New methods for calculating the commencement of a table in the numerical integration of ordinary differential equations

NEW METHODS FOR CALCULATING THE COMMENCEMENT OF A TABLE IN THE NUMERICAL [NTEGRATIONOF ORDINARY DIFFERENTIALEQUATIONS* Yu. V. RAKITSKII Lenningrad (Received 4 October 1966)

FOR the numerical

integration

dzi

-=ff(t,Zi,Z2

,...,

dt

by the convergent

difference

fn-x

=

f (L--X,

the previous Zik =

zi(t,,

. * . , hh

ZZrt,

t,

&l-x),

calculation

+ kh)

zio

(i =

1,2,... ,m)

(0.1)

ib,=i), x=--i

x=--i {Zln,

a(to)=

Gl),

method

z.+~=zn+h&&.( zn =

of the set of equations

=

fn-x

=

to +

nh,

of additional

(k=p,

{fl,

initial

p-i,

n--X, f2, n-x,

* * * I fm, n-x>,

(0.2)

h > 0, values

. . . . 1, -1,

. . . . p-r),

r>p>o.

(0.3) is necessary.

A complete

set of initial

values

together with those given in (0.1)

ZiO (i = 1, 2, . . . ) m) is said to be the commencement The number p in (0.3) indicates buted with respect To calculate instance

how the commencement

to the given initial

initial

values

of the table is distri-

point.

iterative

methods

are most frequently

used, for

the method of A. N. Krylov [l].

The most complete

list of formulae for calculating

table is given in [21.

*

of the table.

Zh. vychisl.

Mat. mat. Fiz. 8, 1, 13 - 27, 1968. 14

the commencement

of the

Numerical integration

All these methods practice

by methods of the Runge-Kutta

of additional

initial

values,

15

equation5

This is one of the reasons

are very laborious.

calculations

the calculation

of ordinary differential

why in

type, which do not require

are preferred to difference

methods.

In fact, if the order of (0.1) is m and the degree of the extrapolation (0.2) used for the solution necessary

to solve

infrequently process

that for sufficiently

we must choose

numerical

is s, to calculate

a set of equations

the commencement

of the m(s-I)-th

order.

large s for the convergence

h less than is necessary

When one considers

of the iterative

for the performance

that in practice

sides

problems

and that it is necessary

of the

(0.1) are often met with disconto calculate

of the table as many times as there are discontinuities, difference

methods

the difficulties

On the other hand it is well known that Runge-Kutta integration.

Therefore,

commencement significance

the commencement of using

are obvious.

mately 4 times more computer time than difference

Theorem

of the table it is It appears not

integration.

tinuous right-hand

formulate

method

the development

of a table,

some results

of new methods

which are both simple

for a wide application whose basis

methods occupy

methods

of difference is contained

approxi-

at each step of the

of calculating

and reliable,

the

has a decisive

methods for integration,

We shall

in the present paper.

A

In the solution

of the set of equations

fourth degree inclusive

(0.1) by Adams’

(the error of the approximate

order not above hs) to calculate

the commencement

methods up to the

solution

on any step has an

of the table we can use the

formula ah = GO + khfi (to, zio, ZP, . . . , zn,,,) (k = 0, -4, -2, -3; i = 1,2,. . . , n).

Theorem

R

In the solution

of the set of equations

(0.1) by Adams’ methods of the fifth

and sixth degree (the error on any step has an order not above h’) the calculation of the initial

values

can be performed by the formula

16

Yu. V. Rakitskii

110+ khfio +

Zfk =

(k =

0, -4,

F(fi+- fi-) + T(fi+- 2fiO+ ft-)

-2,

-3,

-4,

-5;

i=1,2

m),

,...,

where

fro =

fitto,ho, 220, . . . , Zmo),

fi+ = fi (to+ h, GO fi- =

hP,+)

f i (to - h, zio - hPi-, 220- hP~-, . . . , ~~0 -

hP,_)

hPi+,

220 +

Pi+ = fr (to + ;,

a0 +

;fio,

Pi-=ff

Qo-

h2 fro,

(

In using

Note. less

. . . , zmo +

+

initial

value

to-;,

Adams’

220 +

interpolation

for each

variable

hPa+,

ff20 ,...,Gno+~

220 -

$f20

methods

than

,a..

h fmo) 9

9 ZmO -

one needs

for extrapolation

, ,

4

fm0)

-

only to calculate

methods

one

of the same

degree. 1. have

In [g] theorems of problem

by methods

of the table.

on the behaviour

(0.2),

Their

solutions

depending

proofs

are given

of approximate

on the errors

solutions

in calculating

below.

