# Full text of "univOfIllinoisUrbana :: illiac :: ILLIAC :: Active Library :: I01-67"

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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