Rapid Poisson series evaluation
William H. Harr, Bernard Kaufman · Astrodynamics Conference · 1982
A method is p r e s e n t e d f o r o p t i m i z i n g t h e e v a l u a t i o n o f Poisson s e r i e s by computer, i n terms o f execu t ion speed and s t o r a g e s i z e . The method is b a s e d on using t h e d i s t r i b u t i v e p r i n c i p l e r e c u r s i v e l y t o remove supe r f luous a d d i t i o n s and m u l t i p l i c a t i o n s , and t r igonomet r i c i d e n t i t i e s t o remove c a l l s t o t h e s y s t e m s i n e and c o a i n e r o u t i n e s . R e s u l t s of t h e a p p l i c a t i o n of t h i s method t o a s e m i a n a l y t i c S a t e l l i t e Ol’bi t p r e d i c t i o n model are presen ted , showing a speed i n c r e a s e by e. f a c t o r of 8 over t h e e v a l u a t i o n of t h e series i n t h e i r Poisson form. I n t r o d u c t i o n With t h e advent of machine automated a l g e b r a i c manipulat ion programs, many problems which were fo rma l ly i n t r a c t a b l e because of t h e extreme amount of a l g e b r a r e q u i r e d h a v e nor g e n e r a t e d new i n t e r e s t . T h e r e are c u r r e n t l y many areas of r e s e a r c h i n which a n a l y t i c me thods a re b e i n g d e v e l o p e d u s i n g t h e s e m a n i p u l a t i o n p rograms t o r e p l a c e o l d e r numeric techniques. With many of t h e s e problems, a new complicat ion has a r i s e n i n t h a t i t o f t en occurs t h a t s e r i e s development i n t h e a n a l y t i c o r s e m i a n a l y t i c methods become very l a r g e i n t h e t o t a l number of terms r equ i r ed . While t h e m a n i p u l a t i o n p rograms can e a s i l y h a n d l e t h e r e s u l t i n g a l g e b r a , t h e p r a c t i c a l a p p l i c a t i o n of t h e s e t h e o r i e s i n a u s e f u l computer program is sometimes very l i m i t e d because of the phys ica l s i z e of the s e r i e s . Th i s paper o u t l i n e s a method of op t imiz ing t h e e v a l u a t i o n of l a r g e Po i s son s e r i e s i n terms of speed, number of unique f a c t o r s , and program s t o r a g e , without d e l e t i n g any of t h e terms i n t h e s e r i e s , or a f f e c t i n g t h e accuracy of t h e e v a l u a t i o n . Po i s son S e r i e s Compression A Poisson s e r i e s , of t h e form: EH/ N a’ bk ... SIN/COS(Aa + BB..) can be r ep resen ted symbolical ly i n t h e form: 5 ai bi ci i = l where a . are t h e numeric coeff ic ients(M/N), bi a r e t h e a l g e b r a i c c o e f f i c i e n t s ( a j bk. .) , and c . .1 are t h e t r i g o n o m e t r i c terms(SIN/COS(Ao+BB..) ). If t h e r e are P unique numeric c o e f f i c i e n t s . y unique a l g e b r a i c c o e f f i c i e n t s , and z unique t r igonomet r i c terms, then t h e maximum s e r i e s l eng th p o s s i b l e is y x Z. Two terms of t h e s e r i e s with t h e same a l g e b r a i c and t r igonomet r i c f a c t o r s , but d i f f e r e n t numeric c o e f f i c i e n t s , a r e assumed t o be combined by t h e d i s t r i b u t i v e p r i n c i p l e t o form a s i n g l e term with a numeric c o e f f i c i e n t equal to the sum of t h e two o r i g i n a l numeric c o e f f i c i e n t s . The maximum number of unique numeric c a e f f i c i e n t s ( r ) i s then also y x z . The minimum s e r i e s l eng th p o s s i b l e is t h e maximum of y and 2. Few series are composed of terms i n which a l l t h r e e f a c t o m are unique t o each term. Therefore , f a r the g r e a t m a j o r i t y of cases, r e p r e s e n t i n g the s e r i e s i n Poisson form is d u p l i c a t i n g information. T h i s paper w i l l show haw d u p l i c a t e information can be removed, r e s u l t i n g i n fewer o p e r a t i o n s r equ i r ed t o eva lua te t h e s e r i e s , and less space necessary t o r ep resen t it. A s a f i r s t s t e p , a Poisson s e r i e s of the form: