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Algorithms: Bessel function I, asymptotic expansion

Algorithms: Bessel function I, asymptotic expansion q u i t e e a s i l y in.to t h e s y s t e m , a n d a m o r e s o p h i s t i c a t e d t n e t h o d of g e n e r a t i n g t e s t d a t a w o u l d b e u s e f u l i n a n u m b e r of i n s t a n c e s . E x t e n s i o n t o a FOWt':aAN t y p e l a n g u a g e a l s o appears to be possible, but this would probably require the t r a n s f e r of a s u b s t a r t t i a l a m o u n t of i n f o r m a t i o n f r o m t h e compiler to the control system. However, even the pro- p o s e d s y s t e m a l l o w s t h e e o m p u t ; e r to i a k e a /a:rge~, t m o r e a c t i v e p a r t i n th.e d i a g n o s t i c t a s k , a~d A clcnowledgme~ ts A number of the details of the diagnostic card f o r m a t s ar~, L~: result of discussio.ns with members o[ the progran~n)i~,~ ~r ~ { responsib!e for the t?KMAD system. { Algorithms C o n t r i b u t i o n s to this d e p a r t m e n t must be in the form s t a t e d in the Algorithms I ) e p a r t m e n t Policy S t ~ t e m e n t (Communications, F e b r u a r y , 1960). C o n t r i b u t i o n s should be sent to J. H. Wegslein, C o i n p u t a t i o n L a b o r a t o r y , N a t i o n a l Bureau of S t a n d a r d s , W a s h i n g t o n 25, 1). C. Algorithms should be in tile P u b l i c a t i o n form of A L G O L and w r i t t e n in a style p a t t e r n e d after the mosb recent algorithms appearing in this d e p a r t m e n t . A l t h o u g h each a l g o r i t h m has been t e s t e d by its c o n t r i b u t o r , no w a r r a n t y , express or implied, is made b y t h e c o n t r i b u t o r , the editor, or the Association for C o m p u t i n g M a c h i n e r y as to the accuracy a n d functioning of t h e a l g o r i t h m and related a l g o r i t h m materiM and no responsibility is a,ssumed by the, c o n t r i b u t o r , the editor, or th<~ Association for C o m p u t i n g M a c h i n e r y in connection therewith. The reproduction of algorithms appearing in this d e p a r t m e n t is explicitly p e r m i t t e d without a n y charge. When reproduction is for publication t)urposes, reference m u s t be made to the algorithm ~mthor and to the Co,nmunications issue bearing the algorithm. :~#, t 5. BESSEL FUNCTION [, SERIES EXPANSION D o r o t h e a S. C l a r k e G e n e r a l E l e c t r i c Co., F P L D , C i n c i n n a t i 15, O h i o Compute the Bessel function I~ (X) when n and X are within the t)ounds of the series expansion. The preeedure calling s t a t e m e n t gives n, X and an absolute tolerance 5 for determining the point at which the terms of the summt~tion become insignificant. Special c'tse : I0(0) = 1 ; I(n, X, ~) = : (Is) S := 0 ; SUln := 0 (n # O) ; go t o S T R T (X = O) ; b e g i n Is := 1 ; r e t u r n e n d summ := I ; g o t o S U R E sfac := 1 (s = O) ; go t o H R E t := 1 ( l ) s sfac := sfac X t snfac := sfac t := s + i ( [ ) s + n snfac := snfac X t sunlIn :~--- s u n l ~ - (X/2),,~×~/(sfac X snfae) (6 < abs (suture - sum)) s := s + l ; sum := summ ; g o t o S T R T e n d I s : = summ ; r e t u r n procedure begin which the terms of the Stlnu~a.ti(:)n ]) :(!( lit ~ significant ; I(n, X, 6) = : ([A) ~2 * i~:~,~ ] ' -~:i sum : = pe ((2X n ) ' - ' - ( 2 X r - - 1) ~) / (r X~, 1 5 : ':~,~ ~:~'~,~, comment Repe'tt : r := r + i pe := p e X ~ ....... ;i begin procedure begin [: if if STRT: if for [IRE : for suIn := sum q- (---i) r X pe ; go t o Repe~.~ e*~,~ IA : = ( l + s u m ) X ( e x p ( ~ ) / s q r t ~ (2 X ~ X ;~' ~'~' l'etul" n end 7. E U C L U ) r X N Ar, GOltITr~5,~ Robert Claussen ( - m n e m l E l e c t r i c Co., C i n c i n n a t i t 5 , O h i o t , :~ ' ~ • eo nllneil procedure begin EUC : if begin il ~J,~ ~ ~~ ¸ Every pair of numbers a f) riot b o t h zero ~'*o(~:' positive greatest c(mnnon divisor: gcd; : ~~,~i EUC ( a , b ) = : (ged) ~'> '¢~ i. S U R E : if" begin end (a = O) ged := b (b = 0) ; return end ; r e t u r n end if begin gcd := a r2 := a ...... :::;~ ;~' • : here : eOIIlIll eIl t 6. BESSEL FUNCTION I, ASYMPTOTIC EXPANSION D o r o t h e a S. C l a r k e G e n e r a l E l e c t r i c Co., F P L D , C i n c i m m t i 15, O h i o Compute the Bessel Function I , ( X ) when n and X are within the bounds of the asymptotic expansion. T h e procedure calling s t a t e m e n t gives n, X and an absolute tolerance ~ for determining the point at Communications o f t h e ACM g := r2/rl Assumption is made t h a t t r u n c a t on ~akc~ pbr:: %:,~; in the above s t a t e m e n t ; r:=r2-rlXg g e d : = rl r2 := r t ; return end if begin begin ? comment integer end go to here e n d (g) /~ ~ 'T~ 7 ; 2 ¸¸ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications of the ACM Association for Computing Machinery

