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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 of 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 the Department of MATHEMATICS We accept, t h i s ' t h e s i s as conforming ,.to:.-the' r e q u i r e d 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 requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h ' C o l u m b i a , 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 reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department o r by h i s r e p r e s e n t a t i v e s . 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 alloived without my w r i t t e n permission. Department of Mathematics The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date_ A p r i l 13, 1962 ABSTRACT The purpose of t h i s t h e s i s i s t o study the fac t o r s , i n v o l v e d : i n determining a most e f f i c i e n t method f o r the numerical i n t e g r a t i o n of the 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 method.requiring a minimum of computation to o b t a i n 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. We o u t l i n e two computational procedures and de r i v e estimates f o r the propagated e r r o r of a general m u l t i - s t e p method when based on e i t h e r procedure. These estimates, l e a d us to conclude t h a t .a s t a b l e s i n g l e - i t e r a t e procedure, i n v o l v i n g one e v a l u a t i o n of f at each step, w i l l determine a s o l u t i o n most e f f i c i e n t l y . In p a r t i c u l a r , t h i s procedure based on Adams formulas, i s recommended. Experimental r e s u l t s support our c o n c l u s i o n i n a l l s t a b l e cases. However, these r e s u l t s a l s o i n d i c a t e t h a t the r o l e of s t a b i l i t y i n the choice of a most e f f i c i e n t procedure i s i n need of f u r t h e r i n v e s t i g a t i o n . I hereby c e r t i f y t h a t t h i s a b s t r a c t i s s a t i s f a c t o r y . i i i . . ACKNOWLEDGEMENTS The author wishes, to acknowledge his•indebtedness to Dr. T-E. H u l l f o r advice and guidance r e c e i v e d i n f o r m u l a t i n g and pre p a r i n g t h i s t h e s i s . Thanks are a l s o due t o the s t a f f of the Computing Centre of the U n i v e r s i t y of B r i t i s h Columbia. T h e i r e f f o r t s made p o s s i b l e the p r a c t i c a l v e r i f i c a t i o n of t h e o r e t i c a l r e s u l t s . i v . TABLE OF CONTENTS CHAPTER I I n t r o d u c t i o n and Summary 1 CHAPTER I I Estimates of the Propagated e r r o r 3 P ( E C ) m Procedure, Flow Chart '. h P E ( C E ) m Procedure, Flow Chart 6 E r r o r Estimate, PE(CE) m Procedure - 7 E r r o r Estimate, P ( E C ) m Procedure . . 12 CHAPTER I I I Comparison and Choice of Methods . . . . . . . . . . 15 CHAPTER IV Numerical R e s u l t s 20 Comparison of S i n g l e - I t e r a t e Procedures . 21 Conclusion 22 "BIBLIOGRAPHY 23 CHAPTER I INTRODUCTION AND SUMMARY An important c l a s s of methods f o r the numerical 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 ( t o ) = X Q i s the c l a s s of i n t e g r a t i o n formulas of the p r e d i c t o r - c o r r e c t o r type. The purpose of t h i s study i s to 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 of t h i s wide c l a s s , w i t h the u l t i m a t e g o a l of choosing the method r e q u i r i n g the l e a s t amount of computing f o r the s o l u t i o n of ( l ) to a p r e s c r i b e d accuracy. Our main concern i s f o r problems i n v o l v i n g a complicated f u n c t i o n f. The computing time f o r the numerical s o l u t i o n of such problems i s almost e n t i r e l y determined by the number of e v a l u a t i o n s of f r e q u i r e d by the i n t e g r a t i o n technique over a s p e c i f i e d i n t e r v a l of t . Under these circumstances, the many c a l c u l a t i o n s of f r e q u i r e d by the Runge-Kutta method would be p r o h i b i t i v e and f o r t h i s reason, i n searching f o r a most e f f i c i e n t means of i n t e g r a t i o n , we c o nsider only the c l a s s of 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 formulas. To achieve our aim, we begin by d e r i v i n g expressions d e s c r i b i n g the behaviour of the e r r o r introduced by the p r e d i c t o r - c o r r e c t o r procedure. These expressions w i l l be employed i n a study of the c h a r a c t e r i s t i c s of such methods and 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 of 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 to consider m u l t i - s t e p i n t e g r a t i o n techniques i n v o l v i n g a s i n g l e e v a l u a t i o n of f at each i n t e g r a t i o n step. At l e a s t as long as there i s no d i f f i c u l t y w i t h s t a b i l i t y , these techniques are found t o be best i n the sense th a t over a s p e c i f i e d i n t e r v a l of t , r e l a t i v e l y few such c a l c u l a t i o n s are r e q u i r e d , i n order t o meet p r e s c r i b e d requirements of p r e c i s i o n . In p a r t i c u l a r , techniques of t h i s type based on Adams formulas of h i g h order, prove to be h i g h l y . e f f i c i e n t and convenient. 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 proposed by Hamming [l] , Milne and^Reynolds [V], Nordsieck [5j , and Ra l s t o n [b] . These methods r e q u i r e only a s i n g l e a p p l i c a t i o n of the c o r r e c t o r formula but u s u a l l y two e v a l u a t i o n s of f at each step of i n t e g r a t i o n . To some extent, these methods a l s o r e q u i r e s p e c i a l techniques which are not g e n e r a l l y a p p l i c a b l e . 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 descr i b e d and.estimates are d e r i v e d . f o r the propagated e r r o r . In Chapter I I I , observations are made on these estimates, to draw.conclusions as to the r e l a t i v e c h a r a c t e r i s t i c s of 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 Chapter XV, experimental evidence- i s o f f e r e d i n support of the accuracy of the e r r o r estimates and,the v a l i d i t y of 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 the need f o r a more complete i n v e s t i g a t i o n of the r o l e of s t a b i l i t y i n the choice of a most e f f i c i e n t method. CHAPTER I I ESTIMATES OF THE PROPAGATED ERROR 3-As mentioned p r e v i o u s l y , should the f u n c t i o n f on the right-hand .side of ( l ) be d i f f i c u l t t o evaluate, the number of 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 c o s t of numerical i n t e g r a t i o n . An e f f i c i e n t m u l t i - s t e p method would be one which r e q u i r e d r e l a t i v e l y few ev a l u a t i o n s of f i n i n t e g r a t i n g ( l ) to w i t h i n a p r e s c r i b e d accuracy,.across a 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 formula w i l l meet p r e s c r i b e d requirements on p r e c i s i o n , we need an estimate of e r r o r , an expression g i v i n g a reasonable approximation .to the e r r o r propagated through numerical c a l c u l a t i o n s . However, our purpose, i s t o s e l e c t an e f f i c i e n t method f o r the i n t e g r a t i o n of ( l ) . For t h i s reason, we s h a l l r e s t r i c t our use of e r r o r estimates to q u a l i t a t i v e a p p r a i s a l s of one method r e l a t i v e to other methods r a t h e r than 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 of accuracy. In d e r i v i n g an e r r o r estimate, we c a r e f u l l y f o l l o w the computational 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 . We s h a l l consider two such procedures, each l e a d i n g t o a d i s t i n c t form of the e r r o r . I t w i l l be seen th a t a f t e r many.iterations of the c o r r e c t o r formula,.these procedures become equi v a l e n t . P ( E C ) m Procedure ( P r e d i c t , 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 best describes the c a l c u l a t i o n and f o r t h i s procedure i s shown i n Figure 1. PREVIOUS STEP PREDICT CORRECT J L TEST P(EG)™,Procedure Figure 1. 5-The 'PREDICT1 and 'CORRECT' stages represent the c a l c u l a t i o n s i n v o l v e d i n a p p l y i n g the r e s p e c t i v e formulas. At the 'EVALUATE f stage, recourse i s made t o some subprogram f o r the c a l c u l a t i o n o f f ( t , y ) , where y i s the approximation t o the t r u e s o l u t i o n x ( t ) , obtained at the immediately preceding 'PREDICT'.or 'CORRECT' stage. The 'TEST', a p p l i e d at the f i n a l stage of the procedure, may determine i f two successive 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 of cl o s e n e s s , or the t e s t may merely represent 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 s h a l l 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 the case. I f m does vary from.step to s t e p , . t h i s assumption o f f e r s a reasonable approximation. P E ( C E ) m Procedure. A fl o w diagram f o r t h i s procedure i s shown i n Figure 2 . The stages of c a l c u l a t i o n shown i n Figure 2 may be rearranged i n var i o u s ways to give e q u i v a l e n t c o n f i g u r a t i o n s . In any case, the P E ( C E ) m procedure, as compared to a P ( E C ) m procedure w i t h the same c r i t e r i o n of '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 of f . In d e r i v i n g an estimate f o r the e r r o r propagated by 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 consider any f i x e d number m of i t e r a t i o n s at e v e r y . i n t e g r a t i o n step. However, our main concern w i l l be the case m=l. As m in c r e a s e s , the accuracy and e f f i c i e n c y of the P ( E C ) m and P E ( C E ) m techniques w i l l d i f f e r only s l i g h t l y . However, f o r small m, the d i s t i n c t i o n becomes important. C o n s i d e r i n g both accuracy and e f f i c i e n c y , i t w i l l be shown t h a t provided both are s t a b l e , the PEC technique i s s u p e r i o r t o PECE. For t h i s purpose, we begin by d e r i v i n g an estimate of the e r r o r propagated by 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 are extended by the P E ( C E ) m procedure. Such an estimate 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 the behaviour of the approximation as m grows l a r g e . PREVIOUS STEP 5_ PREDICT ( C E ) m Procedure Figure 2. 7-E r r o r estimate,-PE(CE) m procedure. Let be approximations to the tr u e s o l u t i o n , obtained from ;the mth . i t e r a t i o n of the c o r r e c t o r formula.at the p o i n t s Then at t = t Q + (ft+k)h, the 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 evaluated .using the ( q . - l ) s ^ i t e r a t e of the corr e c t o r , formula, the . q^1 i t e r a t e i s given by (3) C = ~2_ *« + ^ X 1 3 ' ^ < x"' y " < : 7 > ) ^ ^ w i t h W « W By f o r m a l l y . s u b s t i t u t i n g the tr u e s o l u t i o n i n t o (2) and (3)> we o b t a i n the - t r u n c a t i o n e r r o r of each formula, and , r e s p e c t i v e l y : However, any numerical e v a l u a t i o n of (2) and (3) r e s u l t s i n numbers &ntv and ^*+h > % = 1^2, . . ., m, such t h a t 8. and (my) w i t h zrn+fe. = n^*-!, . The q u a n t i t i e s n^w, Av • i n equations (6) and (7) have been determined at the k+1 previous steps, whereas X>+k. and Xm denote the e r r o r due to•round-off i n e v a l u a t i n g the p r e d i c t o r and qth . i t e r a t e of the c o r r e c t o r at t = t + (n+k)h. We define and :and subtract (1+) from (6) and (.5) from (7) to o b t a i n the e r r o r equations xte-i K CO In equations (8) and (9)> ( 1 0 ) = 4(t«.x„j - J7*..,, _„i), 1,-so t h a t iL.^ i s some value of > . at l e a s t when the d e r i v a t i v e e x i s t s , at ~t»v+j , and on an open., i n t e r v a l between and v^»ti . I f i n equation (9)> we set 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 , q = m, m-l> . . ., 1, then f i n a l l y , we o b t a i n •*— o+i^  i n terms of C;n<.| , i = 1,2, n+k-1. I f f s a t i s f i e s 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 in c r e a s e s , the r e s u l t a n t equation w i l l g i ve an expression f o r the e r r o r had the c o r r e c t o r been i t e r a t e d i n d e f i n i t e l y . We-now.make the assumption t h a t a constant 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 equation w i t h constant c o e f f i c i e n t s . I f we take q = m i n the corresponding homogeneous equation, 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) „(»*»>) + e *»0 (2 + 4 L ? In equation ( l l ) , © ~ h ^ t , , and i n p r a c t i c e , j@| < I I f i n equation ( l l ) i s re p l a c e d by ^ , the general s o l u t i s determined by.the r o o t s o f the polynomial, i o n (12) .ft) We now assume t h a t the q u a n t i t i e s 'n«-h , 'n+b , > and VJ^ k are constants i n n and q , denoted by T c , Tp , r c , and rp r e s p e c t i v e l y . Then a p a r t i c u l a r s o l u t i o n of the.;.nonhomogeneous equation-corresponding 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 that our i n t e r e s t i s con f i n e d t o c o r r e c t o r formulas which are c o n s i s t e n t i n the sense t h a t the tr u e s o l u t i o n w i l l s a t i s f y (3) t o w i t h i n terms t h a t are o(h). For our purposes, the p r e d i c t o r formula should at l e a s t supply an exact s o l u t i o n to ( l ) when f = 0. I f CK^~ ^t,^ ' ' t h e n l t i s necessary (as w e l l as s u f f i c i e n t ) f o r the c o e f f i c i e n t s of any p r e d i c t o r - c o r r e c t o r p a i r to s a t i s f y : i n order to be c o n s i s t e n t i n the sense described ( H u l l , Luxemburg [2] ). I f we apply the f i r s t of r e l a t i o n s (lh) to (13), we o b t a i n , a f t e r regrouping terms: (15; U>- — j /._ ; Z _ r — ' I f the k+1 r o o t s of (12) are denoted by which f o r s i m p l i c i t y , are assumed d i s t i n c t f o r a l l m, the complete s o l u t i o n to (10) when q = m, w i l l have the form (16) e T - A ~ . , X 1 , + A-xA-t + ••• + A ^ , / w , " ° ^ • The use of a k^*1 order m u l t i - s t e p formula i n v o l v e s r e p l a c i n g the f i r s t order d i f f e r e n t i a l equation ( l ) , by a k**1 order d i f f e r e n c e equation. This process introduces k-1 extraneous s o l u t i o n s . However, the coupled p r e d i c t o r -c o r r e c t o r p a i r introduces 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 to the e r r o r w i l l vanish as the e f f e c t of the p r e d i c t o r i s suppressed. That i s , as m i s increased. To show t h i s to be the case, we note t h a t e x a c t l y one root of (12), \ I- A - o. say f o r def i n i t e n e s s , has the property t h a t 11. I t remains t o show t h a t i n (l6), the m u l t i p l i e r of t h i s root remains bounded wi t h i n c r e a s i n g m. Let £ Q , £, , . .., G-b, be the e r r o r s i n . t h e k+1 o r d i n a t e s used t o s t a r t the i n t e g r a t i o n procedure. Also,.define For any m, the m u l t i p l i e r s , i n (l6) are determined by a system of equations of the form: r — A _. _\ -4- • ' • • - + • _V E =A~,X~, +A~_X,*a + • • • + A - . f c t l X ^ s sA^X +A«Xx + - • + A~^ A - - * . w i t h determinant I I • - . 1 x«,( X ^ i • * • « • * I f — i s expanded by minors of the ( k + l ) s t column, then •I. A =• I» -ji? l inn. "^!*>,n +| I I ivyi ^ We consider A-By use of equation (15), 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 small A Thus remains bounded as m . increases. 12. We conclude t h a t the component of the e r r o r due t o one root of (12) w i l l . d i m i n i s h w i t h r e p e a t e d . i t e r a t i o n of the c o r r e c t o r formula. F u r t h e r , the remaining roots, tend.towards those of the c h a r a c t e r i s t i c polynomial of the c o r r e c t o r a c t i n g alone. I f we l i m i t our choice to s t a b l e methods which are c o n s i s t e n t i n the sense t h a t r e l a t i o n s (lh) h o l d , then f o r any m, equation (12) w i l l have one 0<3h 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 the components of the e r r o r due to the remaining 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 choice of i n t e g r a t i o n formulas, equation (l6) gives ( i 7 ) = £ e s r t" ' +-' \m approximately, as the propagated e r r o r of the PE(CE) procedure. In d e r i v i n g (l7)> the e r r o r s i n the s t a r t i n g values are assumed equal and denoted by € .-In summary, equations (12) and (17) i n d i c a t e t h a t repeated i t e r a t i o n has the e f f e c t of suppressing ;the i n f l u e n c e of the. p r e d i c t o r on the 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 of the method. As m.increases, equations (12) and (17) become r e s p e c t i v e l y , the c h a r a c t e r i s t i c polynomial and propagated e r r o r estimate d e r i v e d by H u l l and Newbery 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. E r r o r estimate, P ( E C ) m procedure. We begin by w r i t i n g expressions f o r the approximations to the tr u e s o l u t i o n obtained from the p r e d i c t o r and each subsequent i t e r a t i o n of the c o r r e c t o r up t o the m^. Since the PEC procedure does not i n c l u d e an e v a l u a t i o n (a) of f u s i n g the f i n a l approximation at each step, the equations f o r V^ *^ } »^o,i,...,/m w i l l be 13-H-l Procgeding as i n the PE ( C E ) m case, we introduce the tr u e s o l u t i o n , i n t o each equation. From the 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 the corresponding equations w i t h J n + j r e p l a c e d by the computed q u a n t i t i e s , j q = 0,1,2,..., We assume t h a t , given by equation (10), as w e l l as the e r r o r s due to t r u n c a t i o n and round-off are constant i n n and q. Then u s i n g the same n o t a t i o n as b e f o r e , we are l e d t o the f o l l o w i n g system of equations f o r the e r r o r s at m. each i t e r a t i o n : K-i K (18) 1=-o K-i 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 the expression f o r W , each preceding equation of ( l 8 ) , then the system reduces t o the p a i r of equations: k-i C—-0 —r— e (19a) + h3> P'^ ' + _ k-i -e Ik. (19b) e „ t R where 0 = ^ ( 3 ^ and \o\ < 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 equation f o r ^-rw^ admits the p a r t i c u l a r s o l u t i o n ^ i - e ~ 0 ^ Z _ f t | + CTP+ er- h 9 7 ft assuming the method i s c o n s i s t e n t i n the sense of (lk). Under the l a t t e r assumption, one of the ro o t s of the c h a r a c t e r i s t i c polynomial of system (19) w i l l , d i f f e r from by terms t h a t are o(h) as n->0 I f we assume th a t the errors, i n the s t a r t i n g values are constant and denoted by £ , then the propagated e r r o r of a s t a b l e , c o n s i s t e n t method based on a P ( E C ) m procedure w i l l be approximately, w i t h W^m^ given by equation (20).. We may v e r i f y from the preceding d i s c u s s i o n , t h a t i f jfc^l^. I , the 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 of the P ( E C ) m and P E ( C E ) m procedures become i n d i s t i n g u i s h a b l e w i t h repeated i t e r a t i o n . For, as m i n c r e a s e s , the P ( E C ) m e r r o r , given by.the homogeneous equations corresponding to (l9)> w i l l s a t i s f y the same r e l a t i o n as the P E ( C E ) m e r r o r given by the l i m i t case of ( l l ) This i n d i c a t e s t h a t 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 . F u r t h e r , both -U^m) andU>( m) become - ^ w i t h i n c r e a s i n g m. CHAPTER I I I COMPARISON AND CHOICE OF METHODS Our purpose i s t o s e l e c t from among the c l a s s of m u l t i - s t e p methods, 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 of ( l ) i n circumstances i n v o l v i n g a complicated f u n c t i o n f . In such circumstances, the time taken at any i n t e g r a t i o n step to evaluate the p r e d i c t o r - c o r r e c t o r formulas may be regarded 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 of f. Thus the s e l e c t e d method and technique of 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 by a need f o r r e l a t i v e l y few eva l u a t i o n s of f , when determining 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 of t , to 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 our choice, the e r r o r estimates d e r i v e d i n the previous chapter w i l l be used to provide 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 estimates w i l l be employed only 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 exact measures of p r e c i s i o n . However, i n the next chapter, some numerical evidence i s o f f e r e d i n support of the accuracy of these estimates. When usin g the estimates, we assume the word l e n g t h adequate t o make the e f f e c t s of round-off s m a l l , compared t o the t r u n c a t i o n e r r o r of any formula considered. In the d i s c u s s i o n t h a t f o l l o w s , a comparison i s made of 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 from the standpoint of 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 considered are both 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 the next chapter that s t a b i l i t y w i l l l i m i t the a p p l i c a b i l i t y of methods w i t h high 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, these methods prove u s e f u l f o r problems solved by 1+th or 5th order formulas and, i n a r e s t r i c t e d sense, when solved by 6th order formulas. We f i r s t compare the e f f i c i e n c y of the P ( E C ) m technique, t a k i n g m = N>1 i t e r a t i o n s per step w i t h the same process having m=l. Provided the methods are s t a b l e , the behaviour of the propagated e r r o r i s de s c r i b e d by equation (21) with m=N or m=l. I f we assume the e r r o r s i n the s t a r t i n g values are small and neglect 1 \N round-off, the propagated e r r o r f o r P(ECj w i l l be determined by 16. l - e ff 7mo i-o S i m i l a r l y , i f the s i n g l e - i t e r a t e technique remains s t a b l e w i t h step s i z e h' , the propagated e r r o r is. determined by (22) U ) -In equation.(22), the terms "Te and "Tp , are expressions of the form X ("t) C^) ' '^le c o e^i c-'- e n" t : > °^ e i t h e r case, i s determined by the choice of i n t e g r a t i o n formula. For most methods of i n t e r e s t , ^ /?}, and are approximately C*0 one*, and .usually |@1 « I • Comparing these expressions f o r U> and oJ" , we see t h a t the accuracy of a s t a b l e PEC technique w i t h h 7 = h would.be comparable t o t h a t of P ( E C ) N . F u r t h e r , such a method would be much l e s s c o s t l y in terms of e v a l u a t i o n s of f. If we set h' = h/N ,,the f a c t o r ( i ) ^ ^ appearing i n (22), would, i n st cases, make a s t a b l e PEC process s u b s t a n t i a l l y more accurate than P(EC)^ w i t h mo step s i z e h . For t h i s choice of h' ; the cost of both techniques would be the same. Between these extremes of accuracy and c o s t , the PEC procedure can be made more e f f i c i e n t than P(EC)' N i f we can choose a step 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 of the Adams type have ^ ^ — ^* ~ I » 2-o •17. We conclude from.the above•discussion,. t h a t , provided.the method remains, s t a b l e , there i s a range of h f o r which the propagated e r r o r of the s i n g l e - i t e r a t e procedure remains w i t h i n prescribed'bounds, and f o r this choice of h,. the method i s more e f f i c i e n t than a s i m i l a r procedure w i t h m >.l . A s i m i l a r comparison of PE ( C E ) m procedures would l e a d to the same c o n c l u s i o n : provided the method remains s t a b l e , the s i n g l e - i t e r a t e procedure 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 the 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 to compare the PECE and PEC i n t e g r a t i o n methods w i t h step s i z e h and h/2 r e s p e c t i v e l y . N e g l e c t i n g round-off, the •propagated e r r o r i s determined by Tc + eTP when.the c a l c u l a t i o n s are extended by the PECE procedure, and.by the case N^2 in.equa t i o n (22), f o r PEC. I f h i s s u f f i c i e n t l y s m a ll to keep the e r r o r of the former method w i t h i n p r e s c r i b e d bounds, the same would be tru e of the PEC procedure unless simultaneously, » __. i _ _ ( |eTP| » y X \ I f the l a t t e r were t r u e , the 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 of i n t e r e s t , since JG| = |^^^»»| • The second i n e q u a l i t y i s by i t s e l f i n v a l i d f o r Adams' formulas of a l l orders. Methods of t h i s type,.of k t h order, have A — —— \ I/LL^^CZU*-^) ••• (.1** +*fc-0 d • Consequently, I ~ k_ ^ P, - 1 and ( 6 h f l > k+l Pk ' The f o r e g o i n g leads us t o conclude t h a t p rovided the PEC method remains s t a b l e , the f a c t o r (•g-)^ '*' i n u/'' > would permit a s u b s t a n t i a l . increase i n h w i t h consequent improvement i n e f f i c i e n c y . Therefore, we are l e d to the PEC procedure 18. as the most' e f f i c i e n t