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Efficient methods for the numerical integration of ordinary differential equations Creemer, Albert Lee 1962

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EFFICIENT .METHODS. FOR THE NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS  "by ALBERT LEE CREEMER B.A., U n i v e r s i t y o f B r i t i s h Columbia,  1956  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n t h e Department of MATHEMATICS  We accept, t h i s ' t h e s i s as conforming ,.to:.-the' required  standard  THE UNIVERSITY OF BRITISH COLUMBIA April,.1962  In presenting  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y British'Columbia,  I agree t h a t the L i b r a r y s h a l l make i t f r e e l y  a v a i l a b l e f o r r e f e r e n c e and f o r extensive  of  study.  I f u r t h e r agree t h a t p e r m i s s i o n  c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may  g r a n t e d by the Head o f my Department o r by h i s  be  representatives.  I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o i v e d w i t h o u t my w r i t t e n  Department o f  Mathematics  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada. Date_  Columbia,  A p r i l 13,  1962  permission.  ABSTRACT The purpose o f t h i s t h e s i s i s t o s t u d y t h e f a c t o r s , i n v o l v e d : i n d e t e r m i n i n g a most e f f i c i e n t method f o r t h e n u m e r i c a l i n t e g r a t i o n o f t h e d i f f e r e n t i a l equation  x' =  f(t,x) .  By "a most e f f i c i e n t method" we mean a  m e t h o d . r e q u i r i n g a minimum o f computation prescribed  to obtain a solution  within  e r r o r bounds. We o u t l i n e two c o m p u t a t i o n a l p r o c e d u r e s  the p r o p a g a t e d procedure.  error of a general multi-step  and d e r i v e  estimates f o r  method when b a s e d on e i t h e r  These e s t i m a t e s , l e a d us t o c o n c l u d e t h a t .a s t a b l e s i n g l e - i t e r a t e  p r o c e d u r e , i n v o l v i n g one e v a l u a t i o n most e f f i c i e n t l y .  o f f a t each s t e p , w i l l determine  a solution  I n p a r t i c u l a r , t h i s p r o c e d u r e b a s e d on Adams formulas, i s  recommended. E x p e r i m e n t a l r e s u l t s s u p p o r t our c o n c l u s i o n  i n a l l stable  cases.  However, t h e s e r e s u l t s a l s o i n d i c a t e t h a t t h e r o l e o f s t a b i l i t y i n t h e c h o i c e o f a most e f f i c i e n t procedure  i s i n need o f f u r t h e r  investigation.  I hereby c e r t i f y t h a t t h i s abstract  i s satisfactory.  iii.  . ACKNOWLEDGEMENTS The a u t h o r wishes, t o acknowledge h i s • i n d e b t e d n e s s  t o Dr. T-E. H u l l  f o r a d v i c e and guidance r e c e i v e d i n f o r m u l a t i n g and p r e p a r i n g t h i s  thesis.  Thanks a r e a l s o due t o t h e s t a f f o f t h e Computing C e n t r e o f t h e U n i v e r s i t y o f B r i t i s h Columbia. verification of theoretical  T h e i r e f f o r t s made p o s s i b l e t h e p r a c t i c a l  results.  iv.  TABLE OF CONTENTS  CHAPTER I  I n t r o d u c t i o n and Summary  1  CHAPTER I I  Estimates o f the Propagated e r r o r  3  P r o c e d u r e , Flow C h a r t  h  P(EC)  m  PE(CE)  m  '.  P r o c e d u r e , Flow C h a r t  Error Estimate, PE(CE) Error Estimate, P(EC)  m  m  6 - 7  Procedure Procedure  CHAPTER I I I Comparison and C h o i c e o f Methods CHAPTER IV  12  . .  Numerical Results  . . . . . . . . . .  15 20  Comparison o f S i n g l e - I t e r a t e P r o c e d u r e s .  21  Conclusion  22  "BIBLIOGRAPHY  23  CHAPTER I  INTRODUCTION AND  An  SUMMARY  i m p o r t a n t c l a s s o f methods f o r the n u m e r i c a l i n t e g r a t i o n of  the  i n i t i a l value problem:  x' = f ( t , x ) , x(to) =  X  Q  i s the c l a s s of i n t e g r a t i o n f o r m u l a s o f the p r e d i c t o r - c o r r e c t o r purpose o f t h i s s t u d y i s t o i n v e s t i g a t e c e r t a i n p r o p e r t i e s g o a l o f c h o o s i n g the method r e q u i r i n g the  computing f o r the  s o l u t i o n of ( l ) to a prescribed  The  The  o f t h i s wide c l a s s ,  w i t h the u l t i m a t e  Our  type.  l e a s t amount o f  accuracy.  main c o n c e r n i s f o r problems i n v o l v i n g a c o m p l i c a t e d f u n c t i o n  computing t i m e f o r the n u m e r i c a l s o l u t i o n of such problems i s a l m o s t  e n t i r e l y d e t e r m i n e d by the number o f e v a l u a t i o n s technique over a s p e c i f i e d i n t e r v a l of t .  o f f r e q u i r e d by the  integration  Under t h e s e c i r c u m s t a n c e s , the many  c a l c u l a t i o n s o f f r e q u i r e d by the Runge-Kutta method w o u l d be p r o h i b i t i v e  and  f o r t h i s r e a s o n , i n s e a r c h i n g f o r a most e f f i c i e n t means o f i n t e g r a t i o n , c o n s i d e r o n l y the c l a s s o f m u l t i - s t e p To a c h i e v e our  aim,  predictor-corrector  e x p r e s s i o n s w i l l be  we  formulas.  we b e g i n by d e r i v i n g e x p r e s s i o n s d e s c r i b i n g  b e h a v i o u r o f the e r r o r i n t r o d u c e d by the p r e d i c t o r - c o r r e c t o r  and  f.  procedure.  the These  employed i n a study o f the c h a r a c t e r i s t i c s of such methods  i n p a r t i c u l a r , t o determine c r i t e r i a f o r the  s e l e c t i o n o f a most e f f i c i e n t  i n t e g r a t i o n procedure. These c r i t e r i a l e a d us t o c o n s i d e r m u l t i - s t e p involving a single evaluation as t h e r e i s no i n the  d i f f i c u l t y with  i n t e g r a t i o n techniques  o f f at each i n t e g r a t i o n s t e p .  At l e a s t as  s t a b i l i t y , t h e s e t e c h n i q u e s are f o u n d t o be  sense t h a t o v e r a s p e c i f i e d i n t e r v a l o f t , r e l a t i v e l y few  are r e q u i r e d ,  i n o r d e r t o meet p r e s c r i b e d  long best  such c a l c u l a t i o n s  requirements of p r e c i s i o n .  In  p a r t i c u l a r , t e c h n i q u e s o f t h i s type based on Adams f o r m u l a s o f h i g h o r d e r ,  p r o v e t o be h i g h l y . e f f i c i e n t and c o n v e n i e n t . S i m i l a r p r e d i c t o r - c o r r e c t o r procedures f o r the numerical s o l u t i o n of ( l ) have r e c e n t l y been p r o p o s e d b y Hamming [l] , M i l n e and^Reynolds Nordsieck  [5j , and R a l s t o n  [b] .  These methods r e q u i r e o n l y a s i n g l e a p p l i c a t i o n  o f t h e c o r r e c t o r f o r m u l a b u t u s u a l l y two e v a l u a t i o n s integration.  [V],  o f f a t each step o f  To some e x t e n t , t h e s e methods a l s o r e q u i r e s p e c i a l t e c h n i q u e s  which are not g e n e r a l l y  applicable.  In the next chapter, c e r t a i n p r e d i c t o r - c o r r e c t o r , techniques are described  and.estimates are d e r i v e d . f o r the propagated e r r o r .  observations  I n Chapter I I I ,  a r e made on t h e s e estimates, t o d r a w . c o n c l u s i o n s as t o t h e r e l a t i v e  c h a r a c t e r i s t i c s o f s e v e r a l p r e d i c t o r - c o r r e c t o r methods.  F i n a l l y , i n C h a p t e r XV,  e x p e r i m e n t a l evidence- i s o f f e r e d i n s u p p o r t o f t h e a c c u r a c y o f t h e e r r o r e s t i m a t e s and,the v a l i d i t y o f a s s o c i a t e d c o n c l u s i o n s .  These r e s u l t s a l s o  i n d i c a t e t h e need f o r a more complete i n v e s t i g a t i o n o f t h e r o l e o f s t a b i l i t y i n the choice  o f a most e f f i c i e n t method.  CHAPTER I I  3-  ESTIMATES OF THE PROPAGATED ERROR  As mentioned p r e v i o u s l y , s h o u l d t h e f u n c t i o n f on t h e r i g h t - h a n d .side o f ( l ) be d i f f i c u l t t o e v a l u a t e , t h e number o f such e v a l u a t i o n s becomes the f o r e m o s t . f a c t o r  i n determining  the cost of numerical  integration.  An  e f f i c i e n t m u l t i - s t e p method would be one w h i c h r e q u i r e d r e l a t i v e l y few evaluations of f i n i n t e g r a t i n g ( l ) t o w i t h i n a p r e s c r i b e d accuracy,.across s p e c i f i e d i n t e r v a l of t .  To determine i f a p a r t i c u l a r m u l t i - s t e p f o r m u l a  meet p r e s c r i b e d r e q u i r e m e n t s on p r e c i s i o n , we need an e s t i m a t e expression g i v i n g a reasonable numerical for  approximation  a will  o f e r r o r , an  .to t h e e r r o r p r o p a g a t e d t h r o u g h  c a l c u l a t i o n s . However, o u r purpose, i s t o s e l e c t an e f f i c i e n t method  the i n t e g r a t i o n of ( l ) .  e r r o r estimates  F o r t h i s r e a s o n , we s h a l l r e s t r i c t our use o f  t o q u a l i t a t i v e a p p r a i s a l s o f one method r e l a t i v e t o o t h e r  methods r a t h e r t h a n as a means f o r p r e c i s e q u a n t i t a t i v e measures o f I n d e r i v i n g an e r r o r e s t i m a t e , we c a r e f u l l y f o l l o w t h e procedure o f . a p p l y i n g a p r e d i c t o r - c o r r e c t o r p a i r .  computational  We s h a l l c o n s i d e r two such  p r o c e d u r e s , each l e a d i n g t o a d i s t i n c t form o f t h e e r r o r . that a f t e r many.iterations  accuracy.  I t w i l l be seen  o f t h e c o r r e c t o r f o r m u l a , . t h e s e p r o c e d u r e s become  equivalent. P(EC)  m  Procedure  (Predict, Evaluate,.Correct,  m i t e r a t i o n s of the c o r r e c t o r ) .  A l o g i c a l f l o w diagram b e s t d e s c r i b e s t h e c a l c u l a t i o n and f o r t h i s p r o c e d u r e i s shown i n F i g u r e  1.  PREVIOUS STEP  PREDICT  CORRECT  JL TEST  P(EG)™,Procedure F i g u r e 1.  5The 'PREDICT  and 'CORRECT' s t a g e s r e p r e s e n t t h e c a l c u l a t i o n s  1  involved i n applying the r e s p e c t i v e formulas.  