implicit runge-kutta processes...implicit runge-kutta processes by j. c. butcher 1. introduction. a...

Implicit Runge-Kutta Processes

By J. C. Butcher

1. Introduction. A Runge-Kutta process is a means of obtaining an approxi-

mation y to the solution at x = x0 + h for the system

-/ = f(y), y = yo at x = x0,

where y is a vector of n elements and f (y) a vector function of these elements.

The equations defining y for a v stage Runge-Kutta process are*

gW =í(yo + h£íaijg(A ii =1,2,

y = yo + A¿ big"',

"),

where the coefficients a,y, &»(¿, j = 1, 2, • • -, v) are numerical constants.

It was shown [1] that the true solution y and the approximation y can be ex-

panded in power series given by the equations t

(i) y = yo + Z<*F^,

(2) y = yo + 2 /3*Fir- 1)1'

The summations are over the different "elementary differentials" F for the

function f, arranged in a sequence of non-decreasing r, the order of F. $ is the cor-

responding "elementary weight" and a, ß are numerical coefficients independent

of the form of f. Some formulae and tables for a and ß are given in [1],

Thus if y, y are to agree to terms in h" we must have

(3) # - i7

where y = rß/a, for all $ for which rip.

A general Runge-Kutta process will be called "implicit" in contrast to those

processes in which em = 0 for i < j; these will be called "semi-explicit." If in addi-

tion a,j = 0 when i = j the process will be called "explicit." It has been traditional

(for example [2, 3, 4, 5, 6, 7]) to consider only explicit processes.

Received November 1, 1962. Revised April 22, 1963.

* If the function f(y) satisfies a Lipschitz condition and h is sufficiently small, then the

equations defining g(1>, g(2), • • • , gw have a unique solution which may be found by iteration

(see Appendix).

t It will be assumed throughout that f (y) and all its derivatives exist and are continuous

so that the Taylor expansions for y and y may be terminated at any term with an error of

the same order as the first term omitted.

50

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IMPLICIT RUNGE-KUTTA PROCESSES 51

For convenience we shall designate the process by an array as follows

an

«21

ai2

Ö22

au

a2,

avi ar2

bi b2 • • • by

where c, = X^=i aa ■A well known example of an explicit process is the following due to Kutta [3];

in this case p = 3.

0 0 0

0 0

■12 0

12 1¥ ¥ ¥

0

In contrast we have as examples of implicit and of semi-explicit processes the

following:

0 0 0 |0

s 1Tí 3

1¥

1"Tí

i6

1 1IS

0 0 0|0

1 i 0

0 1 0

The first of these is equivalent to a process due to R. F. Clippinger and B. Dims-

dale and quoted by Kunz [8]. On the other hand the semi-explicit process appears

to have been previously overlooked even though it is of comparable accuracy and

more convenient for practical use. Using results proved in section 2 of this paper

or by simply verifying (3) for the appropriate 3> it is easy to verify that p = 4 in

each case.

It is well to consider what we might hope to gain by relaxing the restriction of

allowing only explicit processes. In explicit processes we have viv + l)/2 (= A7E ,

say) coefficients o,-y, 6¿ to choose while in the semi-explicit case the number is in-

creased to JVS = viv + 3)/2. However, for implicit processes we have available

the complete set of Nj = viv + 1) coefficients. It is reasonable to hope that with

N variables we can satisfy M restrictions as long as M ^ N.

The numbers NE , Ns , Ni are shown in Table 1 as are the numbers of restric-

tions M corresponding to different values of p. This number is, of course, the num-

ber of elementary differentials with orders not exceeding p, or what is equivalent,

the number of rooted trees with no more than p nodes [9].

Basing our comparison on this table we see that the gain in accuracy that can

be achieved by the use of implicit rather than explicit processes is not more than

one power of h and in some cases semi-explicit processes do not give even this gain.


52 J. C. BUTCHER

Table 1

NE Na Nj

12

34

5

67

1

36

10152128

2

59

14202735

2

61220304256

M

124

817

3785

However, the type of argument based on merely counting the number of equa-

tions of the form (3) that must be satisfied, ignores the relationships between

them. In fact, it happens that the M restrictions of Table 1 can often be satisfied

with considerably fewer than M variables.