means of a p p l y i n g a p a i r of p r e d i c t o r - c o r r e c t o r - formulas for. the s o l u t i o n , of ( 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 rovide a s o l u t i o n of d e s i r e d accuracy and do so w i t h a minimum number of e v a l u a t i o n s of f. Having e s t a b l i s h e d .the most e f f i c i e n t means of extending c a l c u l a t i o n s , we must now adopt a s u i t a b l e p a i r of m u l t i - s t e p formulas. The c r i t e r i a governing t h i s choice are accuracy and s t a b i l i t y . To meet the accuracy requirement, we may r e s o r t to h i g h order formulas, since the time taken to evaluate the 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 complicated. Thus the s t a b i l i t y requirement remains as the formost f a c t o r governing our choice. 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 those of the Adams type; having extraneous s o l u t i o n s only at the o r i g i n .of the complex plane when h=o. Besides t h i s d e s i r a b l e property, these methods are simple t o program f o r computation, r e q u i r i n g at each step only the o r d i n a t e determined.at 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 summary, the estimates of propagated error, have l e d us to the PEC procedure 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 of ( l ) . In the next chapter, a comparison i s made between PEC and PECE procedures'using Adams formulas. CHAPTER IV EXPERIMENTAL RESULTS The t h e o r e t i c a l d i s c u s s i o n of the f o r e g o i n g s e c t i o n s has l e d us to s e l e c t the PEC procedure "based on Adams formulas, as a most e f f i c i e n t and convenient means of s o l v i n g ( l ) , to 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 of e r r o r . In t h i s chapter, we o f f e r experimental evidence i n support of our c h o i c e . S p e c i f i c a l l y , our aims are to e s t a b l i s h the accuracy of the e r r o r estimate d e r i v e d i n Chapter I I , and to demonstrate the h i g h e f f i c i e n c y of the s e l e c t e d methods. The d i f f e r e n t i a l equation used i n a l l experiments was (23) w i t h o * t 6 S r a d i a n s . The exponential component of the 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 choice of i n i t i a l c o n d i t i o n s , so t h a t OtSiryb't. + b co% b't X(-t) -z. Various values were assigned to a and b^ and the problem was solved u s i n g Adams formulas w i t h k=3,^ ,5> o r 6. In each case, we used a p r e d i c t o r and c o r r e c t o r of the same order. The t r u n c a t i o n e r r o r s of these formulas may be expressed as w i t h fcl" and given i n Table 1 f o r formulas of both types. Table 1 Truncation E r r o r C o e f f i c i e n t s , Adams Methods. k 3 1+ 5 6 H k 251 95 19O87 5257 P r e d i c t o r 720 258 60I+80 17280 H k -19 -3 -863 -275 r\. Corrector 720 16*0 60480 24192 .20. In Table 2, the row labeled'"Max. Abs. E r r o r " shows the magnitudes of the maximum e r r o r observed u s i n g the PEC procedure. The a p r i o r i bounds have been c a l c u l a t e d from equation (2l).with m=l, g=a, and \C\t [T"p| t jv^j < Icf ' t In a l l cases we have taken a=b and h=0.04. . Table 2 PEC, m=l; a=b, h=0.0U, 0 i S t - 8 . k . 3 h 5 6 a 1 -1 l -1 1 -1 1 -0.625 Max. Abs. E r r o r x,10' O.53 3-8 0.018 •0.26 0.0006 0.25 0.000U A p r i o r i . Bound x 10? . 658 0-73 81 3-025 •5 0.0017 .5 0.003 R a t i o 0.08 0.7 O.05 0.7 0.05 O.k 0.05 0.1 In experiments w i t h ;5th and 6th order formulas, the e r r o r was predominantly due to round-off. Consequently, i n c a l c u l a t i n g the e r r o r bounds f o r these methods, we have allowed f o r the randomness of the e r r o r by co n s i d e r i n g the mean e r r o r to be zero, and re c o r d i n g the standard d e v i a t i o n , which i s .(computed bound) V 200 very•approximately, for.