self-starting multistep methods for the numerical

N A S A TECHNICAL NOTE N A S A TN D-2936 --- i' f

SELF-STARTING MULTISTEP METHODS FOR THE NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS

I

by WilZiam A. Mersman

Ames Research Center Mofett Field, CdliJ:

NATIONAL AERONAUTICS A N D SPACE A D M I N I S T R A T I O N WASHINGTON, D. C. 0 JULY 1965

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TECH LIBRARY KAFB, NM

I Illill 11111 llll lllll lllll Ill# lllll Ill1 Ill 02547b8

NASA T N D-2936

SELF-STARTING M U L T I S T E P METHODS FOR T H E NUMERICAL

INTEGRATION O F ORDINARY D I F F E R E N T I A L EQUATIONS

By Wi l l i am A. M e r s m a n

A m e s R e s e a r c h C e n t e r Moffet t Field, Cal i f .

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical lnformotion Springfield, Virginia 22151 - Price $2.00

SELF-STAXTING MULTISTEP METHODS FOR THE NUMERICAL

IIYIEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS

By W i l l i a m A. Mersman Ames Research Center

Classical , multistep, predictor-corrector procedures f o r the numerical integrat ion of systems of ordinary d i f f e r e n t i a l equations a re generalized t o provide compatible, s e l f - s t a r t i ng methods. Expl ic i t algorithms and t ab le s of numerical coeff ic ients a r e presented,

INTRODUCTION

The numerical in tegra t ion of systems of ordinary d i f f e r e n t i a l equations on modern automatic computers i s usually accomplished by means of so-called multistep methods, pa r t i cu la r ly t h e predictor-corrector methods associated with the names Adam, Bashforth, Moulton, StErmer, and Cowell. It i s usually assumed tha t these methods a re not self-s tar t ing, and recourse i s had t o s ingle-s tep methods l i k e t h a t of Runge-Kutta t o obtain s t a r t i n g values. This leads t o cumbersome computer programs requiring what amounts t o unessent ia l t a l l y ing t o determine whether enough s t a r t i n g values have been obtained.

The purpose of t h e present report i s t o derive simple generalizations of t he c l a s s i c a l predictor-corrector formulas t h a t i m e d i a t e l y y ie ld compatible s e l f - s t a r t i ng procedures t h a t produce a l l t h e required backward differences d i r ec t ly from the i n i t i a l conditions.

STATEDENT OF THE PROBUM

The problem i s t o devise a se l f - s ta r t ing , multistep procedure f o r the numerical solut ion of t h e i n i t i a l value problem

(1) I d x = y

- = f ( X , Y , t )

a t

dY at

a t t h e discrete , equally spaced points tn , n = 1, 2, 3, . . . The var i - ables x, y, and f a re vectors, a l l of t he same ( f i n i t e ) dimension.

Let t h e common i n t e r v a l of t h e independent var iable be denoted by h, so t h a t

t n = t o + nh and introduce t h e usual notat ion

The index, n, w i l l be r e s t r i c t e d t o i n t e g r a l values, usual ly posit ive, although negative values w i l l be introduced i n some of t he s t a r t i ng procedures t o be discussed l a t e r .

The general problem, then, i s t o devise algorithms f o r calculat ing x,, yn, fn, f o r and the i n i t i a l values ( 2 ) . ignoring t h e variable, x, throughout t h e general theory.

n = 1, 2, 3, . , given s imply the d i f f e r e n t i a l equations (1) The theory f o r f i r s t - o r d e r systems i s obtained by

The procedure t o be used i s the conventional one of approximating the function, f , by a polynomial i n t of degree q. The problem i s then s p l i t i n t o two: t h e forward in tegra t ion problem and t h e s t a r t i n g problem.

The Forward Integrat ion Problem

The problem of backward difference xn, Yn, fn, and t h e

in tegra t ing forward one s t e p w i l l be solved by means of formulas of t he Adams type. Here it i s assumed tha t t n , first q backward differences of fn :

1 vofn = f n

k = l(1)q

are known. quant i t ies a t t = tn+l .

Several algorithms w i l l be derived f o r computing a l l these

The Star t ing Problem

Before the forward in tegra t ion algorithms can be applied, it i s necessary t o compute i n i t i a l values of the backward differences of preferably a t t = t o .

f a t some point, This w i l l be done by developing i t e r a t i v e algorithms

f o r t he backward calculat ion of X-n, Y-n, f -n at n = l ( l ) q , from which t h e backward differences of f o a re eas i ly calculated.

It sometimes happens t h a t t he i n i t i a l values l i e near a s ingular i ty . I n t h i s case t h e backward s t a r t e r may f a i l . For t h i s reason, i t e r a t i v e forward s t a r t e r s w i l l a l so be derived f o r t h e calculat ion of Xn, yny fn, a t n = l (1 )q .