1

Let us make a)

of approximate

been formulated

(O.l), obtained

the commencement

Theorem

Asymptotics

a vector

conditions

the following function

(0.1) written

assumptions:

z(t) of’the in vector

variable

region

exists,

satisfying

form:

z(to) = 5;

$=f(V). b) in the closed

t (m-dimensional)

r[, which

is convex

(4.1)

with respect

to z and defined

by

the inequalities to -

z<1
IzrW

-ztI

GR

(R >

0,

i=

4, 2, . . . . m), (4.2)

Numerical

the given tives

vector

function

with respect c)

integration

f(t,

of ordinary differential

z) is continuous

to all arguments

for numerical

d) (L, zn)

E

integration

II;

and has

up to the third method

the quantity

(0.2)

17

equations

continuous

order

partial

deriva-

z(t_,),

. . . . ,

inclusive;

is used;

r is such

that

~(t_~),

~(t_~),

z(t_,) exist; e)

the supplementary

ZP =

initial

values

satisfy

the conditions

~o+~iP~+~2*~2+~(~3),

fP

=

f&9

zp),

(1.3) fP--X

where f)

a yp, $,,

=

error,

the approximate

is defined

for all n ‘/p

Here

wn(t,) ”

permitted solution

on each of problem

+ 2r by the asymtotic

we(h) +hW(h)

zn =

the vector

(x=

p-Z (y = 0, 1, 2) do not depend

the round-off

Then

do,*-_T+&P-xJc+o(~2)

is equal

to the exact

GL..,r),

on h; step, (l.l),

f’n = O(h4). permitted

(0.2),

h-+0.

+h2wz(tn) +O(h3), solution

by method

expansion

of problem

(l,l),

taken

(1.4)

with

t = L,;

.-

functions

-(&+P)

w,(t,)

and w?(t,)

(n-dimensional)

satisfy

z(

at~~or),,(~o)+[b-,+.,+

A4

r

the equations

&,_,)]X p-1

k-l

Yu. V. Rakitskii

18

x

+

i

Fk

B, = -y and 8, = I/, are Bernoulli

Proof.

We write

sum (0.2)

with

the difference

~f~~or)(fo-fp-k,lk=

(X-k+l)b,

i

k=i

where

1 ]+ h=O

[-kf’(t,,n)-*

numbers. (0.2)

equations

n = p, p + 1, . . . . . 5ince

in summary

the method

(0.2)

form.

For this

converges,

we

we have

yb,+ibp=1. x=R

x=--I

If n &p + r we obtain

the required

formula (1.7)

n-i fp) +

-

h

5

( ‘i

k=i We stress equation

the fact (0.2).

the integral

that

has

equation,

the summary

equation,

one form for the description obtained

in accordance

b, -

derived

(fn-ta -

from the given

of (1.7) with

1)

fp-tt)

*

x=--i

only

difference

if n >/p + r, whereas

(1.1)

z(t)=zo+]f(7,Z)dZ to

has

one form of description

This

is explained

is higher

than

It must the invariance

by the fact

the original

be noted

for all t >/to.

that

that

differential by the choice

of the descriptive

the order

of the difference

equation

(0.2)

equation. of additional

form for the summary

initial

values

equation

we can obtain

for any

n >/p.

Numerical

Ye

shall

put

integration

fp_k= fp (Ii = I,

2,

of ordinary

differential

Then formula (1.7)

. . . , r).

n-l

zn=zp+h~fv+hb-i(fnW=p and is true for any n >cp. approximate

solution

%I =

UO(b,

n)

h $;v

-

case

the expansion

of the

n)

+

h2v2(kt,

only for fairly

n)

+

h-+0,

0 (hs),

(1.8)

large n.

(1.7):

hb_* (fn -

-

h ;f ( g

b, -

‘xc_*

k=i

We choose

flzUi(G17

exists,

Ve now transform zp -

in the general

1) (fn--k-fp)

in powers of h must take the form

and if formula (1.4)

zn -

Therefore

has the form

r lk4 \ &-k=i x=--i

fp)+hr,

19

equations

rtO sufficiently

fP) -

(l-9)

1) (L-k ,

-

fp--k) =

fv.

L$t “Pp

large and such that

(lim nob = 0).

n-r>n0

h-+0

On the left-hand

side of (1.9)

th ere are vector

or equal to no, and on the right-hand The expansion

in powers of h of the left-hand and the right-hand

v = p, p + 1, p + 2, . . . , i.e.

in the form (1.8).

If a formula of the form (1.4) terms of the expansion

(1.9)

exists

formula

than

is found by means

side by using directly

an no is found beginning

we use Euler’s

greater

than nO.

side of (1.9)

do not depend on no.

side of (1.9)

of numbers

side of numbers less

of a formula of the form (1.4)

On the left-hand

functions

(0.2)

with

with which the

Yrc. v. Rakitskii

20

where

B, + 1( p) is the (s + 2f-t.h Reruoalli

are Bernoulli

polynomial,

8,,(r = 1, 2,

l

. . , s + 2)

numbers (R, = -y, R, = &+ B, = 0, R,=-Y30, B, =O, . . . ), ad

blacfaurin’s.4 formula

Considering

the conditions

of h in the expansion

I we equate

of nearem

the coefficients

of iike powers

11.9) and obtain

@%@n) =

z, t

(1.10)

Numerical integration

=Q%,,_,

follows

it

tion

with

of (1.1) 41~0,

21

equations

no2f($0,20)+ &noP(to,20)

+f From(l.lO)

of ordinary differential

that

given

according

BOO

zo.

I z(t,)

by virtue vO(tn,

In addition

to the conditions

(1.3)

n)

of the uniqueness 3 z(c,)

of the soIu-

for any n >,p.

and equation

(O-2), if p < v \< no-

1

we have

=(no--Plfo.

(1.13)

h=O

Therefore

Since the step,

from (1.11)

(1.23)

equation

is valid

beginning

with

(1.5)

follows.

for n0 >,p and we must which

v,(t,,

We cut out the sum on the right-hand p 4 v 4 n0 - 1.

Here

we consider

and the formula

for w,(t,),

have

n) 1 q(t,), side

n - r >~so,

is equal

of (1.12)

the number

of

to p + r.

by using

(0.2)

with

the equality

obtained

from (1.5)

with

tn = to.

For no >,p + r we

find

(1.14)

x Substituting

Since

(1.14)

(1.14)

In concluding la (1.4).

kt(fO-f,_k)]

in (1.12)

is valid

we determine

in the left-hand

* (1. , p + 3r the expression

(1.6).

for n0 >,p + r, then

the proof

Cubstituting

we obtain

“(“,-’ In;o’.

+

v,(t,,

the order side

of (1.7)

n) 2 w*(tn)

for n >p

of the remainder

+ 2r.

term

in formu-

with n = p + 2r, p + 2r + 1,

Yu. V. Rakitskii

22

q&t) + hw1 (L) + h2wL’2(ln) +

Zn =

where

q(r,)

and w,(t,)

satisfy

t = t, and cn = IcIn

with

If condition differentiable,

Substituting

z(t,),

9

f) of Theorem

1 is satisfied

into account =

w,(t,),

is the exact

and the vector

equations

Mjh3

in (0.2)

(1.15)

es, by virtue

z($)

solution

of (1.1)

I

(1.5)

(h-to,

ljMill < M (M is a constant

function

and (1.6),

en,., . e . , cmnL

on taking %+2r+j

where

(1.5)

(1.15)

En,

j=

not depending

we obtain

of the convexity

fit,

z) is

0,

1, . . ..r).

on h as h + 0).

a difference

of the region

z.u,(t,) and the restrictions

function

and (1.6) we find that

equation

for the vector

II, the equations

on the round-off

error

for

permitted

on the

step

(1.16)

(L-a, u+jJ En,

Since on using

by the conditions the well-known

difference obtain

equations

Ildflt,,,,

of the theorem majorizing

h-+0.

formulae

of the form (1.16)([1],

~~_~)/du~_~ll < K = const,

to evaluate pp.

solutions

of linear

362 - 364, or 143, p. 333),

we

the evaluation En =

forn>,p+2r.

This

proves

O(h3),

h-+0,

the theorem.

Theorem 2 If the conditions

of Theorem fp--k)h=cl

are satisfied, Proof.

formula If (1.17)

(1.4)

1 are satisfied -

is valid

are satisfied,

fe

(A? =

and, 1, 2, .

in addition, l

. , r),

for all n >/p + r. then

for n >/no >/p the equalities

the equalities (1.17)

integration

Numerical

of ordinary differential

h-i

T

k-i

T

z(z bx-

k=l

1> (fn-k - h--k) h=O t=t

x=-l

20

=

n

k=i

23

equations

1> V(blJ(ba))--fol

hi

X=--I

and

=

P)x

(no -

wowi(te)+

no@0- 1)

V=p

P(P- 1)

2

0

2

3

fo’.

are valid. The last formula is obtained ‘&J

z q&,

from (1.14)

n) for n >/p and tu,(t,)

with conditions

(1.17).

Therefore

E v2(tn, 02! for n >/p + P. Consequently

we c:in Ijeter::iiric &p+r+j = and then in this case

evaluate En

Theorem

(h+O, j = 0,1,....r),

0(h3)

=

O

(r~ >

(h3)

r, h-+0).

p+

3

If the conditions

of Theorems

&+b_I+

i;( kxl

1 and 2 are satisfied

5kq=o, x=--i

and also the conditions

gq4=Pfol

(1.18)

then u),(tn) E o and if p < n < p c r

+0(M), zn= z(tn) Proof.

If conditions

(1.17)

and (1.18)

h-0.

are observed

(1.S) is transformed

tn

w (b) =

s af (%4

to Consequently Theorem

wl(t,)

= 0 and (1.29)

da

wi (T) &.

is true.

4

If the conditions

of Theorems

1, 2, 3 are satisfied

and the formulae

(1.19) into

Yn. V. Rakitskii

24

I =(P -

dfp--R

Qfo’

dhlmzo hold,

the asymptotic

Proof. taking

expansion

(1.14)

We find the second

into

considerations

(k=

(1.17),

with respect

(1.18),

h=l

(0.2)

it has

Pfd) +

with n = p, p + 1, p + 2, . . . , we are convinced

formula

form for all n hp.

descriptive

0 (hS)

ep+j =

Theorem

h=O

p=-1

in this

a unique

and consequently

(1.7)

i( ~bz-l)[~-(p-k)fo’]} .

+2

that

to h of equation

and (1.20):

=dfn

Substituting

(1.20)

r),

for all n >/p.

is valid

derivative

I,2 ,...,

Therefore

j = 0, 1, . . . , r)

(h-4

E,, = O(h’) if n >p.

5

If a)

for the approximate

degree

solution

of equation

(1.1)

we use

method

(0.2) of

s >, 2:

Ci = B, _1-b-l,, +

c,2+

i

(

yb,,

-

1)

0,

=

k--i x=--i

(1.21)

mx_lbxls r

I

A-l

1

)

(Y- 1)lkEl

(--k)V-1 = 0

(y=2,3,..,,s-i), R, = -‘/I, By (y = 2, 3, . . .) are Bernoulli

where b)

in the closed

with respect partial (S >,

region

II, defined

to z, the vector

derivatives

function

with respect

numbers;

by the inequalities f(t,

z) is continuous

to all arguments

the additional

initial

values

satisfy

and has

up to the order

2);

c)

(1.2) which

the conditions

is convex

continuous

s + 2 inclusive

Numerical integration of ordinary differential

r-i pYhY zo+z - f,,(-) y5i 9

2p =

equations

25

+ asp ha + a,*, p hati + 0 (h6S2),

fP = f (Wp),

r-zhv(p

z

fp-h =

-

k)v

Y!

V=dl

f,,‘y’ + d,-i,

(k=

dimensions)

mate solution

!Jere z(t,) satisfy

d,,

p_-k

for all n >/p + 2r by the asymptotic

‘+ h8w,$n)

is the exact

solution

hs +

0 (h*+i)

are vector constants

error permitted on each step lYn = O(hS

is defined z(L)

+-

&2,...,r),

a

the round-off

2 n, 8 =

h-1

d d SP’ as + I, p’ s- 1, p-k* s, p-k and do not depend on h;

the coefficients

el

p--k

(123

+ h8+iws+t,&n) of (1.1)

for t = t,;

l

(of m-

3), then the approxi-

formula

+ C+P+~),

h-+0.

(1.23)

ws, s(tnI and W, + 1, s(tnI

the equations

(1.25) 8-h

-2

&++i+p k!(s+

S==l

1

-k)!

- (Bi + p) af(;;:O) h-0

w8:8

(h)

+

yu. V. Rakicskii

26

c

x wwvn,

(p

2) -dhr

1

I

1

h=o

s-i fj-

k)

_

dSfp--ti

-

__

Theorem 6 If the conditions

deif

p--k

dhei formula

(1.23)

Theorem

of Theorem

I =

(p

5 are satisfied k)

-

s-if

r”’

and,

in addition,

(k = 1,2,.

. . , r;

the conditions

k+p),

h=O

is valid

when

R > p + ;.

7 of Theorem

If the conditions

6 and the equalities

-

I

B,

dsz,

=

dhs are satisfied,

w S, s + ,($J

we have

Z,l,s+*(tn)

1

)(--Q-1 =

0,

psfy)

Ih=O

q

0, and if p < n < p + r, the formula

=,2(&J

h-to.

+O(hs+‘),

is valid.

Theorem 8 If the conditions

d”fp-A dhs hold,

the asymptotic

Theorems

I

of Theorem

= (p -

7 are satisfied

k)sfO(S)

and the formulae

(k =

1,2,.

. . , r;

k$=P),

h=O

expansion

5 - 8 are proved

(1.23)

is valid

by the method

for all n >,p#

of complete

mathematical

induction.

of ordinary differential

Numerical integration

The basis

of the proof is the method

the approximate

solution

remainder

term of formula

evaluated

in the same

To simplify following

as n + N, which (1.23),

is used

way as in the proofs

supplementary

the dominating

formulae

components

in the proof of Theorem

with the conditions

the very cumbersome

1) for methods

of choosing

27

equations

of Theorems

of Theorems

expressions

of

1.

The

5 - 8, is

1 - 4.

encountered

in the proofs

the

are used:

(0.2) of degree

s )2

the equalities

(_I)‘-I

----

=

i

V k-i

(-klV --

are valid, of powers

2

-=1’

operator

2,...,s),

r

z (X--qv-%

by the expansion

form of the description

.

(v-1,2

..)

s),

in powers

of h of the formula

of method

which

is

(0.2) [S], and the summation

161:

2) when conditions

Corollary

1

(v=

i=o x=-i

and are proved

the inverse

p-‘bx,8

x=-t

(1.22)

are satisfied

1

Let us use the notation

Bl c1,=r+ We introduce

where

1 (z_.4)lk,

the matrix function

E is a unit matrix

can be written

r

k-i

_%.-I’ B(z

(mX m).

U(t) (mXm)

(-k)1-i

which

Then the solution

satisfies

V=s++l). the equation

of equations

(1.24)

and (1.25)

in the form

(I =s,sj-

1).

Yu. V. Rokitskii

28

In this formula

(I = s, s + 1) are determined

~1, s(to)

from (1.24) and (1.23

with tn = to. In [S, 7, g] the expansion with the condition above

the degree

that

of the approximate

the order

of the method 2.

of the error of numerical

solution

in powers

in the calculated integration

The! method of calculating

of h is studied

initial

values

is

used.

the additional

initial values The proposed based

method

on the properties

1 - 8 of the preceding

in Theorems

In accordance c01011aly The

of calculating

the commencement

of the asymptotics

with Theorems

of approximate

section

of this

5 - 8 we have

of the table solutions

is wholly

put forward

paper.

the following

corrollaries.

2 order

of the error

(0.2) of degree additional

initial

ZP-It,

of the approximate

s >,2 is equal values

to the degree

are found

eiN (p -

--%+C

6 -

which

ii) j

j!

j=l

solution

of problem

of the method

satisfy

(1.1)

when n ‘,p

by method + r if

the conditions

(j-i)

fo + qq

(k = 0,1,. , . ,r; p #

O),

or

*;*hj(-

Z-h,8 =

zo+‘T?Jj!

k\j

(j--f)

fo

+0(w)

(k=1,2

,...,

r).