-200 i n t e g r a t i o n steps. With f i x e d a and b i n Table 2,,the r a t i o of e r r o r t o bound remains approximately constant i n :k, at l e a s t r e l a t i v e to the wide v a r i a t i o n i n the bounds themselves. This, i n d i c a t e s t h a t when comparing methods, the estimate 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 of the e r r o r . A study, of the e r r o r estimate f o r the PECE procedure gives r e s u l t s very s i m i l a r to the PEC case, and.are shown.in Table 3-21. Table 3 k 3 k 5 6 a 1 . -1 1 - l 1 -1 .1 -0.625 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 p r i o r i Bound x 105 150 .11 60 0.007 33 D .000^ 0.36 0.000:015 R a t i o 0.1 0 . 7 0.05 0 .7 0.06 0 . 7 0 .03 0 . 2 Comparison of s i n g l e - i t e r a t e procedures. We f i r s t d efine the cost of i n t e g r a t i o n per u n i t l e n g t h of i n t e g r a t i o n i n t e r v a l as ^ s , where h i s the step s i z e and ^ " the number of eva l u a t i o n s of f at each step. For the PEC and PECE procedures, V i s one and two r e s p e c t i v e l y . To compare the r e l a t i v e e f f i c i e n c y of these procedures, we have•integrated.equation ( 2 l ) u s i n g both, on an equal cost b a s i s . That i s , we i n t e g r a t e d the - d i f f e r e n t i a l equation u s i n g Adams formulas of v a r i o u s orders w i t h step h . f o r the PECE procedure and h/2 f o r PEC Tables k and 5 show the r a t i o of the magnitude of the maximum observed e r r o r i n a PECE i n t e g r a t i o n to t h a t of PEC, f o r the same case. In a l l cases a=b, and the Adams p r e d i c t o r and c o r r e c t o r formulas are of the same order. Table k Max. PECE e r r o r a k +1 -1 -2 -3 3 35 15 Ik k 67 27 27 •* 5 8 52 •* .6 k * * * PEC procedure unstable. 22. Max. PECE e r r o r J  Max. PEC e r r o r c=20, h=0.10> \ \ a k +i - l -2 -3 3 35 12 Ik ih •h 170 2.5 * •* 5 50 1* * -* 1 6 7 •* •* * PEC procedure unstable. In each.stable case,.the r a t i o of the maximum PECE e r r o r to the maximum-PEC error, 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 give an i n d i c a t i o n of the improvement i n e f f i c i e n c y t o be obtained by i n c r e a s i n g the step s i z e used f o r the PEC i n t e g r a t i o n . Conclusion Both t h e o r e t i c a l and experimental 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 , the PEC procedure based on Adams formulas i s a most e f f i c i e n t means of determining a s o l u t i o n t o w i t h i n e r r o r t o l e r a n c e s . The experimental r e s u l t s have a l s o exphasized the•importance of s t a b i l i t y when choosing a most e f f i c i e n t method. Our o r i g i n a l assumption had been t h a t i f a m u l t i - s t e p formula based.on the PECE procedure was s t a b l e w i t h step s i z e h, then.the corresponding PEC procedure would a l s o be s t a b l e w i t h h/2 I f t h i s was t r u e , the c r i t e r i o n of choice would be-the r e l a t i v e s i z e of the propagated e r r o r . However, our numerical experiments have shown otherwise and consequently, a new p o s s i b i l i t y has a r i s e n . To achieve a p r e s c r i b e d accuracy, 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 of PECE co u l d permit the use of'higher order formulas at lower i n t e g r a t i o n c ost than a s t a b l e PEC procedure. The i n v e s t i g a t of t h i s p o s s i b i l i t y remains as a problem f o r the f u t u r e . BIBLIOGRAPHY 23-Hamming, R.W. Stable P r e d i c t o r - C o r r e c t o r Methods f o r Ordinary D i f f e r e n t i a l Equations, J.Assoc.Comput.Mach..vol.6, 1959, PP-37-^7-Hull,-T.E. and Luxemburg, W.A.J. Numerical Methods and Existence Theorems, f o r Ordinary D i f f e r e n t i a l Equations, Numerische Math, v o l . 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 Minimize Propagated E r r o r s , J.Soc.Indust. Appl.Math. vol-9> 19^1, pp.31-47. M i l n e , W.E. and Reynolds, R.R. F i f t h - O r d e r Methods f o r the Numerical S o l u t i o n of Ordinary D i f f e r e n t i a l Equations, J.Assoc.Comput.Mach. v o l . 9 , 1962, pp.64-70. Nordsieck, Arnold. On Numerical I n t e g r a t i o n of Ordinary D i f f e r e n t i a l Equations, Math, of Comp..vol.16, 1962,.pp.22-49-R a l s t o n , A. Some T h e o r e t i c a l and Computational Matters R e l a t i n g to P r e d i c t o r - C o r r e c t o r Methods of Numerical I n t e g r a t i o n , Computer J . vol.4, 1961, pp.64-67. 

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