Section 2 of this paper will be devoted to some general results on the relation-

ships between the different $, and the following sections will contain a study of a

set of processes which are generalizations of the Gauss-Legendre quadrature formu-

lae.

2. Some General Results. It is convenient to restrict ourselves in this paper to

processes in which ci, c2, • • •, c, are all distinct and none of 6i, b2, ■ • •, 6, vanishes.

If these restrictions are relaxed some results of this section will require slight

modifications.

We shall use the symbols A, B, C, D, E to represent certain statements about

the numbers oy , &¡. These statements will depend on one or, in the case of E, two

integral parameters which we will write as though they were arguments and the

statement symbol a function of them. In the definitions of the symbols which now

follow, k and I will always denote positive integers.

Ait):

Bit):

Cit):

Dit):

$ = — whenever r ^ £,7

[♦*"*]-¿ to «*"*-1 far **«,¿=i k

2Z Gti c/-1 = ~ for i = 1, 2, • • • , v and k ^ £,y—i k

¿ 6, ct^ij = 6y(1 7 Ci ) for j = 1, 2, • • -, v and k £ Í,¿=.1 K

1EiS, r,): [**-V-1]] - ZZbic'-'atjc/-1 = T.

t-l 3=1 t(K -\- I)for k g S and î g ij.

A number of theorems which are needed in the later sections of this paper can be

expressed conveniently in terms of these symbols. After a statement of these the

proofs will follow.

Theorem 1. If Ait), then Bit).

Theorem 2. If Ait + v), then Eit, v)-



Theorem 3. If Bit + v) and Civ), then Eit, ij).

Theorem 4. If Bit + v) and Dit), then E(t, »).

Theorem 5. If Biv 4- n) andEiv, v), then Civ).

Theorem 6. If Bit + v) and Eit, v), then Dit).

Theorem 7. //5(f), Civ) and Dit), where f è t + V + 1, f ^ 2v + 2, then

A(t).To prove Theorem 1, we note that r = k for $ = [<p* ]. Also from equation

(31) of [1] we see that y = k. Similarly for $ = fo*-V-1]] we find r = k + I,y = lik 4-1) so Theorem 2 follows.

Theorem 3 may be verified immediately. If k á S, l á v then of the three

expressions

(4) ZZbid *-l !-l

i=l 3=1

1+t-l(5) ft hd

(6) ifJ+Tb)'

(4) and (5) are equal by Civ) whereas (5) and (6) are equal by Bit + v)-

Theorem 5 is a sort of converse to theorem 3. If k ^ v, I ^ i? (5) and (6) are

equal by Biv + v) while (4) and (6) are equal by E{v, v). Hence we have

¿^cWZaiW-'-ic/Uo.i-l \3-l I /

For k = 1, 2, • • • , v this constitutes a set of v homogeneous linear equations in the

v variables (T^Liam:*'1 — r"'c- ). * ■» 1,2, •• • , v. Since the matrix of co-

efficients is non singular, these variables all vanish, a result equivalent to Theorem

5.To prove Theorem 4 we see that if k á t, I ^ V then (4) and (6) are each equal

(by D it) and B it + v) respectively) to

(7) liibjcr-bjCj™-1)K y=i

so the result follows.

Theorem 6 bears the same sort of relationship to Theorem 4 as Theorem 5

bears to Theorem 3. If Jb á í, l è » (6) and (7) are equal by Bit + v) and (4)

and (6) are equal by E(t, v). Thus

Z c/-1 fe 6.- ct^ij - \ bj(l - Cj")) = 0,y=i \i=i k /

for I = 1,2, • ■ • , v and the result follows in the same way as for Theorem 5.

Finally, assuming that fis £ + ij 4- 1, f ^ 2rç 4- 2 we will, prove Theorem 7,

that A (f) follows from B(Ç),C(v) and D(t). We shall use the symbol m to denote

the identity of two expressions when regarded as linear functions oîbi ,b2, • ■ • ,bv.

Thus if <i> = Zí=i biXi, *' = Zi=i 6.x/ where x¿, X¿' are functions of

an , <*i2, • • • , ayy then $ = $> will mean x< = Xi > i'■ — 1, 2, • • • , v.


54 J. C. BUTCHER

Using this symbol we see immediately that if $i = $< (i = 1, 2, • • • , s) then

[$!$2 ... $j = [$/cí>2' • • • $,']. As a preliminary to the proof of the theorem we

show by induction the lemma that

7$ s r[<pr_1]

when r ^ t? 4- 1. If we suppose 4> = [$i$2 •••$»] and that

(8) 7i*i - r^-'lf

since the orders n , r2 • • • r, of $i, 4?2 •••,*» are each less than r, we have

y = ryi72 • • • 7.

so that

7$ = ryi 7j • • • 7„[<i>1 $2 ... $,]

-mm---fJI#'1"lI^] ••• [*r'_1]]

■ rSUiínE «í3 c/1_1 jf r2 Z «¿y cPj ••• \r»^l aa tyu)

Using Civ) and that fact that none of n , r2 ■ • • , rs can exceed v we find

7$ = r Z &<c«nH1 +r2 H-h-,^1^1

i—1

proving the lemma.

In this result we may now replace = by = and, by B(Ç) and the lemma we

have A(v + 1 ). To prove the result of the theorem for f ^ r > v + 1 we write

$ = [<ï>i<ï>2 •••$«] and suppose that $ is one of $i, $2, • • • , $» chosen so that its

order r is not exceeded by any of ri, r2, • • • , r¡. For r ^ v the lemma still holds

so we need consider only the cases r > v- We now use induction on /, r assuming

the result true for all lower values of r and with a given r for all lower values of r .

No two of n , r2, ■ • ■ ,rs can exceed v for otherwise we would have

r = 1 + n + r2 + ■ ■ • + rs ^ 2V 4- 3 > f.

Using the same sort of calculation as in the proof of the lemma we find

7$ = ry [i> <pr-r ]

= 17 L o¿ci a<yXyi-i

where

»'■Eto'¿=i

f Throughout this paper the convention will be adopted that if an elementary differential

is assigned a subscript, superscript or other distinguishing mark, the same marks will be used

with the corresponding elementary weights and the numerical constants a, ß, y.



and, by the induction hypothesis,

$ = —.7

Now r — r é it + V + 1) — iv + 1) = t- Hence, using Dit) we have

y* - ¿¿-¿m - eT')x!r — r i=i

; (*' - [«pr-r */*»' • • • «CDr _ r'

where $' being of order greater than 1 can be written

3> = [$i$2 ■ • • $.']•

The orders of $i', $2', • ■ • , 3v are each less than r so, using the induction hy-

pothesis, a short calculation enables us to evaluate

Hence

proving Theorem 7.

[<j>r r $1'$2/ • • • $'..] = —.

7 T

7$ = -^(i7--!Í) = l,r — r \y yr/

3. Integration Processes Based on Gaussian Quadrature Formulas. For the

system

(9)dy2 ,^ - 1, 2/2 - x„

• at x = £o,

the integration process reduces to a means of evaluating I = fxl+h f(x) dx using

the quadrature formula I = ft/JLi 6¿/(x0 + Ac,-). For this formula to be accurate

to terms in h", it is not necessary for all the conditions implied by A (p) to be satis-

fied but only those contained in the statement Bip). Hence to each Runge-Kutta

process there corresponds a quadrature formula characterized by the values of

bi,b2, ■ ■ • ,b„ ; Ci ,c2, • • ■ , c„. In this section we concern ourselves with the well

known Gauss-Legendre quadrature formulae [10]. It will be found that to each

such formula, as adapted for integrating in the range (x0, x0 + h), there corresponds

a unique Runge-Kutta process with the same order of accuracy.

For the form of the Gauss-Legendre formula that we use, c\,c2, ■ ■ • ,c, are the

roots of the equation P„(2c — 1) = 0, where P„ix) is the Legendre polynomial of

degree v. The consequence of this choice is that when any v of the equations of the

set B (2v) hold the rest do also so that bi ,b2, • • • , b, can be found as solutions to

these equations.

The important results of this section are the following:


56

v = 1:

J. C. BUTCHER

Table 2. Processes based on Gauss-Legendre quadrature formulas

v = 2:1_

41 a/34 6

4 "r 6 4

1 V32 6

i 4- V?2^6

36

1 1

2 2

2 VT5 j^ VIE9 "

_5_ \/Ï5 236 24 9

15 36 30

_5_ \/Ï536 24

A 4- VÍ5 2 a/Ï5 _5_36 + 30 9 15 36

_5_18

4

9_5_18

1 yl52 10

12

1 \/l52 ^ 10

* = 4: «i 0)1 — <l>3 + 0>4 0>1 — 0)3 — W4 Oil — 0)5

/ . / II I0)1 — 0)3 + 0)4 0)1 0)1 — 0)5 0)1 — 0)3 — 0)4

0)1 + 0)3 -J- 0)4 0)1+0)5 0)1 0)1 + 0)3 — 0)4

0)1 + 0)5 0)1 + 0>3 + 0)4 0)1+0)3 — 0)4 0)1

20)1 w 2o)/ £0)1

2~'°2

0)2

2 + W2

1,

2+0)2

[0,1 = i

V3Ü / = 1 , a/30144 ' Wl 8 ~t~ 144 ' 0)2

1 /15 + 2V3Ö2T 35

0)2 =

0)4 = 0)2

1 /15 - 2V3Ö A , V3Ö\ / _ / A a/30\2I/-35-' <ü,-w,x6+-247' W3 "^ Ve-^rJ'

/1 , 5v/30\ , / /1 5\/3Ö\ 0X2T + ^6r;' "* =w* V2î"i68"j' °»-"»-2«"

0)6 = 0)2 — 2o>3


IMPLICIT RUNGE-KUTTA PROCESSES

Table 2.—Continued

57

v = 5:

0)1/ . / 32 / /

0)1 — 0)3 + 0)4 Kñc — "6 0)1 — 0)3 — 0>4 0)1 — 0)»

0)1 — 0)3 +0)4 0)132225

— 0)6 0)1 — 0)6 0)1 — 0)3 — 0)4

0)1 + 0)7 0)1 + 0>732

225

320)1 + 0)3 + 0)4 0)1 + 0)6 j^- + fa>6'

0)1 — 0)7

0)1

0)1 — 0)7

0)1 + 0)3 — 0)4

. / . . / 32 / /0)1 + 0)6 0)1 + 0)3 + 0)4 ™£ +0)6 0)1 + 0)3 — 0)4 0)1

— 0>2

0)2

1

2

1 -L. '

2+0)2

1 ,

2+«2

20)!225 2W1'

322 + 13\/70

2o)i

_ 322 - 13\/7Ô , _ 322 + 13\/7Ü _ 1 /LW1-36ÖÖ ' Wl-36ÖÖ-' W2 ~ W'

35 + 2V7Ö63

452 - 59y

3240, 1 t /35 - 2V7Ö /452 + 59\/7Ü\ > >(

= 2V -63-' u° = <*\-3240-)' Ui = a*\

/64 + 11\/7Ö\ / / /64 - 11\/7Ö\ _ /23 - \/70\

I-ÏÔ8Ô-)' <*=«»x—1Ô8Ô-j' "» = ««1^ 405 j.0)4 = 0)2

' s 'f23_+_V7Ô\ 9 i , 0 i ,= 00)2 I --jnË- ) ' 0)6 = 0)2 — ^0)3 — 0)6 , 0¡6 = 0>2 — Z0>3 — 0)6 ,

/308 - 23\/7Ö\ / = , /308 + 23y/7Ö\"|

\ 960 / ' ** '= ** \ 960 )\ "0)7 = 0)2

Theorem 8. If A(2v) then Bi2v), Civ), D{v).