I n e i the r case the f i n a l "output" of t h e s t a r t i n g procedure w i l l be xo, yo, fo, and t h e f irst q backward differences of f,, s o t h a t t h e s t a r t e r w i l l be compatible with the forward integrat ion procedures.

In the following sect ion t h e basic backward difference equations w i l l be

It i s derived. These a re general izat ions of t he equations usually ascribed t o Adam, Bashforth, Moulton, StGrmer, and Cowell ( r e f . 1, chs. 5 and 6 ) . t h e generalization of t he c l a s s i c a l formulas t h a t makes se l f - s t a r t i ng procedures possible.

GENEFKGIZED BACKWARD DIFFERENCE EQUATIONS

Normal Form

The basic difference equations, r e l a t ing Vy,.Vx, and v2x a t t = t j t o f and i t s backward differences a t t = tn , both J and n being arb i t ra ry , a r e

W

k=o

These a re wr i t ten formally as i n f i n i t e s e r i e s t o simplify ce r t a in index manip- ulat ions. They terminate and a re exact whenever f i s a polynomial i n t .

The proof i s a straightforward generalization of Henrici 's ( r e f . 1, pp. 191-194 and 290-293). Write y j as a Taylor's s e r i e s with remainder centered a t t j -1 ( r e f . 2, p. 95):

Now note t h a t

3

Approximating f by means of Newton’s in te rpola t ing polynomial ( ref . 1, PP. 190-1911

where the symbol

r(s + 1) (’1 = k!r(s + 1 - k)

yields equation (4), with y given by

Equation ( 5 ) i s obtained s imi la r ly by wri t ing the Taylor’s s e r i e s f o r x j and X j - 2 centered a t tj-1

adding and inser t ing Newton’s in te rpola t ing polynomial yields equation ( 5 ) , with cr given by

Before deriving equation (6 ) it i s convenient t o discuss equations ( 4 ) and ( 5 ) and some of t he properties of y and cr.

Equation (4 ) with j = n + 1 i s t h e Adam-Bashforth predictor ( r e f . 1, pp. 192-193). Henrici uses t h e notation

yk = y-l,k

Equation ( 4 ) with j = n i s t h e Adam-Moulton corrector ( r e f . 1, pp. 194-195). Henrici uses the notation

yk* = Yo,k

Equation ( 5 ) with j = n + 1 i s the StGrmer predictor (ref. 1, pp. 291- 292). Henrici uses the notation

4

Equation ( 5 ) with j = n is t h e Cowell corrector (ref. 1, pp. 292-293). Henrici uses the notation

o,k Ok* = 0

A s w i l l be seen later, equations ( 4 ) t o (6) with j = n - q ( l > n - 1 form t h e bas is f o r t h e se l f - s t a r t i ng procedures t o be developed.

The most important property of y and CT i s t h a t each row i s the first backward difference of t h e preceding r o w :

1 bp+i,k = ap,k - ap,k-i

while yp, = ap,o = l. These follow immediately from the def ini t ions, equa- t i ons (7) , ( 8 ) , and the well-known recurrence r e l a t i o n f o r t h e binomial coef- f i c i e n t s :

Equations (9) can be rearranged i n the useful form

where the subscript on V j i s used t o emphasize tha t V here operates on j , not on n or k.

The t ab le s of y and a a t t h e end of t he report were computed by means of t he def ini t ions, equations (7) and ( 8 1 , f o r t he f i r s t row ( p = -l), and the difference equations (9) f o r subsequent rows.

To re turn t o the der ivat ion of equation (61, note f i rs t t h a t x bears t he same re l a t ion t o y as y does t o f . Hence, we can wri te equation ( 4 ) i n t h e t r ans l i t e r a t ed form

00

Writing equation ( 4 ) with j = n gives

and taking the ( m - 1)s t backward difference gives 03

5

Subst i tut ing i n equation (4a) and rearranging gives

k=o r=o Applying t h e backward difference operator

V j and using equations (10) gives

k v2xj = h2 f Yo,rYn-j,k-rV f n k=o n=o

Comparison with equation ( 5 ) gives t h e important i d e n t i t y

k

U r = o

Hence,

and t h i s reduces equation (11) t o equation (6 ) . Q.E.D.

Equation (12) cons t i tu tes a valuable check on t h e t ab le s of y and o and has been used. t h e exp l i c i t formulas

I n addition, f o r small values of k, it i s convenient t o have

Yp,o = 1

Equation (6), with j = n + 1 o r n, cons t i tu tes a new predictor- corrector scheme, t o be discussed l a t e r , which appears t o have some advantages over t he St8rmer-Cowell scheme of equation ( 3 ) . j = n - q(1)n - 1, equa- t i o n (6), l i k e equations ( 4 ) and ( 5 ) , forms the bas i s f o r an i t e r a t i v e s t a r t e r .

For

This completes the der ivat ion of t h e basic backward difference equations Closely r e l a t ed t o these are similar equations using the i n t h e normal form.