Algorithms: Bessel function I, asymptotic expansion

Communications of the ACM , Volume 3 (4) – Apr 1, 1960

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Publisher
Association for Computing Machinery
Copyright
Copyright © 1960 by ACM Inc.
ISSN
0001-0782
DOI
10.1145/367177.377894
Publisher site
See Article on Publisher Site

Abstract

q u i t e e a s i l y in.to t h e s y s t e m , a n d a m o r e s o p h i s t i c a t e d t n e t h o d of g e n e r a t i n g t e s t d a t a w o u l d b e u s e f u l i n a n u m b e r of i n s t a n c e s . E x t e n s i o n t o a FOWt':aAN t y p e l a n g u a g e a l s o appears to be possible, but this would probably require the t r a n s f e r of a s u b s t a r t t i a l a m o u n t of i n f o r m a t i o n f r o m t h e compiler to the control system. However, even the pro- p o s e d s y s t e m a l l o w s t h e e o m p u t ; e r to i a k e a /a:rge~, t m o r e a c t i v e p a r t i n th.e d i a g n o s t i c t a s k , a~d A clcnowledgme~ ts A number of the details of the diagnostic card f o r m a t s ar~, L~: result of discussio.ns with members o[ the progran~n)i~,~ ~r ~ { responsib!e for the t?KMAD system. { Algorithms C o n t r i b u t i o n s to this d e p a r t m e n t must be in the form s t a t e d in the Algorithms I ) e p a r t m e n t Policy S t ~ t e m e n t (Communications, F e b r u a r y , 1960). C o n t r i b u t i o n s should be sent to J. H. Wegslein, C o i n p u t a t i o n L a b o r a t o r y , N a t i o n a l Bureau of S t a n d a r d s , W a s h i n g t o n 25, 1). C. Algorithms should be in tile P u b l i c a t i o n form of A L G O L and w r i t t e n in a style p a t t e r n e d after the mosb recent algorithms appearing in this d e p a r t m e n t . A l t h o u g h each a l g o r i t h m has been t e s t e d by its c o n t r i b u t o r , no w a r r a n t y , express or implied, is made b y t h e c o n t r i b u t o r , the editor, or the Association for C o m p u t i n g M a c h i n e r y as to the accuracy a n d functioning of t h e a l g o r i t h m and related a l g o r i t h m materiM and no responsibility is a,ssumed by the, c o n t r i b u t o r , the editor, or th<~ Association for C o m p u t i n g M a c h i n e r y in connection therewith. The reproduction of algorithms appearing in this d e p a r t m e n t is explicitly p e r m i t t e d without a n y charge. When reproduction is for publication t)urposes, reference m u s t be made to the algorithm ~mthor and to the Co,nmunications issue bearing the algorithm. :~#, t 