Theorem 9. // jB(2k) {hence a Gauss-Legendre formula) then either of C(v),

D (v) implies the other and either implies A (2v).

Theorem 10. Given values of bi ,b2, ■ ■ • , b,, ci, c2, • • • ,cy either C(v) or D (v)

defines a¿y (i,j = 1,2, ■ • • , v) uniquely.

To prove Theorem 8 we use Theorems 1 and 2 to deduce Bi2v) and Eiv, v)

and then Theorems 5 and 6 to deduce Civ) and ö(v).

Theorem 9 also follows easily from the previous results. From Theorems 3 and

4, Bi2v) with either of Civ), Div) implies Eiv, v); whereas from Theorems 5 and

6, Bi2v) and Eiv, v) imply both Civ) and Div). Finally we use Theorem 7 with

the values v = t = v, f = 2v to deduce A (2v).

Theorem 10 follows from the fact that Civ), Div) each consist of v sets of v

equations in v unknowns and since bi ,b2, • • ■ ,b„ are all non-zero and Ci ,c2, • • • , c,

are all distinct, the Matrices of the coefficients in the various sets are non-singular.

The sets of equations symbolized by C iv) are slightly simpler than those sym-

bolized by D i v ) so it is the former sets that we use for practical evaluations of


58 J. C. BUTCHER

flu, «12, ■ • • , o„. To find the parameters in a v stage Runge-Kutta process of

order 2v we may thus perform the following steps

(1) Evaluate the roots of P, (2c - 1) =0

(2) For each value of » (t — 1, 2, • • • , *) find an (j = 1, 2, • • • , v) as solutions

to the linear system

■ 1Z Clij Cy*_1 = 7 Ci (k = 1,2, ■ ■■ ,v).y=i k

(3) Find 6y ij = 1, 2, • • • , v) as solutions to the linear system

¿W_1 = £ (*= 1,2, ••-,,).y=i k

In Table 2 values of the parameters are given for v — 1, 2, 3, 4, 5. It will be

noticed that for the case v = 2 the process is the same as one suggested by Hammer

and Hollingsworth [11].

4. The Error Term. For a process of order p, the two series (1) and (2) are

identical for rgp, Hence the error vector y — y is given by

7 y k K ir-l)! " r\) kir- 1)! V y)

and we shall suppose that h is sufficiently small for this to be approximated by

£ 77^1T-(#-1)F = ^ £ ^F = A>+1 £ eFr-p+l (r — 1)! \ 7/ p! r=p+l r=p+l

where

5 = 4»--, e = —.7 p!

The coefficient of hp+1 in y — y is called by Henrici [12] "the principal error func-

tion" and our approximation is to assume that the principal error function is the

only important contributor to the truncation error.

We now restrict ourselves to the processes considered in the previous section

so that p = 2v. We shall study properties of 5 for the different F of order 2v + 1

so that a simple procedure can be found for evaluating e in these cases.

Suppose for an elementary differential F of order 2v + 1 we have

(10) F= {FiF2--- Fa}

and the orders of Fi, F2, • • • , F, are n , r2, • ■ • , rs. Then since

l + ri + r2+ ■■■ +rs = 2v+l

it is clear that the number of n , r2, ■ ■ ■ ,r„ which exceed v is either 0 or 1. In the

former case we describe F as a "central" elementary differential and in the latter

case a "non-central" elementary differential.

If F is non-central, suppose rx > v and that

(11) Pi- {TÄ •••¥,).Consider

(12) F' = {TO--- F„ÍF2F3--- F,}}



so that the orders ñ , f2, ■ ■ ■ fc, 2v + 1 — n of Fi, F2, • • • , F,, {F2F3 • • • F„}

are each less than ri while the order of F is the same as that of F.

Either F' is central or we may form F" from F' in the same way as F' was

formed from F. We may continue this process, forming in turn F, F', F", • • ■ until

the sequence is terminated at a central elementary differential F* say. It will be

proved that S = — ô' and hence that 5 = —5' = 8" = • • • = ±S* where 5* corre-

sponds to F* and the sign to be used is determined by the parity of the number of

members in the sequence F', F", • • • , F* .