6

first and second sums. and fu r the r confirmation of t h e i r effectiveness i s offered by Henrici ( r e f . 1, pp. 327-339).

Astronomers have long used these concepts (ref. 3 ) ,

These s m e d forms w i l l be derived i n the next section.

Summed Form

The basic difference equations, i n t h e summed form, r e l a t ing x and y a t t = t j t o f , i t s backward differences and i t s backward sums at j and n being arb i t ra ry , a re

t = tn , both

J k=o where the backward sums j = n:

A, and Bn a re defined by these same equations with

1 W

An = yn - 1 yo,k+l.Dkfn

J k=o The names f i rs t and second sum for An and h, respectively, a r e j u s t i f i e d by t h e property

1

The proof i s qui te straightforward. Replace t h e coeff ic ient , y, i n equa- t i o n ( 4 ) by i t s equivalent value from equations (10) :

where, again, t h e subscript j i s aff ixed t o v t o emphasize t h a t t h e opera- to r , especial ly i n t h e r i g h t member, a c t s on j ra ther than n or k. This i s a simple, l i n e a r difference equation whose solut ion i s obviously t h e first of equations (l?), i n which An i s an a r b i t r a r y constant vector, independent of j, whose value i s determined by se t t i ng j = n, equations (16).

If y i s now eliminated from equation (6) by means of t h e first of equations ?16), and i f t h e coeff ic ient , 0, is replaced by means of equa- t ions(10) , t h e r e s u l t i s again a simple, l i nea r difference equation:

7

h2 1 an- j,k+z*n ( : o ) V j X j = hAn + V j

whose solut ion i s c l ea r ly t h e second of equations (l?), where a r b i t r a r y constant vector independent of s e t t i n g j = n (eqs. (16)) .

% i s another j, whose value i s determined by

Equations (17) follow immediately from equations (16) on taking t h e back- w a r d difference and applying equations (4) and (6) with j = n.

As w i l l be seen i n l a t e r sections, equations (16) w i l l be used only once,

j = n +l

n replaced by LI -k 1 and j = n + 1, w i l l

i n connection with the s t a r t i n g procedure, t o obtain i n i t i a l values, A0 and BO, of t he f irst and second sums. w i l l y ie ld the summed versions of t he Adams-Bashforth and Stzrmer predictors, while equations (13) and (l7), with y ie ld the summed versions of t he Adams-Moulton and Cowell correctors.

Subsequently, equations (13) with

Before discussing s t a r t i n g procedures, it i s necessary t o convert t h e basic difference equations ( k ) , ( 3 ) , ( 6 ) , (l?), and (16) t o the backward ordinate form, i n which e a r l i e r values of f are used ra ther than i t s backward differences.

GENERALIZED BACKWARD OFDINATE EQUATIONS

The backward difference equations of previous sect ions have been wr i t ten formally as i n f i n i t e s e r i e s merely f o r manipulative convenience. it i s assumed t h a t f i s approximated by a polynomial i n t of degree q, say, and the se r i e s are a l l truncated a t If, then, t h e backward d i f - ferences i n equations (15) and (16) a re eliminated by means of t h e well-known

I n practice,

k = q.

i den t i ty ( r e f . 1, p. 190) k

&, = E ( - 1 ) m m=o

and t h e order of su"a,tion i s reversed i n following backward ordinate equations a r e

6) fn-m

the resu l t ing double sums, t h e obtained i n summed form:

I y j = An + h t Yq,n-j,m f n-m

m=o q

x = Bn + ( j - n - 1)- + hzCDq,n-j,m f n-m j

m=o

where the coef f ic ien ts y and cr are

8

k=m I

1 k=m

Again the sums An and are given by setting j = n: q 1

(20) I m=o

Bn xn + hAn - h2 f 09, o,mfn-m m=o

Eliminating An and Bn from equations (18) and (20) then gives the normal form of the backward ordinate equations :

q

= yn - h 1 aq,n- j ,dn-m m=o

Y j

q

x j = Xn - ( n - j)hyn + h2CPq,n-j,mfn-m m=o

where the coef f ic ien ts a and P a r e

(22) I aq,p,m = Yq,o,m - Yq,p,m

Pq,p,m - Oq,p,m - Gq,o,m + PYq,o,m

The t ab le s of y , 6, a, and P a t the end of t he report were computed

-

using equations (19) and (22) together with the e a s i l y proved re la t ions :

9

Taking f ( t ) = tr i n equations (21), r = O(l)q, gives the i d e n t i t i e s q

which were used as a f i n a l check on t h e tab les .

All t h e basic formulas, summed and n o m l , i n backward difference and backward ordinate forms, have now been derived. The remaining sect ions of t h e report a r e devoted t o t h e presentation of general and spec i f ic algorithms f o r s t a r t i n g and continuing the integrat ion.

EUICKNARD STARTING ALGORITHMS

A backward s t a r t i n g algorithm i s one t h a t produces the backward d i f f e r - ences of Po, up t o order q, given merely the i n i t i a l conditions, equa- t i ons ( 2 ) , and, of course, t he d i f f e r e n t i a l equations (1). t o an algorithm t h a t w i l l produce backward values of t he ordinates, f,p, f o r p = l (1 )q ; it i s then a simple matter t o obtain t h e backward differences a t fo.

This i s equivalent

(See the appendix for de ta i l s . )

The desired algorithm, i n normal f o r m , i s impl ic i t ly contained i n equa- t i ons (21) with n = 0, j = -p:

q - Y-p = Yo -

m=o

U m=o

f-p = f ( t o - Ph,X-p,Y-p) J f o r p = l ( 1 ) q . This i s a s e t of 3q implici t equations f o r t he 3q unknowns Y-p, x-p.' f -p*

If an approximate solution i s known, an improved solution i s readi ly obtained by i t e r a t i n g equations (23); inser t ing "old" values i n the r igh t members produces "new" values in the l e f t members. exhib i t s the a f o r q = 2, 3 and discusses the convergence of the i t e r a t i o n , but does not derive the general formulas. Two methods of obtaining an i n i t i a l approximation w i l l now be given.

Collatz ( r e f . 4, pp. 99-101)

10

Bootstrap S t a r t e r

Since only one value of f , namely fo, i s known i n i t i a l l y , t he simplest possible procedure i s t o take Set t ing p = 1 gives t h e "predicted" values of x, y, and f a t t-1. Then se t t i ng q = 1, p = 1, gives corrected values. Keeping q = 1 and now se t t i ng p = 2, gives predicted values a t t-2. This bootstrapping procedure can be repeated u n t i l t he desired value of q i s reached. The exp l i c i t algorithm i s

q = 0 i n equations (23).

Mult i g l e correct or k = l(1)p P 7

Y-k = Yo - h l a p , k , m f -m m=o

Experienced computer programers w i l l recognize t h i s as a simple, nested DO LOOP.

To make the procedure more tangible, t h e f irst few algorithms are writ ten exp l i c i t l y below:

p = 1; predictor y-l = yo - hfo

h2 X-1 = xo - hyo + e f o

f-1 = f ( t - l , X - L , Y - d

h y-1 = yo - E ( f o + f-1) k = 1; corrector

h2 X - 1 = xo - hyo + 7 (2fo + f-1)

f-1 = f ( t - l ,x - l ,Y- l )

11

p = 2; predictor y-2 = yo - 2hf-1

h2 3 X-2 = xo - 2hyo + - (2fo + 4f-1)

f-2 = f(t-2+2,y-2)

h Multiple corrector y-1 = yo - (3f0 + 8f-1 - f-2)

h2 24

k = 1 X-1 = xo - hyo + - (7fo + 6f-1 - f-2)

f-1 = f(t-1yx-1,y-d

h 3

y-2 = YO - - ( f o + 4f-1 + f-2)

X-2 = xo - 2hyo + - (2fo + 4f-1) h2 3

I [

[ I I f-3 = f(t-3JX-3.9y-3)

k = 2

f-2 = f(t-2+-2,y-2)

h

h2

(3f0 + 9f-2)

X-3 = XO - 3hy0 + 8 (gfo + 18f-1 + 9f-2)

y-3 = yo - p = 3; predictor

f-3 = f(t-3+3,y-3)

h 24

Multiple corrector y-1 = yo - - (gfo + lg f -1 - 3f-2 + f-3)

h2 k = 1 X-1 = XO - hyo + - (97fo + 114f-1 - 39f-2 + 8f-3) 360

f-1 = f ( t - l y x - l y Y - l )

Y-2 = Yo - - ( f o + 4f-1 + f-2)

x-2 = xo - 2hyo + - (28fo + 66f-1 - 6f-2 + 2f-3) 3

h2 k = 2 45

f -2 = f(t-2+-2,y-2)

h Y-3 = Yo - 8 (3fo 9f-1 gf-2 + 3f-3)

h2 X - 3 = X o - 3hYo + - (331fo + 972f-1 + 243f-2 + 54f-3). k = 3

360

The bootstrap s t a r t e r seems t o be qui te e f f i c i e n t i n prac t ice , but it i s awkward and space-consuming when programmed for automatic computers, because of t he mul t ip l ic i ty of matrices and algorithms required. following logica l ly simpler method.

This suggests t he

12

I t e r a t ed S ta r t e r

The bootstrap starter i s e s sen t i a l ly an e f f i c i en t method of obtaining first approximations f o r use i n t h e r igh t members of equations (23), which a re then i te ra ted . i z e by se t t i ng t h e s ingle set of equations (23).

A l og ica l ly simpler, but less e f f i c i en t method i s t o i n i t i a l - f - m = fo, m = l ( l ) q , i n t he r igh t members, and then i t e r a t e

Backward Star te r , Summed Form

Star t ing with equations (18) and ( 2 0 ) , instead of (23) and again s e t t i n g n = 0, j = -p gives t h e impl ic i t , swnmed form of t h e backward s t a r t e r :

Y v

Bo = XO + hAo - m=o

I I n i t i a l values can be obtained by t h e obvious bootstrap procedure or by s t a r t - ing with f - m = fo, m = l (1 )q .

h A 0 = Yo + f o

L

Bo = xo + hA0 - - h2 = xo + hyo + (&) h2fo 12 f o Equations ( 2 5 ) can then be i t e r a t e d u n t i l they converge.

This algorithm i s mentioned mainly f o r t h e sake of completeness. The pr inc ipa l reason f o r introducing the first and second sums i s t o obtain b e t t e r control of t he accumulated round-off e r ro r during a long integration, but t h i s consideration may be i r re levent t o a s t a r t i n g procedure.

Before turning t o t h e subject of forward s t a r t i n g procedures, it may be noticed t h a t t h e backward s t a r t e r (eqs. ( 2 3 ) ) produces t h e ordinates p = l (1 )q .

fmp, f o (see t h e These are e a s i l y converted t o backward differences a t

appendix). If summed forms of t he forward in tegra t ion procedure a re t o be used subsequently, t h e i n i t i a l values of A. and Bo are eas i ly computed from equations(16), with n = 0. Thus even here there i s no compulsion t o use the summed form of the s t a r t e r .

FORWARD STARTING ALGORITHMS

The der ivat ion of forward s t a r t i n g algorithms i s almost t r i v i a l . Each of t he backward s t a r t e r s discussed previously becomes a forward s t a r t e r by means of t h e simple transformation

h 4 -h

f - m --* f m

This produces t h e forward ordinates, fa, m = O(1)q.

J

The conversion t o backward differences a t tq i s straightforward. How- ever, t o do t h i s on an automatic computer would involve e i the r losing a l l information a t the points between requirements excessively. Furthermore, e i t h e r choice leads t o a procedure t h a t i s d i f fe ren t , i n i t s external appearance and mode of usage, from t h e backward s t a r t e r .

to and tq, or else increasing t h e storage

A preferable procedure i s t o compute the backward difference t ab le a t tq and then extend it back t o to by holding Vqf constant; t h i s , of course, i s consistent with t h e s t a r t i n g procedure. This i s e a s i l y done, and programming d e t a i l s a re given i n the appendix.

Choosing the l a t t e r a l te rna t ive provides computer programmers with a ba t t e ry of s t a r t i ng procedures, forward or backward, bootstrap o r i t e r a t i v e , . i n normal or summed form, with iden t i ca l ex terna l appearance. I n every case the input data consis t of the i n i t i a l conditions, and the-output data consis t of t h e t a b l e of backward differences (and sums) at t h e i n i t i a l point, i n a form compatible with the forward in tegra t ion procedures t o be discussed next.

PEIEDICTOR-COFXECTOR ALGORITHMS FOR FORWARD INTEGRATION

The purpose of t h i s sect ion i s t o present a va r i e ty of algorithms f o r the forward integrat ion from t n t o tn+l . Specif ical ly , it i s assumed t h a t t h e input consis ts of tn , xn, yny fn, and t h e first q backward differences of fn, together with t h e sums An and Bn when appropriate. The output i s t o be

14

the same l i s t of quant i t ies a t tn+l . r i thms with any of t h e s t a r t e r s provides t h e complete solut ion of t he i n i t i a l - value problem.

The combination of any of these algo-

All t he algorithms t o be presented involve the use of a predictor f o l - lowed by a corrector, requiring two calculat ions of f (x ,y , t ) . Conflicting philosophies regarding the need f o r a corrector can be found i n references 3 and 5 . The general consensus among automatic computer users seems t o favor the use of one corrector. In t h e algorithms given below t h e user can, of course, omit t h e corrector i f he chooses.

The backward difference equations (4 ) through (6) and (15) through (17) give predictor formulas when j = n + 1. Taking j = n and then replacing n by n + 1 yields corrector formulas. I n every case, i f both predictor and corrector a re truncated a t t he same value k = q, then subtracting the pre- d ic tor from t h e corrector and using t h e recurrence re la t ions (eqs. ( 9 ) ) f o r t he coef f ic ien ts y and 0 gives a shorter formula f o r t h e corrector.

Throughout, predicted values a re indicated by an a s t e r i s k (*).

Normal Form

Firs t -order system.- The Adams-Bashforth predictor i s

q r 7

= yn + h l Y - l , k v k 'n %+l

k=o and t h e Adam-Moulton corrector i s

Simple second-order system.- If t h e f i r s t derivative, y, does not occur i n the d i f f e r e n t i a l equation and i s not required, t he formulas f o r x are:

StGrmer predictor q

k=o

General second-order system.- When the f i r s t derivation, y, i s present, equations ( 4 ) and (6) y ie ld the predictor

15

The predictor i s

The corrector i s

Summed Form

- I * - B, + h2 1 xn+i -

I k=o

and, of course,

I n a l l these predictor-corrector algorithms, t he calculat ion of t h e d i f - ference t ab le i s f a c i l i t a t e d by noting tha t , on defining

* E = fn+i - fn+l

t h e differences a re

V k fn+l = &",+I + E , k = l(l)q

(see r e f . 1, p. 196). course, d i r e c t l y from the def in i t ion :

The predicted differences, $f",+,, are obtained, of

16

CONCLUDING REMARKS

The present report displays a var ie ty of algorithms f o r s t a r t i ng and continuing the numerical in tegra t ion of systems of ordinary d i f f e r e n t i a l equa- t ions . The exhaustive t e s t i n g of these algorithms, f o r t h e purpose of compar- ing t h e i r effectiveness, would be an expensibe process. deal of relevant experience has been obtained i n recent years i n computing in s t a l l a t ions throughout t h e world. compromise between t h e conf l ic t ing desiderata of speed, accuracy, and programming compactness can be achieved by the following choice:

Fortunately, a grea t

I n t h e present writer's opinion t h e bes t

(1) Use t h e fourth-order methods f o r f i r s t -o rde r equations, sixth-order methods f o r second-order equations ( q = 4, 6, respect ively) .

Use the i t e r a t e d s t a r t e r , i t e r a t e d eight times.

Use the swnmed form of t h e predictor-corrector algorithm, i n backward difference form.

( 2 )

( 3 )

(4) Carry four ex t ra s ign i f icant decimal d ig i t s , i n floating-point form, t o control round-off e r rors .

The effectiveness of t h e summed form of the predictor-corrector algorithms has long been known t o astronomers ( r e f . 3 ) , and addi t ional evidence i s furnished by Henrici ( r e f . 1, pp. 336-339). l e s s e f f i c i e n t than t h e boots t rap version, but i s far simpler t o program and i s much more modest i n i t s storage requirements.

The i t e r a t e d s t a r t e r i s somewhat

The use of backward differences i n the forward integrat ion i s preferable t o the use of backward ordinates f o r t w o reasons: formula tends t o add near ly equal quant i t ies of a l te rna t ing sign, whereas the backward difference formula adds monotonically decreasing quant i t ies ; and ( 2 ) t he ava i l ab i l i t y of t he difference t ab le makes e r ro r estimation and automatic adjustment of t he in t e rva l s ize a straightforward procedure.

(1) the backward ordinate

Ames Research Center National Aeronautics and Space Administration

Moffett F ie ld , C a l i f . , Apri l 19, 1965

CONVERSION O F ORDINATES TO DIFFEXENCES

The calculat ion of a t ab le of backward differences from a t ab le of ordinates, and i t s extension i n e i t h e r d i rec t ion when higher order differences a re neglected, is a familiar procedure t o computers working with penci l and paper. The programming of such procedures f o r automatic computers i s l e s s familiar, and the purpose of t h i s appendix i s t o give some t y p i c a l algorithms i n FORTRAN format, t h e most widely used s c i e n t i f i c programming language.

The algorithms exhibited here a re wr i t ten as though x, y, f , A, and B were scalars . When they are vectors, t h e algorithms a re e a s i l y generalized by t h e addi t ion of a second subscript and su i tab le addi t iona l "DO LOOPS" over t he range of t h e second subscript ( t he dimension of t h e vectors) .

Since the FORTBAN language does not permit subscr ipts nor indices i n loops t o assume nonpositive values, ce r t a in log ica l a r t i f i c i a l i t i e s appear i n these algorithms. Experienced programmers should have no d i f f i c u l t y i n removing them f o r other, less r e s t r i c t ed , programing languages.

Backward S t a r t e r

All t h e versions of t h e backward s t a r t e r discussed i n the main body of t he report produce t h e backward ordinates, f-p, p = o(1)q. placed i n the a r ray L:

Suppose these a re

~ ( p + m) = f - p

where m i s an a rb i t ra ry , posi t ive integer . Then t h e nested DO LOOP

DO k = l , q r

w i l l y ie ld VPfo i n L ( p + m), p = O(l)q, g _> 1. The s t ruc ture of t he algorithm is i l l u s t r a t e d by t h e following diagram, i n which v io la t ion of t h e FORTRAN res t r ic t ion! ) :

q = 3 and m = 0 ( i n

18

I n i t i a l values

L(3) = f-3

- . -

L(2) = f-2

L ( 1 ) = f -1

. .. -

k = l , 2 = 3

j = l , n = 2

= f -2 - f-3

= vf-2 ~ _ _ _ _ _ _

j = 2, n = 1

L(2) = L ( 1 ) - L(2)

= f -1 - f-2

= vf-1

j = 3 , n = O

L ( 1 ) = L ( 0 ) - L ( 1 )

= f o - f -1

= v f o

k = 2 , 2 = 2 I k = 3 , 2 = 1

j = 2 , n = 1

L(2) = L ( 1 ) - L(2)

= v f o - vf-1

= F f o

It i s evident from t h e diagram t h a t increasing q by uni ty adds one row t o each column and adds an addi t iona l column on t h e r igh t with one row.

Forward S t a r t e r

i All t h e versions of t h e forward s t a r t e r discussed i n the main body of t he

Suppose these a re placed report produce the forward ordinates, fp, p = O(1)q. i n t h e a r ray L :

~ ( p + m) = f p

Then the nested DO LOOP

DO k = l , q

Z = q + l - k

. DO j = 1 , 2

n = m + q - j

- ~ ( n + 1) = L(n + 1) - L(n)

w i l l y ie ld vpf, i n L(p + m), p = O(1)q. The diagram i l l u s t r a t e s t h e case q = 3 , m = 0 .

~


L ( 3 ) = f3

L(2) = f2

~

L ( 1 ) = f l

L(0) = f o

k = 3 , 2 = 1 k = l , 2 = 3

j = l , n = 2 j = 1, n = 2 j = l , n = 2

L(3) = L(3) - L(2)

= vf3 - v f2 = v-3 - v-2

= V2f3 = v3f3 -

j = 2 , n = 1

= vf2 - vf l

= Ff2 -

= f 3 - f 2

= v f3

j = 2 , n = 1

L(2) = L(2) - L ( 1 ) L(2) = L(2) - L ( 1 )

= f2 - f l

= vf2

j = 3 , n = O

L(1) = L ( 1 ) - L(0)

= f l - f,

= vp1 _ _ _ _

To obtain backward differences of f , it i s necessary t o make t h e assumption t h a t Vq+lf = 0. Then Vqf i s constant:

vqfp = $-fo = vqf , p = l(1)q Then t h e addi t ional nested DO LOOP

r = q - 1

DO k = l , r

2 z q - k

DO j = l , l

n = m + q - j

L(n) = L(n) - L(n + 1) yields @fo i n L(p + m), p = O(l)q, t he desired r e s u l t . The diagram i l l u s t r a t e s the case q = 4, m = 0.

20

~


L(4) = V4.f

L(2) = f f 2

L(0) = f,

k = l , 2 = 3

j = l , n = 3

L(3) = L(3) - L(4)

= a”f3 - V4f3 = v3f2

~

j = 3 , n = l

L(1) = L(1) - L(2)

= v f 1 - ffl = Vf,

k = 2 , 2 = 2 k = 3 , 2 = 1 I I

21

REFERENCES

1. Henrici, Peter: Discrete Variable Methods i n Ordinary Di f f e ren t i a l Equa- t ions . John Wiley and Sons, Inc., 1962.

2. Whittaker, E. T.; and Watson, G. N.: A Course of Modern Analysis. Fourth ed., Cambridge Univ. Press, 1940.

3. Herrick, Samuel: Step-by-step Integrat ion of x = f(x,y,z, t) Without a "Corrector ." no. 34, Apri l 1931, pp. 61-67.

Mathematical Tables and Other Aids t o Computation, vol. 5 ,

4. C o l l a t z , L.: The Numerical Treatment of D i f f e ren t i a l Equations. Third Edition, Springer, 1960.

5 . H u l l , T. E.; and Creemer, A. L.: Efficiency of Predictor-Corrector Pro- cedures. Journal of the Association f o r Computing Machinery, vol . 10, no. 3, Ju ly 1963, pp. 291-301.

22

P/k -1 0 1 2 3 4 5 6 7

60480 y 6 19087 -863 2 71 -191 271 -863 19087 198721 943759

-

0

1

0 -1 -2 -3 -4 -5 -6 -7 -8

120960 y 7 36799 -1375 351 - 191 191 -351 13 75

-36799 -434241

Y 0

1 1 1 1 1 1 1 1 1

-

73 121 181 253 337

2Y 1

1 -1 -3 -5 -7 -9

- 11 - 13 - 15

-51- -124 -245 -426 -679

12Y 2

5 -1 5 23 53 95 149 215 293

120 1. 120 . - . 2 I.