5. BESSEL FUNCTION [, SERIES EXPANSION D o r o t h e a S. C l a r k e G e n e r a l E l e c t r i c Co., F P L D , C i n c i n n a t i 15, O h i o Compute the Bessel function I~ (X) when n and X are within the t)ounds of the series expansion. The preeedure calling s t a t e m e n t gives n, X and an absolute tolerance 5 for determining the point at which the terms of the summt~tion become insignificant. Special c'tse : I0(0) = 1 ; I(n, X, ~) = : (Is) S := 0 ; SUln := 0 (n # O) ; go t o S T R T (X = O) ; b e g i n Is := 1 ; r e t u r n e n d summ := I ; g o t o S U R E sfac := 1 (s = O) ; go t o H R E t := 1 ( l ) s sfac := sfac X t snfac := sfac t := s + i ( [ ) s + n snfac := snfac X t sunlIn :~--- s u n l ~ - (X/2),,~×~/(sfac X snfae) (6 < abs (suture - sum)) s := s + l ; sum := summ ; g o t o S T R T e n d I s : = summ ; r e t u r n procedure begin which the terms of the Stlnu~a.ti(:)n ]) :(!( lit ~ significant ; I(n, X, 6) = : ([A) ~2 * i~:~,~ ] ' -~:i sum : = pe ((2X n ) ' - ' - ( 2 X r - - 1) ~) / (r X~, 1 5 : ':~,~ ~:~'~,~, comment Repe'tt : r := r + i pe := p e X ~ ....... ;i begin procedure begin [: if if STRT: if for [IRE : for suIn := sum q- (---i) r X pe ; go t o Repe~.~ e*~,~ IA : = ( l + s u m ) X ( e x p ( ~ ) / s q r t ~ (2 X ~ X ;~' ~'~' l'etul" n end 7. E U C L U ) r X N Ar, GOltITr~5,~ Robert Claussen ( - m n e m l E l e c t r i c Co., C i n c i n n a t i t 5 , O h i o t , :~ ' ~ • eo nllneil procedure begin EUC : if begin il ~J,~ ~ ~~ ¸ Every pair of numbers a f) riot b o t h zero ~'*o(~:' positive greatest c(mnnon divisor: gcd; : ~~,~i EUC ( a , b ) = : (ged) ~'> '¢~ i. S U R E : if" begin end (a = O) ged := b (b = 0) ; return end ; r e t u r n end if begin gcd := a r2 := a ...... :::;~ ;~' • : here : eOIIlIll eIl t 6. BESSEL FUNCTION I, ASYMPTOTIC EXPANSION D o r o t h e a S. C l a r k e G e n e r a l E l e c t r i c Co., F P L D , C i n c i m m t i 15, O h i o Compute the Bessel Function I , ( X ) when n and X are within the bounds of the asymptotic expansion. T h e procedure calling s t a t e m e n t gives n, X and an absolute tolerance ~ for determining the point at Communications o f t h e ACM g := r2/rl Assumption is made t h a t t r u n c a t on ~akc~ pbr:: %:,~; in the above s t a t e m e n t ; r:=r2-rlXg g e d : = rl r2 := r t ; return end if begin begin ? comment integer end go to here e n d (g) /~ ~ 'T~ 7 ; 2 ¸¸

Journal

Communications of the ACMAssociation for Computing Machinery

Published: Apr 1, 1960

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