We now summarize the principal result of this section.

Theorem 11. // F = {f2"}, 5 = - (x!)7{ (2?) !f> + 1)1}.Theorem 12. If F is central, Ô = - (vOVifi» !] 7}.Theorem 13. If F is non-central, and F' is as defined by (12), 5 = —S'.

To prove Theorem 11 we write the polynomial P„(2c — 1) as

Py (2c - 1) = po + pic + p2c2 +

and consider the equations B (2¡0,

+ PvC,

jL, bi Ci ' = ■=-,<=i k

together with the equation

Z bi Ci" =2v+ 1

+ S.

Multiplying the equations of B (2r) for fc = 1, 2,

and adding we find

k = 1,2, ••• ,2v

,v+lbypo,Pi

Po + Pi/2 + p2/3 + •

and similarly using the values fc = 2, 3, ■

find

po/2 + pi/3 + pi/4 +

Po/3 + pi/4 + p2/5 +

P*

+ pf/iv+l) =0

, v + 2 in the same way and so on we

• + pv/iv + 2) = 0

• + V,/iv + 3) = 0

Po/i v + i) + pi/iv + 2) + P2/iv + 3) + ■■■ +p, V^r+i + s~] = °-

We now eliminate po ,Pi,

and find

1

py from this set of homogeneous linear equations

1

v+ 1 v + 2 v+ 3

v+ 1

1

v + 2

1

27+1 +5

= 0


60 J. C. BUTCHER

so that

5 = -Dy+i/Dy = -

where we have written (see for example [13])

M)4i2v)l(2v+ 1)1

DNN

h \ N+ 1

N N + 1 N+2 2N - 1

[1!2!3! ••• jN - l)!]3

iV!(Ar + !)!••• (2tf - 1)!

This completes the proof of Theorem 11.

Since n ,r2, • • • ,r, are no more than v for a central elementary differential (8)

holds and the proof of the lemma that was introduced before that of Theorem 7

holds in this case as well. We thus have

7$= i2v + l)[<t>2'] = 1 - (,!)4/[(2.)!]2

and Theorem 12 follows on dividing by y.

Using the notation of (10), (11) and (12) we compute using, as usual, equation

(31) of [1]

7l = »*i7l72 ■ • • 7»

7 = ni2v + 1)7172 • • • 7V7273 • • ■ 7«

7' = i2v + 1) i2v — n + l)fi72 • • • 7VY273 • • • 7.

so that

1 + 1 = _I_.7 7' (2v — n + 1)7j 72 • • • 7«

Again employing the methods of Theorem 7 we arrive at the results

$ =7i 73 • • • 7,

[*! <t>¿¡

$■' =

i2v — r» + 1)72 73 • • • 7

r; T T r ,2y—r

($1 - [#1 #2 • • • *, <p2'-ri+1])

72 73[#!«»••• ^t*1"1]]

(2v — n + 1)7273 •••7.

[#i#2---W-ri+i



Hence

$ + $' = 75--7—rr-$i(2c — n + 1)72 73 •••7«

-1 + i7 7

which is equivalent to the result of Theorem 13.

Although in principal these results enable us to write down the coefficients for

each elementary differential of order 2c + 1 occuring in y — y, the practical value

of this is lessened by the large numbers of such terms which actually occur. How-

ever, as some guide to the truncation error we shall evaluate the coefficients «i and

e2t, say, for Fx = jf2"} and F2 = {2yf}2y.

These may be regarded as extreme cases; Fi is the only elementary differential

of order 2v + 1 which involves derivatives of order 2v of the elements of f (y) while

F2 is the only elementary differential of the same order involving only first deriva-

r)f f)ftives. If — is the matrix whose U, j) element is —— then we can write

3y ¿>yy

n n n ¿fiv4

Fi = Z Z ••• Z -—-—fhfit -•' fiu>n-i 12-1 «2,=i dyil dyiz • • • dyi2.