1 1 l l 0

37 13 I -14 -1

24Y ~

3 9 -1 1 -9 -55 -161 -351 -649

- 10 79

72OY 4 - 251 - 19 11

- 19 25 1 1901 6731 17261 36731

1440 y 5 475 -27 11

- 11 27

- 475 - 4277 -17739 -52261

2400 4 19 -I -I 19 299 13 19 3 799 8699 17219

2400

5 18 -1 0 1

- 18 -317

- 1636 -5435

- 1413 4

604800 6 43 15 -221 31 31

-221 43 15 84199

1866091 496471

604800 7

~~

4125 -190 31 0

-31 190

- 4125 -88324

- 5 84795

3 6288000

23 7671 - 9829 1571 -289 -289 15 71 -9829 23 76 71 5537111

8

g = 0: 2y P/" 0

0 -1 1 -3 2 3

- ~ -- ~~ ~~

g = 1: 12y I q =- 2: 24y- . . ~

0 1 0 - 1-2 12- -7 1 -15 4 -1

-13 -5 -25 -12 1 -7 -23 -23 -28 -9

-33 4 -55 -_ ~ - - ~- -

- 1

462 - 830

-1522 - 13 74 -1586

962 _ ~

P/m 0 1 2 3 4 5

~ ___- -

3 -

0 1 2 3

-469 177 -87 19 -739 -393 63 -11 -709 -783 -327 1-9 -739 -633 -897 -251

_I_

-469 -1743 873 -1901

. - . . ..

g = 6: 1209603. 0

- 84161 -122335 - 120609 - 121151 -120769 - 1213 11 -119585 - 15 775 9

1 55688 - 74536

-119272 - 122488 -118312

13 8008

- 124792

-130936

2 ~ -.

-66109 26813

-67165 - 128803 -115261 -129859 -8943 7

- 903 715 _ - _

3

-17984

-60480

- .

5 7024

14528

-135488 - 1029 76 - 177984 1198528

- +

4

-31523 8899

-5699 7843

-53795 -147773 -54851

-1465949

5 . -_ -

9976

1528 -1688 3832

-2648

-46424 -176648 717928

.-

6 -

- 13 75 351 - 191 191

-351 13 75

-36 799 - 43 4241 ~-

2

-4234 244 5730 60104 120526 184476 178874

3 4

-434 186 -376 31 5730 -376 60724 5265 116726 63574 266976 54851

3036 -1171 4125 60290

i 120991 1 181440 241889 302590 I358755

' P/m 0 0 1

0 1 1 0 1 1.3 12 1 2 23 14 3

q = 3: 2400 I q = 4: 604800 1 1 2 3 4 2144 5 420 60104 120 83 6 1825 76 220124

-2334 - 66 5730 60414 118626 225 726

1136 - 124 -3 76 5 420 62624 75476

-221 31 31

-221 43 15 84199

= 5: 604800 1 q 1

- 190 4 88324 4125

.-

p/m 0 1 2 3 4

0

3094 5265

181626

327340

60104 120991

240749 5 6

I q = 6: 36288000 1 0 1 2 3 4 5 6

244614 306474 3607974 7261194 10 88813 4 14503914 182143 74 20099274

- 401475 3 8205 339465 3601905 7255125 10921125 14297505 2320 5 465

3 78740 -57460 - 16 780 3 495 80 3612020

11265140 5979020

7200140

-217695 3 4725 -2475 -26895 339465 366 7005 6 856 125 19583 625

703 74 - 11286 173 4 3594

-20826

3 873 414 306474

186503 4

~~

- 9829 15 71 -289 -289 15 71 -9829 23 76 71 553 7111

7259171 10886111 14514911 181455 71 21762971 25639271

5 6 7

25

~ - .

q = 0: a ] g = 1: 2a

q = 3 : 24a p / m ~ 1 2 3 1 9 19 -5 1 2 8 32 8 o 3 9 27 27 9 4 0 64 -32 64 5

q = 4: 720a 2- 1 3 1 - 4 ~-

o 1

251 646 -264 106 -19 232 992 192 32 -8 243 918 648 378 -27 224 1024 384 1024 224 475 -250 3000 -1750 2125

P/" 0 1 1 19087 65112

3 18495 87480

5 18575 87000

2 18224 90240

4 18304 89088

6 17712 93312 7 36799 -41160

26

__ ~

2 3 4 5 6 -46461 37504 -20211 6312 -863

31347 58752 -19683 5832 -783

31875 80000 58125 28200 -1375

528 21248 -12912 4224 -592

24576 96256 11136 3072 -512

11664 117504 11664 93312 17712 418803 -570752 717213 -353976 216433

1 2 3 4 5 6 7

6 -1 16 32 27 54 27

I q = 4: 14 1 2

2048

-

0 1 1 - 2 3 4 5 7386 12945 -9132 5646 -2046 321

17040 52224 -17760 13056 -4848 768

36096 136704 16896 58368 -7680 1536 45750 178125 37500 93750 18750 4125 53136 233280 23328 176256 11664 46656

26568 94527 -1944 23490 -7776 1215

0

28549 65728

102465 139264 176125 212544 257593

NASA-Langley, 1965 A-2116

-

57750 223 488 400950 577536 753 750 933120

1051638 . ~~~

q = 6: 120960~ 2

-51453

-64881 - 10 7520

-9216 46875 93312

324135

3 42484

100864 170100 335872 512500 705024 585844

4

-23109 -55872 - 88209

- 28125 46656

4393 83

- 10 7520

5 ~~

725 4 17664 27702 36864 5 7750

186624 129654

-2432 -3807 -5120 -6875

175273

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