At t h e 'EVALUATE f  stage,  r e c o u r s e i s made t o some subprogram f o r t h e c a l c u l a t i o n o f f ( t , y ) , where y i s the approximation t o t h e t r u e s o l u t i o n x ( t ) , p r e c e d i n g 'PREDICT'.or 'CORRECT' s t a g e . of  t h e p r o c e d u r e , may determine  obtained at the immediately  The 'TEST', a p p l i e d a t t h e f i n a l  stage  i f two s u c c e s s i v e i t e r a t e s , y^ ''"^ and y ^ ^ , -  s a t i s f y some c r i t e r i o n o f c l o s e n e s s , o r t h e t e s t may m e r e l y r e p r e s e n t a count of  i t e r a t i o n s t o some f i x e d number, m.  Throughout t h i s d i s c u s s i o n , we  assume f o r s i m p l i c i t y , t h a t t h e • l a t t e r i s t h e c a s e . to  s t e p , . t h i s assumption  PE(CE)  m  Procedure.  shall  I f m does v a r y from.step  o f f e r s a reasonable approximation.  A f l o w diagram f o r t h i s p r o c e d u r e i s shown i n F i g u r e 2 .  The s t a g e s o f c a l c u l a t i o n shown i n F i g u r e 2 may be r e a r r a n g e d i n v a r i o u s ways t o g i v e e q u i v a l e n t c o n f i g u r a t i o n s . p r o c e d u r e , as compared t o a P ( E C )  m  I n any c a s e , t h e P E ( C E )  m  p r o c e d u r e w i t h t h e same c r i t e r i o n o f 'TEST',  i n v o l v e s an e x t r a e v a l u a t i o n o f f . In d e r i v i n g an e s t i m a t e f o r t h e e r r o r p r o p a g a t e d b y a m u l t i - s t e p p r e d i c t o r - c o r r e c t o r method, we c o n s i d e r any f i x e d number m o f i t e r a t i o n s a t e v e r y . i n t e g r a t i o n step.  However, o u r main c o n c e r n w i l l b e t h e case m=l.  i n c r e a s e s , t h e a c c u r a c y and e f f i c i e n c y o f t h e P ( E C ) d i f f e r only s l i g h t l y .  m  and P E ( C E )  m  As m  techniques w i l l  However, f o r s m a l l m, t h e d i s t i n c t i o n becomes i m p o r t a n t .  C o n s i d e r i n g b o t h a c c u r a c y and e f f i c i e n c y , i t w i l l be shown t h a t p r o v i d e d b o t h a r e s t a b l e , t h e PEC t e c h n i q u e i s s u p e r i o r t o PECE.  For t h i s  purpose, we b e g i n b y d e r i v i n g an e s t i m a t e o f t h e e r r o r p r o p a g a t e d b y a p r e d i c t o r c o r r e c t o r method when c a l c u l a t i o n s a r e extended b y t h e P E ( C E )  m  procedure.  Such  an e s t i m a t e i s a l s o i n s t r u c t i v e , i n t h a t i t o f f e r s an i n s i g h t i n t o t h e b e h a v i o u r of t h e a p p r o x i m a t i o n as m grows l a r g e .  PREVIOUS STEP  5_ PREDICT  (CE)  m  Procedure  F i g u r e 2.  7Error estimate,-PE(CE)  procedure.  m  Let  be a p p r o x i m a t i o n s t o t h e t r u e s o l u t i o n , o b t a i n e d from t h e mth . i t e r a t i o n o f ;  the c o r r e c t o r f o r m u l a . a t  Then a t t = t  Q  the p o i n t s  + (ft+k)h, t h e p r e d i c t e d o r d i n a t e i s k-i-  »«  (2) I f f i s e v a l u a t e d .using t h e ( q . - l ) ^  i t e r a t e o f t h e c o r r e c t o r , f o r m u l a , t h e . q^  s  1  i t e r a t e i s given by  C = ~2_ *«  (3)  with  W  +  ^ X ' ^ "' 1 3  <x  y  "  <  :  7  >  )  ^ ^  « W By f o r m a l l y . s u b s t i t u t i n g t h e t r u e s o l u t i o n i n t o ( 2 ) and (3)> we  o b t a i n t h e - t r u n c a t i o n e r r o r o f each f o r m u l a ,  However, any n u m e r i c a l  and  , respectively:  e v a l u a t i o n o f ( 2 ) and ( 3 ) r e s u l t s i n numbers  ^*+h > % = 1^2, . . ., m, such t h a t  & n t v and  8. and (my)  zr+fe. = ^n*-!,  with  n  .  The q u a n t i t i e s  ^nw, Av  •  i n equations  (6) and (7) have been d e t e r m i n e d a t t h e k+1 p r e v i o u s s t e p s , whereas X>+k. and  Xm  denote t h e e r r o r due t o • r o u n d - o f f i n e v a l u a t i n g t h e p r e d i c t o r and q t h  . i t e r a t e of the corrector at t = t We  + (n+k)h.  define  and :  and  subtract  (1+) from (6) and (.5) from (7) t o o b t a i n t h e e r r o r e q u a t i o n s  te-i  K  x  CO In e q u a t i o n s (8) and (9)>  = 4(t«.x„j - J7*..,, _„i), 1,-  (10)  so t h a t at  iL.^ i s some v a l u e o f  ~t»v+j  > . a t l e a s t when t h e d e r i v a t i v e e x i s t s ,  , and on an open., i n t e r v a l between I f i n e q u a t i o n (9)>  and  ^v»ti .  we s e t q = m and r e c u r s i v e l y s u b s t i t u t e to (1'0  corresponding, r e l a t i o n s f o r obtain  •*— o+i^ i n terms o f  , q = m, m-l> . . ., 1,  C;<.| , i = 1,2, n  n+k-1.  t h e n f i n a l l y , we  I f f satisfies a  L i p s c h i t z c o n d i t i o n and h i s s u f f i c i e n t l y s m a l l , as m i n c r e a s e s , e q u a t i o n w i l l g i v e an e x p r e s s i o n indefinitely.  the r e s u l t a n t  f o r t h e e r r o r had t h e c o r r e c t o r been i t e r a t e d  We-now.make t h e assumption t h a t  a c o n s t a n t f o r q = 0 , l , 2 , . . . , m and n .= 1,2, Then equation.(9) becomes  The above-.is a . l i n e a r , d i f f e r e n c e e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t s .  