-(ÏÏôi is given by Theorem 11 as — iv !)*/{ (2»') ! (2c + 1) !j while ßi is given immediately

from Theorem 7 of [1] as 1.

Hence, we have

fc«i -(c!)4«i =

(2c)! [(2c)!]3(2c+ 1)

On the other hand, F2 is non-central and the central F* corresponding to it is

{Í,_! f}2_j}. 7* is iv !)2(2c + 1 ) so that

=F(c!)252 — ±5* —

(2c)!(2c+ 1)!

and the upper (or lower) sign is to be used if v is even (or odd). It is easy to see

that ß2 is (2c) ! Thus

ß2S2 +(c!)262 =

(2c)! (2c)!(2c+l)

-*0-The first few values of n and e2 may be found from Table 3.

It is natural to compare the integration processes described here with methods

based on quadrature formulae such as the method of Stoller and Morrison [14]. In

t The convention concerning subscripted constants a, ß, y will be extended to S and «.


62 j. c. butcher

Table 3

1A 1/62

1 -24 +122 -4320 -7203 -2 016000 +1008004 -1778 112000 -25 4016005 -2 534876 467200 +10059 033600

this type of method y¿ is estimated at x0 + cth (z = 1, 2, • ■ • , c) by some convenient

integration process and y is "finally found at x0 + h by the quadrature formula

y = yo + h Z M(y<)¿=i

with an error supposed due only to the quadrature formula.

In the cases where Gauss-Legendre quadrature is used it turns out that the

principal error function is

-a"( E «f>r=2»+l

and no coefficient eia exceeds the corresponding e found for the processes studied

in this paper. Although no general rule can be made this suggests that there is a

tendency for the methods of this paper to be the less accurate. On the other hand

the lack of suitable integration formulae of sufficiently high accuracy for estimating

ji (i = 1, 2, • • ■ , c) in the cases c = 3, 4, • • • is a disadvantage of the other method.

5. A Numerical Example. We now illustrate the process with c = 3 using it to

find i/(0.3) where

y = y, 2/(0) = 1.

For this equation all elementary differentials of order 2c + 1 = 7 vanish identically

except F2 so we find a principal error function i//i7/100800.

In the vicinity of y = 1 the value h = .3 gives as an estimate of the truncation

error (.3)7100800 « 22 X 10~10.Using the starting values ga) = g(2) = gl3) = 1 and working to 10Ö, 8 iterations

are required for convergence. The final values are gm = 1.0343925937, </2) =

1.1618293499, gw = 1.3049861726 giving 2/(0.3) = 1.3498588105. This exceedsthe true result, exp(0.3) = 1.3498588076 by 29 X 10~10 in reasonable agreement

with our error estimate.

Appendix. For a non-explicit process, g(1), g<2), • • • , g(F> would normally be

found by an iterative procedure. If gj/0 denotes iterate number N for g(,) then a

possible process would be

(13) g»w = f(yo + h (Z aij gN(i) + g aij g(Jlty .



We define I as the greatest of

| 02i |, | Osi I + I O321, • • • , I drt I + I ay21 + • • • +| ayy-i I,

u as the greatest of

I On I + I Ok I + • • • + I 01,1,1 0221 + I o«i I + • • • + I a* I, ■ ■■ ,\a„\

and a as I + u. For a vector v. = (vi ,v2, • • ■ , vn) we denote by || v || the greatest

of I Vi |, I v21, • • • , I v„ I (any other norm could be used in the discussion which

follows but we choose this one for definiteness). We now prove the following result:

Theorem 14. // f (y) satisfies a Lipschitz condition,

l|f(y)-f(z)ll = £||y-z||

and I A I < 1/ (La) then the equations defining g(1), g(2), • • • , g'"' have a unique

solution and gw(1), g//2>, • • • , gy(,,) defined by (13) fewd io ¿Az's solution as N tends to

infinity.To prove that there is no more than one solution we assume, on the contrary

Mthat there are two and denote these by g(1>, g(2), • • • , gc,) and g(1), g(2>,

so that,g

(14) g(i) = f(yo + AZo.yg(y))y=i

Hence we have

gW - iW

g(i) =f(yo + AZa.yg<;'))3=1

f (yo + A g aij g0)) - f ^y0 + A g an ¿A

ÊMgw-t0))||AIL

Hence

H |Ä|LZ(|ai3|.||g<,',-g°',||)3=1

á I A I La max II g0) - gU) ||.

g(<)-g(i)|| <max||gw-g°')

for all i, a contradiction.

