 



[(d->2) h/2] of x^ or [(d) h/2] oT x^ if d is even, 
Neville's method of successive linear int^erpolations is 



^(^i^T 4.1^14.?* 



X, 



. T X. , ) = 

.1 -1 1+.1 ^ 



1+J-l 1+j 

- - J (x. - 

1+J-l' ^ 1+J 



[f(x.x,^^, 0,0, x.^.^^) (x._^.-x) -f (x.^ix.^2-"- 
(x.-x)]/(x. .»x.) 

^ 1 ^" ^ 1+J J' 



x._,.) 

1+J 



This expression is applied successively with j = 1^2j35G.«ad, 



Igr 



DATE 1/5/53 


Rr: 


5/20/59 


PROGRAMME) BY 
APPROVED BY 


J 


No Snyder 


J» ] 


Po Nash 





LOCATION 


ORDER 


iroras PAflE 1 Ti 





00 K(ii; 

kO F 

Ifl 6f 




Store argunent 
Clear 6 




1 


35 y 

i^6 6y 




9rlng In link 
Plant d X 2"^^ 




2 


]A 18L 
42 12L 




Plant table address 




3 


i;^ laL 

42 5L 




Plant h*8 address 




k 


lA 18L 
42 USL 




Plant link address 




5 


41 7P 




Waste 






L5 (p^)T 


By 3 


Call in h to 1 Aiirt 4 




6 


40 IF 
40 4f 








7 


L5 6f 
LO 49L 








8 


10 ?1F 

42 TF 








9 


L5 F 
10 19F 




Ccnnpute initial entry addresses to 




10 


66 IF 
35 F 




table and plant in the linear 
interpolator . 




I] 


10 20F 










LO 7? 






12 


40 2F 

L4 (ih-1)F 


B3r2 






13 


42 22L 
L4 i8L 




■ 




Ik 


42 2QL 
L5 47L 








15 


42 24l 
50 2F 








16 


75 IF 




Conpute initial (x^-x) and (x ) 






00 59F 









17 
18 

19 

20 
21 
22 

25 
2k 

■ 25 
26 

27 
28 

29 

51 
52 



ORDER 
LO F 
^0 F 
kO 3F 

l4 if 

hQ 5F 
^1 7F 

50 5F 
75 ( )F 
ho 2F 

51 F 
50 5F 
7^ ( )F 
LO 2F 
66 i|F 
S5 F 

i^O ( )F 
L5 7F 
L4 lf9L 
kO 7F 
LO 6f 
32 35L 
L5 20L 
Lk 18l 
h2 20L 
L5 22L 
Lif 18l 
42 22L 
L5 2ifL 
L^ 18L 
k2 2kh 
L5 3F 
Lk IF 
40 5F 
L5 5F 
lA IF 



From 35 
By l4,28 
and UO 



NOTKS 


PAGE 2 


in 3 and 5 respectively. 

> 


• 


Clear 7 for counter 





THE LINEAR INTERPOLATOR 



By 13,30, 
mnd 39 



By 15,31 



and 39 



'I 



Finisheai with this order of inter- 
polation 



Step addresses in the linear inter- 
polator 



Advance [x -x] and (x -x) hy h, 
and repeat. 



LOCATION 


ORDER' ■" 


^OT^ ^ ^ ^Mm3 : 




40 5F 




. 


55 


26 20L 




„ 


■ , 


L5 6f 


Frcjm 27 


Have we reached desired depth of 


% 


LO i^9L 
i^O 6f 




Interpolation? ■ 


37 


LO if9L 
32 58L 


■ 


■ . ■ ' 


:>Q 


22 krL 








L5 l^TL 


ftm 37 


3 

l^eset adiiresses in the llnotiq: 


Z'9 


42 22L 




Interpolatoir * ; 




,- k2 24L 




' ' ' 


ko 


L4 18L 
42 20L 




J 


i^l. 


L5 4f 




- 


k2 


l4 if 
ko 4f« 




Ad-vance (x :,-x. ) "by h 


' 


L5 F 




Reset (x.-x)-lk 3 . 


^3 


40 3F 






kk 


L4 4f 
lio 5F 
L5 S5 




Compute (x. ^-x) and place in 4 


^5 


iP¥^ 




Test difference of succeBeire 


.■! 


Waf 




interpolates against e 


k6 


L7 25" 




" 




LO 50L 




C ' 


U7 


32 1% 








L5 S3 


From 31 


Put f(x) m A and Q ani leftir6 


48 


' 50 S3 








26 (p+3)F By 4 


' 


k9 


00 IF 
00 F 




Unit of count 


50 


00 F 




Error parameter = e 




00 34 




■^' 








9 

i 



II