I f we  t a k e q = m i n t h e c o r r e s p o n d i n g homogeneous e q u a t i o n , a f t e r r e c u r s i v e s u b s t i t u t i o n s , we o b t a i n  „(»*»>)  (11)  In equation ( l l ) , © ~ h ^ t , If  +e  (2  +  *»0  L?  4  , and i n p r a c t i c e , j@| < I  i n equation ( l l ) i s replaced by ^  , t h e g e n e r a l s o l u ti o n  i s determined by.the r o o t s o f the polynomial,  (12)  We now assume t h a t t h e q u a n t i t i e s VJ^k  .ft) n ' «-h  a r e c o n s t a n t s i n n and q , denoted b y T  Then a p a r t i c u l a r s o l u t i o n o f  c  ,  'n+b  , Tp , r  , c  , and r p  > and respectively.  the.;.nonhomogeneous e q u a t i o n - c o r r e s p o n d i n g t o ( l l )  when q.= m,.is  (13)  .1-©  ^ fegj - ^ Pi) ^ ( £ < - h £ ^  10. At t h i s p o i n t we a s s e r t t h a t our i n t e r e s t i s c o n f i n e d formulas which are c o n s i s t e n t  to corrector  i n t h e sense t h a t t h e t r u e s o l u t i o n w i l l  (3) t o w i t h i n terms t h a t a r e o ( h ) .  satisfy  F o r our p u r p o s e s , t h e p r e d i c t o r f o r m u l a s h o u l d  a t l e a s t s u p p l y an e x a c t s o l u t i o n t o ( l ) when f = 0. I f  CK^~ ^t,^  ' '  t  h  e  n  l t  i s  n e c e s s a r y ( a s w e l l as s u f f i c i e n t ) f o r t h e  c o e f f i c i e n t s o f any p r e d i c t o r - c o r r e c t o r p a i r t o s a t i s f y :  i n o r d e r t o be c o n s i s t e n t i n t h e sense d e s c r i b e d  ( H u l l , Luxemburg [2] ).  I f we a p p l y t h e f i r s t o f r e l a t i o n s (lh) t o (13), we o b t a i n , a f t e r r e g r o u p i n g terms:  (15;  U>-  —  j  /._  ;  Z_r—  '  I f t h e k+1 r o o t s o f (12) a r e denoted b y  which f o r s i m p l i c i t y , a r e assumed d i s t i n c t f o r a l l m, t h e c o m p l e t e s o l u t i o n t o (10) when q = m, w i l l have t h e form  e T - A ~ . , X 1 , A-xA-t + ••• + A ^ ,  (16)  /w,  +  " °^  •  The use o f a k^* o r d e r m u l t i - s t e p f o r m u l a i n v o l v e s r e p l a c i n g t h e 1  f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n ( l ) , by a k** o r d e r d i f f e r e n c e e q u a t i o n . 1  process introduces  k-1 e x t r a n e o u s s o l u t i o n s .  corrector p a i r introduces the e r r o r w i l l v a n i s h m i s increased. say  \  fo r  This  However, t h e c o u p l e d p r e d i c t o r -  an a d d i t i o n a l " i t e r a t i v e r o o t " whose c o n t r i b u t i o n t o  as t h e e f f e c t o f t h e p r e d i c t o r i s s u p p r e s s e d .  That i s , as  To show t h i s t o be t h e c a s e , we n o t e t h a t e x a c t l y one r o o t o f (12),  def i n i t e n e s s , has t h e p r o p e r t y  that  I-  A  - o.  11.  I t remains t o show t h a t i n (l6), t h e m u l t i p l i e r o f t h i s r o o t remains bounded w i t h i n c r e a s i n g m. Let  £  , £, , . .., G-b, be the e r r o r s i n . t h e k+1  Q  t o s t a r t the i n t e g r a t i o n procedure.  o r d i n a t e s used  Also,.define  F o r any m, t h e m u l t i p l i e r s , i n (l6) are d e t e r m i n e d by a system o f e q u a t i o n s o f the f o r m :  r  — A  _.  _\  E  =A~,X~, +A~_X,*a + • • • + A - .  sA^X  +  -4-  • ' • • - + • _V  fctl  X ^  s  A « X x + - • + A~^ A - - * .  w i t h determinant I I  •  x«, X ^ i • (  If —  We  i  .1  -  *  •  «  •  s expanded by m i n o r s o f t h e ( k + l )  s t  *  column,  then  consider  •I.  A  linn.  "^!*>,n |  =• +  I»  -ji?  I I ivyi  ^  A-  By use o f e q u a t i o n ( 1 5 ) , we f i n d t h a t i f h i s s u f f i c i e n t l y s m a l l  Thus  A  remains bounded as m . i n c r e a s e s .  12. We c o n c l u d e t h a t t h e component o f t h e e r r o r due t o one r o o t o f (12) w i l l . d i m i n i s h with r e p e a t e d . i t e r a t i o n o f the c o r r e c t o r formula.  Further, the  r e m a i n i n g r o o t s , tend.towards t h o s e o f t h e c h a r a c t e r i s t i c p o l y n o m i a l o f t h e c o r r e c t o r a c t i n g alone. I f we l i m i t o u r c h o i c e t o s t a b l e methods w h i c h a r e c o n s i s t e n t i n t h e sense t h a t r e l a t i o n s (lh) h o l d , t h e n f o r any m, e q u a t i o n (12) w i l l have one <3h  0  root which i s approximately  . The s t a b i l i t y p r o p e r t y i n s u r e s t h a t t h e  components o f t h e e r r o r due t o t h e r e m a i n i n g r o o t s w i l l be n e g l i g i b l e .  For t h i s  c h o i c e o f i n t e g r a t i o n f o r m u l a s , e q u a t i o n (l6) g i v e s  = £ e "' -'  (i )  srt  7  +  a p p r o x i m a t e l y , as t h e p r o p a g a t e d  \m e r r o r o f t h e PE(CE) p r o c e d u r e .  In  d e r i v i n g (l7)> t h e e r r o r s i n t h e s t a r t i n g v a l u e s a r e assumed e q u a l and denoted by € .In  summary, e q u a t i o n s (12) and (17) i n d i c a t e t h a t r e p e a t e d  iteration  has t h e e f f e c t o f s u p p r e s s i n g ;the i n f l u e n c e o f the. p r e d i c t o r on t h e s t a b i l i t y and a c c u r a c y c h a r a c t e r i s t i c s o f t h e method.  As m . i n c r e a s e s ,  equations  (12) and  (17) become r e s p e c t i v e l y , t h e c h a r a c t e r i s t i c p o l y n o m i a l and p r o p a g a t e d e s t i m a t e d e r i v e d b y H u l l and Newbery  error  Q[) f o r s t a b l e and c o n s i s t e n t c o r r e c t o r  formulas. Error estimate, P(EC)  m  procedure.  We b e g i n b y w r i t i n g e x p r e s s i o n s f o r t h e a p p r o x i m a t i o n s  t o the true  s o l u t i o n o b t a i n e d f r o m t h e p r e d i c t o r and each subsequent i t e r a t i o n o f t h e c o r r e c t o r up t o t h e m^.  S i n c e t h e PEC p r o c e d u r e  does n o t i n c l u d e an e v a l u a t i o n  (a) o f f u s i n g t h e f i n a l a p p r o x i m a t i o n a t each s t e p , t h e e q u a t i o n s f o r V^^* ^»o,i,...,/m }  w i l l be  13-  H-l  P r o c g e d i n g as i n t h e P E ( C E ) i n t o each e q u a t i o n . equations w i t h  J  n +  m  c a s e , we i n t r o d u c e t h e t r u e s o l u t i o n  ,  From t h e r e s u l t i n g r e l a t i o n s , we s u b t r a c t t h e c o r r e s p o n d i n g j  We assume t h a t  r e p l a c e d b y t h e computed q u a n t i t i e s ,  j q = 0,1,2,...,m.  , g i v e n b y e q u a t i o n ( 1 0 ) , as w e l l as t h e e r r o r s due t o  t r u n c a t i o n and r o u n d - o f f a r e c o n s t a n t i n n and q.  Then u s i n g t h e same n o t a t i o n  as b e f o r e , we a r e l e d t o t h e f o l l o w i n g system o f e q u a t i o n s f o r t h e e r r o r s a t each i t e r a t i o n :  K-i  K  1=-o  (18)  K-i  W , each p r e c e d i n g  I f we r e c u r s i v e l y s u b s t i t u t e i n t o t h e e x p r e s s i o n f o r  e q u a t i o n o f ( l 8 ) , t h e n t h e system reduces t o t h e p a i r o f e q u a t i o n s : k-i  e (19a)  +h  _  3> P'^'  k-i  C—-0  —r—  +  -e  Ik.  (19b)  e„  where  t R  0 = ^(3^  \o\ <  and  I  I f we e l i m i n a t e  from (19),.the r e s u l t a n t e q u a t i o n f o r ^-rw^  admits t h e p a r t i c u l a r s o l u t i o n  i-e  ~  ^  0^Z_ft|  + CT P +  er- h  9  7 ft  assuming t h e method i s c o n s i s t e n t i n t h e sense o f (lk). Under t h e l a t t e r a s s u m p t i o n , one o f t h e r o o t s o f t h e c h a r a c t e r i s t i c p o l y n o m i a l o f system (19) w i l l , d i f f e r from  b y terms t h a t a r e o ( h ) as  n->0  I f we assume t h a t t h e e r r o r s , i n t h e s t a r t i n g v a l u e s a r e c o n s t a n t and denoted by £ P(EC)  , t h e n t h e p r o p a g a t e d e r r o r o f a s t a b l e , c o n s i s t e n t method b a s e d on a m  p r o c e d u r e w i l l be  a p p r o x i m a t e l y , w i t h W^ ^ m  g i v e n b y e q u a t i o n (20)..  We may v e r i f y from t h e p r e c e d i n g d i s c u s s i o n , t h a t i f jfc^l^. I , t h e s t a b i l i t y and,accuracy c h a r a c t e r i s t i c s o f t h e P ( E C ) become i n d i s t i n g u i s h a b l e w i t h r e p e a t e d i t e r a t i o n . P(EC)  m  m  and P E ( C E )  m  procedures  F o r , as m i n c r e a s e s , t h e  e r r o r , g i v e n b y . t h e homogeneous e q u a t i o n s c o r r e s p o n d i n g t o (l9)>  s a t i s f y t h e same r e l a t i o n as t h e P E ( C E )  m  will  e r r o r g i v e n b y t h e l i m i t case o f ( l l )  This i n d i c a t e s that the c h a r a c t e r i s t i c polynomials of the procedures w i l l c o i n c i d e a f t e r many i t e r a t i o n s . w i t h i n c r e a s i n g m.  F u r t h e r , b o t h -U^ ) andU>( ) become m  m  ^  CHAPTER I I I  COMPARISON AND CHOICE OF METHODS  Our purpose i s t o s e l e c t from among t h e c l a s s o f m u l t i - s t e p methods, the most e f f i c i e n t f o r t h e i n t e g r a t i o n o f ( l ) i n c i r c u m s t a n c e s complicated f u n c t i o n f .  involving a  I n such c i r c u m s t a n c e s , t h e t i m e t a k e n a t any i n t e g r a t i o n  s t e p t o e v a l u a t e t h e p r e d i c t o r - c o r r e c t o r f o r m u l a s may be r e g a r d e d as n e g l i g i b l e when compared t o even a s i n g l e c a l c u l a t i o n o f f .  Thus t h e s e l e c t e d method and  t e c h n i q u e o f a p p l i c a t i o n w i l l be c h a r a c t e r i z e d b y a need f o r r e l a t i v e l y few e v a l u a t i o n s o f f , when d e t e r m i n i n g a s o l u t i o n over a s p e c i f i e d i n t e r v a l o f t , t o w i t h i n p r e s c r i b e d e r r o r bounds. As a guide i n making o u r c h o i c e , t h e e r r o r e s t i m a t e s d e r i v e d i n t h e p r e v i o u s c h a p t e r w i l l be u s e d t o p r o v i d e c r i t e r i a f o r comparing v a r i o u s i n t e g r a t i o n methods.  These e s t i m a t e s w i l l be employed o n l y i n t h i s q u a l i t a t i v e sense, r a t h e r  than as e x a c t measures o f p r e c i s i o n . numerical evidence  However, i n t h e n e x t c h a p t e r , some  i s o f f e r e d i n support o f t h e a c c u r a c y o f t h e s e e s t i m a t e s .  When u s i n g t h e e s t i m a t e s , we assume t h e word l e n g t h adequate t o make t h e e f f e c t s o f r o u n d - o f f s m a l l , compared t o t h e t r u n c a t i o n e r r o r o f any f o r m u l a c o n s i d e r e d . In t h e d i s c u s s i o n t h a t f o l l o w s , a c o m p a r i s o n i s made o f v a r i o u s p r e d i c t o r - c o r r e c t o r methods f r o m t h e s t a n d p o i n t o f e f f i c i e n c y .  