To prove the convergence of the iterative scheme a similar calculation to the

above shows that

II &/° - gJ& || Û I h I L(l max || &/» - g^ || + « max || g^i - g£22 ||)3 y

and hence that


64 J. C. BUTCHER

ii (il (i) m ^ \h\ Lu n (y) (y)max ||gV" - gtf-i || û - ' , , T1 max || gj& - g^i,

y i — 111 •"' i

. , , j. / a — a | A | Ll — u\ h «i (j=|Ä|LVa- i-iail¿ ;mfx||glrii-g"

^ fc max || gj& - ¿& ||

(3)2

where fc = | A | La < 1.From this it easily follows that every element of girM — g^-i has modulus less

than MkN for some fixed M and this ensures the convergence of the sequence

gwM)AT=l)2, •••.Suppose the limit of the sequence is gw. It is trivial to show that

ga), g<2), • • • , g(r) satisfy the equation (14).

Finally it may be noted that the transformation gir-i —> g//° is a "contraction

mapping" [15] and Theorem 14 is a special case of a fundamental theorem on such

mappings in complete metric spaces.

University of Canterbury

Christchurch, New Zealand

1. J. C. Butcher, "Coefficients for the study of Runge-Kutta integration processes,"J. Aust. Math. Soc, v. 3,1963, p. 185-201.

2. C. Rtjnge, "Über die numerische Auflösung von Differential-gleichungen," Math. Ann.,v. 46, 1895, p. 167-178.

3. W. Kutta, "Beitrag zur naherungsweisen Integration von Differentialgleichungen,"Zeit. Math. Physik, v. 46, 1901, p. 435-453.

4. E. J. Ntström, "Über die numerische Integration von Differentialgleichungen," ActaSoc. Sei. Fennicae, v. 50, 1925, p. 1-55.

5. S. Gill, "A process for the step-by-step integration of differential equations in anautomatic digital computing machine," Proc. Cambridge Philos. Soc, v. 47, 1951, p. 96-108.

6. A. Huta, "Une amélioration de la méthode de Runge-Kutta-Nyström pour la résolutionnumérique des équations différentielles du premier ordre," Acta Foc. Nat. Univ. Comenian.Math., v.l. 1956, p. 201-224.

7. A. Huta, "Contribution à la formule de sixième ordre dans la méthode de Runge-Kutta-Nyström," Acta Fac. Nat. Univ. Comenian. Math., v. 2, 1957, p. 21-24.

8. K. S. Kunz, Numerical Analysis, 1st edition, McGraw-Hill, New York, 1957, p. 206.9. R. H. Mekson, "An operational method for the study of integration processes," Pro-

ceedings of Conference on Data Processing and Automatic Computing Machines at WeaponsResearch Establishment, Salisbury, South Australia, 1957, paper No. 110.

10. Z. Kopal, Numerical Analysis, 1st edition, Chapman and Hall, London, 1955, p. 368.11. P. C. Hammer & J. W. Hollingsworth, "Trapezoidal methods of approximating

solutions of differential equations," MTAC v. 9, 1955, p. 92-96.12. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, 1st edition,

John Wiley & Sons, Inc., New York, 1962, p. 131.13. T. Muir, A Treatise on the Theory of Determinants, Dover, New York, 1960, p. 437.14. L. Stoller & D. Morrison, "A method for the numerical integration of ordinary

differential equations," MTAC, v. 12, 1958, p. 269-272.15. A. N. Kolmogorov & S. V. Fomin, Elements of the Theory of Functions and Functional

Analysis, v. 1, Graylock Press, Rochester, 1957, p. 43.


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