To advance t h i s  d i s c u s s i o n , we assume t h a t a l l methods c o n s i d e r e d a r e b o t h c o n s i s t e n t and s t a b l e . I t w i l l be seen i n t h e n e x t c h a p t e r t h a t s t a b i l i t y w i l l l i m i t t h e a p p l i c a b i l i t y o f methods w i t h h i g h e f f i c i e n c y c h a r a c t e r i s t i c s .  However, t h e s e methods prove  u s e f u l f o r problems s o l v e d b y 1+th o r 5th o r d e r f o r m u l a s and, i n a r e s t r i c t e d when s o l v e d b y 6th o r d e r  formulas.  We f i r s t compare t h e e f f i c i e n c y o f t h e P ( E C ) i t e r a t i o n s p e r s t e p w i t h t h e same p r o c e s s h a v i n g m=l. s t a b l e , the behaviour o f the propagated m=N o r m=l.  sense,  m  technique, t a k i n g  m  = N>1  P r o v i d e d t h e methods a r e  e r r o r i s d e s c r i b e d by equation (21) w i t h  I f we assume t h e e r r o r s i n t h e s t a r t i n g v a l u e s a r e s m a l l and n e g l e c t  round-off, the propagated  1 \N e r r o r f o r P ( E C j w i l l be d e t e r m i n e d b y  16.  l-e  ff 7mo  i-o  S i m i l a r l y , i f t h e s i n g l e - i t e r a t e t e c h n i q u e remains s t a b l e w i t h s t e p s i z e t h e p r o p a g a t e d e r r o r i s . determined  (22)  by  U)In equation.(22), t h e terms "T  and "Tp  e  form  h' ,  X ("t)  C^)  '  '^  le  c o e  ^i -'- " c  e n  , a r e e x p r e s s i o n s o f the  > °^ e i t h e r c a s e , i s d e t e r m i n e d  t :  b y the c h o i c e o f i n t e g r a t i o n f o r m u l a . F o r most methods o f i n t e r e s t , ^ one*, and . u s u a l l y |@1 « we  I  •  /?, }  and  are a p p r o x i m a t e l y  Comparing t h e s e e x p r e s s i o n s f o r U>  see t h a t t h e a c c u r a c y o f a s t a b l e PEC t e c h n i q u e w i t h h = h 7  comparable t o t h a t o f P ( E C ) . N  C*0  and  oJ"  ,  would.be  F u r t h e r , such a method would be much l e s s c o s t l y  i n terms o f e v a l u a t i o n s o f f . I f we s e t h' = h/N ,,the f a c t o r ( i ) ^ ^  a p p e a r i n g i n (22),  would, i n  mo s t c a s e s , make a s t a b l e PEC p r o c e s s s u b s t a n t i a l l y more a c c u r a t e t h a n P ( E C ) ^ w i t h step s i z e h  .  F o r t h i s c h o i c e o f h' ; the c o s t o f b o t h t e c h n i q u e s would be  the  same. Between t h e s e extremes o f a c c u r a c y and c o s t , the PEC p r o c e d u r e can be made more e f f i c i e n t t h a n P(EC)'  N  i f we can choose a s t e p s i z e h'> h/N  ,  f o r which  the former i s s t a b l e .  b I B -  * P r e d i c t o r s and c o r r e c t o r s o f t h e Adams t y p e have ^  ^  —  2-o  ^* ~  I »  •17. We c o n c l u d e from.the a b o v e • d i s c u s s i o n , . t h a t , p r o v i d e d . t h e method remains, s t a b l e , t h e r e i s a range o f h  f o r which the propagated e r r o r o f t h e  s i n g l e - i t e r a t e p r o c e d u r e remains w i t h i n p r e s c r i b e d ' b o u n d s , and f o r t h i s c h o i c e o f h,. t h e method i s more e f f i c i e n t t h a n a s i m i l a r p r o c e d u r e w i t h A s i m i l a r comparison o f P E ( C E ) conclusion:  m >.l .  p r o c e d u r e s would l e a d t o t h e same  m  p r o v i d e d t h e method remains s t a b l e , t h e s i n g l e - i t e r a t e p r o c e d u r e  w i l l determine a s o l u t i o n w i t h i n p r e s c r i b e d e r r o r bounds, i n t h e most e f f i c i e n t manner. These c o n c l u s i o n s l e a d us t o compare t h e PECE and PEC i n t e g r a t i o n methods w i t h s t e p s i z e h and h/2 r e s p e c t i v e l y .  Neglecting round-off, the  •propagated e r r o r i s d e t e r m i n e d b y  T  eT  c +  P  when.the c a l c u l a t i o n s a r e extended b y t h e PECE p r o c e d u r e , and.by t h e case N^2 i n . e q u a t i o n ( 2 2 ) , f o r PEC.  I f h i s s u f f i c i e n t l y s m a l l t o keep t h e e r r o r o f t h e  former method w i t h i n p r e s c r i b e d bounds, t h e same would be t r u e o f t h e PEC procedure u n l e s s s i m u l t a n e o u s l y ,  »  i__  __.  (  |eT | » y X \ P  I f t h e l a t t e r were t r u e , t h e f i r s t i n e q u a l i t y would be u n l i k e l y t o h o l d . f o r most methods o f i n t e r e s t , s i n c e  JG|  =  |^^^»»|  • The second i n e q u a l i t y i s b y  i t s e l f i n v a l i d f o r Adams' f o r m u l a s o f a l l o r d e r s . A  o r d e r , have Consequently,  — —— \  P,  ••• (.1** +*fc-0 d  I/LL^^CZU*-^)  I  ~  - 1  and  Methods o f t h i s t y p e , . o f k t h  (6  •  k_ ^ h f l  > k+l Pk '  The f o r e g o i n g l e a d s us t o c o n c l u d e t h a t p r o v i d e d t h e PEC method remains s t a b l e , t h e f a c t o r (•g-)^'*' i n u/'' consequent  > would p e r m i t a s u b s t a n t i a l . i n c r e a s e i n h w i t h  improvement i n e f f i c i e n c y .  T h e r e f o r e , we a r e l e d t o t h e PEC p r o c e d u r e  18.  as the most' e f f i c i e n t means o f a p p l y i n g a p a i r o f p r e d i c t o r - c o r r e c t o r - f o r m u l a s for. t h e s o l u t i o n , o f ( l ) .  That i s , i f the method.remains s t a b l e , t h i s  technique  w i l l p r o v i d e a s o l u t i o n o f d e s i r e d a c c u r a c y and do so w i t h a minimum number o f evaluations of f. H a v i n g e s t a b l i s h e d .the most e f f i c i e n t means o f e x t e n d i n g we must now  adopt a s u i t a b l e p a i r o f m u l t i - s t e p f o r m u l a s .  g o v e r n i n g t h i s c h o i c e are a c c u r a c y and s t a b i l i t y . r e q u i r e m e n t , we may  calculations,  The  criteria  To meet t h e  accuracy  r e s o r t t o h i g h o r d e r f o r m u l a s , s i n c e t h e time t a k e n t o  e v a l u a t e t h e p r e d i c t o r and c o r r e c t o r i s n e g l i g i b l e when f i s c o m p l i c a t e d . the s t a b i l i t y requirement  Thus  remains as t h e f o r m o s t f a c t o r g o v e r n i n g our c h o i c e .  Formulas w i t h good s t a b i l i t y p r o p e r t i e s are t h o s e o f t h e Adams t y p e ; having extraneous  s o l u t i o n s o n l y a t t h e o r i g i n .of t h e complex p l a n e when  h=o.  B e s i d e s t h i s d e s i r a b l e p r o p e r t y , t h e s e methods a r e s i m p l e t o program f o r computation,  r e q u i r i n g a t each step o n l y t h e o r d i n a t e d e t e r m i n e d . a t  the  immediately previous step i n a d d i t i o n t o the d e r i v a t i v e s In procedure  summary, the e s t i m a t e s o f p r o p a g a t e d  as the most e f f i c i e n t f o r the- i n t e g r a t i o n o f ( l ) .  c h a p t e r , a c o m p a r i s o n i s made between PEC formulas.  e r r o r , have l e d u s t o the I n the  next  and PECE p r o c e d u r e s ' u s i n g Adams  PEC  CHAPTER IV  EXPERIMENTAL RESULTS  The s e l e c t the PEC convenient  t h e o r e t i c a l d i s c u s s i o n o f t h e f o r e g o i n g s e c t i o n s has p r o c e d u r e "based on Adams f o r m u l a s ,  l e d us  as a most e f f i c i e n t  to  and  means o f s o l v i n g ( l ) , t o w i t h i n a p r e s c r i b e d t o l e r a n c e o f e r r o r .  t h i s c h a p t e r , we Specifically,  o f f e r experimental  e v i d e n c e i n s u p p o r t o f our  our aims are t o e s t a b l i s h the a c c u r a c y  In  choice.  o f the e r r o r  estimate  d e r i v e d i n C h a p t e r I I , and t o demonstrate the h i g h e f f i c i e n c y o f the  selected  methods. The  d i f f e r e n t i a l e q u a t i o n used i n a l l e x p e r i m e n t s  was  (23) with o * t 6 S  radians.  The  e x p o n e n t i a l component o f t h e s o l u t i o n may  be  e l i m i n a t e d by s u i t a b l e c h o i c e o f i n i t i a l c o n d i t i o n s , so t h a t  X(-t)  -z.  OtSiryb't. +  b co%  b't  V a r i o u s v a l u e s were a s s i g n e d t o a and b^ and t h e p r o b l e m was u s i n g Adams f o r m u l a s w i t h k=3,^,5> c o r r e c t o r o f t h e same o r d e r .  The  o r  6.  I n each c a s e , we used a p r e d i c t o r and  t r u n c a t i o n e r r o r s o f t h e s e f o r m u l a s may  e x p r e s s e d as  with  and  fcl" given i n Table 1 f o r formulas of both types. Table 1 Truncation Error C o e f f i c i e n t s , Adams Methods. k k Predictor H  H  k  r\.  Corrector  3 251 720 -19 720  solved  1+  5  6  95  19O87 60I+80  5257 17280  -3 16*0  -863 60480  -275 24192  258  be  .20. I n T a b l e 2, t h e row labeled'"Max. Abs. E r r o r " shows the magnitudes o f t h e maximum e r r o r o b s e r v e d u s i n g t h e PEC p r o c e d u r e . been c a l c u l a t e d from e q u a t i o n ( 2 l ) . w i t h m=l,  The a p r i o r i bounds have  g=a, and \C\ [T" | t  p t  jv^j < Icf't  I n a l l c a s e s we have t a k e n a=b and h=0.04. . Table 2 PEC, m=l; a=b, h=0.0U, 0 i S t - 8 . k .  h  3 1  -1  l  Max. Abs. E r r o r x,10'  O.53  3-8  A priori. 658 Bound x 10? .  0-73  81  3-025  0.7  O.05  0.7  a  Ratio  0.08  6  5 -1  1  -1  1  -0.625  0.018 •0.26 0.0006 0.25 0.000U  •5 0.05  0.0017  .5  0.003  O.k  0.05  0.1  I n e x p e r i m e n t s w i t h ;5th and 6th o r d e r f o r m u l a s , t h e e r r o r was p r e d o m i n a n t l y due t o r o u n d - o f f .  C o n s e q u e n t l y , i n c a l c u l a t i n g the e r r o r bounds  f o r t h e s e methods, we have a l l o w e d f o r t h e randomness o f t h e e r r o r b y considering  the mean e r r o r t o be zero, and r e c o r d i n g  the standard d e v i a t i o n ,  which  is .(computed bound)  V 200 v e r y • a p p r o x i m a t e l y , for.-200 i n t e g r a t i o n  steps.  W i t h f i x e d a and b i n T a b l e 2,,the r a t i o o f e r r o r t o bound remains a p p r o x i m a t e l y c o n s t a n t i n :k, a t l e a s t r e l a t i v e t o t h e wide v a r i a t i o n i n t h e bounds t h e m s e l v e s .  This, i n d i c a t e s t h a t when comparing methods, t h e e s t i m a t e w i l l  at l e a s t e x h i b i t a q u a l i t a t i v e behaviour s i m i l a r t o t h a t o f the e r r o r . A study, o f t h e e r r o r e s t i m a t e f o r the PECE p r o c e d u r e g i v e s r e s u l t s v e r y s i m i l a r t o the PEC c a s e , and.are  shown.in T a b l e 3-  21. Table 3 k  6  5  k  3 1  1  -1  .1  -0.625  1  . -1  Max. Abs. E r r o r x 1CK  16  O.O78 3-2 0:001*9 0.021 O.COO30 0.0090 0.000003  A priori Bound x 105  150  .11  60  0.007  33  Ratio  0.1  0.7  0.05  0.7  0.06  a  -l  D.000^  0.36 0.000:015 0.03  0.7  0.2  Comparison o f s i n g l e - i t e r a t e p r o c e d u r e s . We f i r s t d e f i n e t h e c o s t o f i n t e g r a t i o n p e r u n i t l e n g t h o f i n t e g r a t i o n i n t e r v a l as  ^  , where h i s t h e s t e p s i z e and ^" t h e number o f  s  e v a l u a t i o n s o f f a t each s t e p . two r e s p e c t i v e l y .  F o r t h e PEC and PECE p r o c e d u r e s , V  i s one and  To compare t h e r e l a t i v e e f f i c i e n c y o f t h e s e p r o c e d u r e s , we  h a v e • i n t e g r a t e d . e q u a t i o n ( 2 l ) u s i n g b o t h , on an e q u a l c o s t b a s i s .  That i s , we  i n t e g r a t e d t h e - d i f f e r e n t i a l e q u a t i o n u s i n g Adams f o r m u l a s o f v a r i o u s o r d e r s w i t h step h . f o r t h e PECE p r o c e d u r e and h/2 f o r P E C T a b l e s k and 5 show t h e r a t i o o f t h e magnitude o f t h e maximum observed e r r o r i n a PECE i n t e g r a t i o n t o t h a t o f PEC, f o r t h e same c a s e .  In a l l  cases  a=b, and t h e Adams p r e d i c t o r and c o r r e c t o r f o r m u l a s a r e o f t h e same o r d e r . Table k Max. PECE e r r o r a k  +1  -1  3  35  15  k  67  27  5  8  52  .6  k  * PEC p r o c e d u r e u n s t a b l e .  -2  -3 Ik  27  •*  •*  *  *  22.  Max. Max.  PECE e r r o r PEC e r r o r  a  \\ k  1  J  c=20, h=0.10>  +i  -l  -2  -3  3  35  12  Ik  •h  170  2.5  *  •*  5  50  1*  *  -*  7  6  ih  •*  •*  * PEC p r o c e d u r e u n s t a b l e . I n e a c h . s t a b l e c a s e , . t h e r a t i o o f t h e maximum PECE e r r o r t o t h e maximum-PEC e r r o r , i s c o n s i d e r a b l e g r e a t e r than. one.  These . r a t i o s g i v e an  i n d i c a t i o n o f t h e improvement i n e f f i c i e n c y t o be o b t a i n e d by i n c r e a s i n g t h e step s i z e used f o r t h e PEC  integration.  Conclusion Both t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s have i n d i c a t e d t h a t when s t a b l e , t h e PEC p r o c e d u r e b a s e d on Adams f o r m u l a s i s a most e f f i c i e n t means o f determining a s o l u t i o n to w i t h i n error tolerances. The e x p e r i m e n t a l r e s u l t s have a l s o e x p h a s i z e d t h e • i m p o r t a n c e s t a b i l i t y when c h o o s i n g a most e f f i c i e n t method.  of  Our o r i g i n a l assumption  been t h a t i f a m u l t i - s t e p f o r m u l a based.on t h e PECE p r o c e d u r e was  had  stable with  step s i z e h, t h e n . t h e c o r r e s p o n d i n g PEC p r o c e d u r e would a l s o be s t a b l e w i t h I f t h i s was  h/2  t r u e , t h e c r i t e r i o n o f c h o i c e would be-the r e l a t i v e s i z e o f the  propagated e r r o r .  However, our n u m e r i c a l experiments have shown o t h e r w i s e and  c o n s e q u e n t l y , a new p o s s i b i l i t y has a r i s e n .  To a c h i e v e a p r e s c r i b e d a c c u r a c y ,  the s u p e r i o r s t a b i l i t y p r o p e r t i e s o f PECE c o u l d p e r m i t t h e use o f ' h i g h e r o r d e r f o r m u l a s a t l o w e r i n t e g r a t i o n c o s t t h a n a s t a b l e PEC p r o c e d u r e . o f t h i s p o s s i b i l i t y remains as a problem f o r t h e f u t u r e .  The  investigat  23BIBLIOGRAPHY  Hamming, R.W.  S t a b l e P r e d i c t o r - C o r r e c t o r Methods f o r O r d i n a r y  D i f f e r e n t i a l E q u a t i o n s , J.Assoc.Comput.Mach..vol.6, 1959, PP-37-^7Hull,-T.E.  and Luxemburg, W.A.J.  N u m e r i c a l Methods and E x i s t e n c e  Theorems, f o r O r d i n a r y D i f f e r e n t i a l E q u a t i o n s , Numerische Math,  vol.2,  I960, pp.30-41. H u l l , T.E. and Newbery, A.C.R.  I n t e g r a t i o n Procedures which  M i n i m i z e P r o p a g a t e d E r r o r s , J . S o c . I n d u s t . Appl.Math. vol-9>  19^1,  pp.31-47. M i l n e , W.E. and R e y n o l d s , R.R.  F i f t h - O r d e r Methods f o r t h e N u m e r i c a l  S o l u t i o n o f O r d i n a r y D i f f e r e n t i a l E q u a t i o n s , J.Assoc.Comput.Mach. vol.9,  1962, pp.64-70.  Nordsieck, Arnold.  On N u m e r i c a l I n t e g r a t i o n o f O r d i n a r y D i f f e r e n t i a l  E q u a t i o n s , Math, o f Comp..vol.16, 1962,.pp.22-49R a l s t o n , A.  Some T h e o r e t i c a l and C o m p u t a t i o n a l M a t t e r s R e l a t i n g t o  P r e d i c t o r - C o r r e c t o r Methods o f N u m e r i c a l I n t e g r a t i o n , Computer J . v o l . 4 , 1961,  pp.64-67.  

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