(24

j-4

In formula for integration

(2.1) the number p = 0.

In other words,

by a method

degree

of the s-th

in calculating

we can use methods

initial of degree

values s-

if p f 0, and s - 2 if p = 0. Corollary

3

We cannot cients

b

xc s and the order the method.

construct for which

a method

(0.2)

the calculation

of the error

of degree

s > 2 by the choice

of additional

of the approximate

solution

initial

values

is equal

of coeffi-

is not required

to the degree

of

1,

integration of ordinary differential

Numerical

Corollary

29

equation

4

If p f 0, with what ever accuracy ws + *, &o) Proof.

we calculate

the additional

values

f O. Suppose

that the initial

20 +x CH(P-k)jhj

Zp-k =

j-i

values have been calculated (k=O

j;H’+O(IP+z)

by the formula , 1,,..,r).

il

Then p” rcr6,6(&l)

=

.(6-i)

--Jo

+

($x,6 - 1)

i

k)

s-ij,- _

(p _

(s -

-x=-i

hat

Expanding

(-

k)

~-1~~

I!)

the quantity (p - k)a- ’ by the binom’ la1 theorem and also considering

the conditions

for the coefficients

of the method of the s-th degree (1.21),

we

obtain Us, .s(to) = For w s + 1, a(to) similar devices

&+,,6(t0)

Theorem

=

0.

give

-p

9

We make a series

of assumptions:

a)

of Theorem

the conditions

degree s + 1 = 2a + 4

is satisfied:

5 are satisfied

in the use of method (0.2)

(a = 0, 1, . . .J, i.e. the additional

condition

of

30

Yu. V. Rakitskii

b) p = 0;

c)

the conditions

(2.2) me satisfied,

where

Gj are bounded

constant

vectors

Vyhich are the same for all

k. Then for all ra >/zt the formula

and in addition

the representations

af the approximate

solution

Numerical integration

of ordinary

differential

31

equations

are valid. Proof.

Let us consider

satisfied.

We consider

formula (1.24)

that the Bernoulli

when the conditions

of Theorem 9 are

numbers R 2 a + 3 = 0 (a = 0, 1, 2, . . .)

and obtain

I’

-

k)~+2jWz,(t,,,z)-

(-

X k=f

53 X

(-

k)2j+2

I 5 =

to

since

x=-i

@f(TZ)

az w-taaa+(4 dz,

for the method of the (2~ + 4)-th degree

Therefore

(2.61, we obtain by means of (1.25) the formula (2.4).

On considering (2.5) follow

directly

For methods solution initial

from Theorem

Formulae

5.

(0.2) of degree 2a + 4 the order of the error of the approximate

is equal to the degree of the method for n > r, if in calculating values

2x2 z-h,za+4

=

20 +

z1P” j-0

Proof.

Substituting

sion ftt_k, 4k, Corollary

6

the

we use the formula

(-M&p

(i + 1) 1

(2.1) in the right-hand

2a + 4) we are convinced

(k=1,2

+O(W*)

,...,

r).

(2.7)

side of (1.1) and finding the expan-

that conditions

(2.2) are satisfied.

32

Ys. V. Rokitskii

For methods additional

(0.2) up to the fourth degree

initial

values

inclusive

Qk, 4 =

zo -

khj (a

(k=1,2,

%I)

so that the order of the error of the approximate of the method

for all n >/r. This

j _k = j(t,, -

kh, Z,I-

khjo)

=

statement jo -

satisfies 1.

Notes.

2.

the conditions Theorem

In calculating

use Euler’s Corollary

formula

is easily

may be equal

proved

to the degree

by the expansion

(k=

+ 0 (W

9. case

of Corollary

6.

for methods

of the fourth degree

conditions

(2.2) are not satisfied

since

r),

I,2 ,...,

we cannot here.

7

In the solution it is sufficient

Z-h,6

of Theorem

values

. . . . r),

solution

-$jo2)

A is a particular

initial

polygonal

to find

khfo’ +

-Ygjo+

which

it is sufficient

by the formula

of problem

to calculate

(1.1) by methods

the initial

=so-khjo+q(j+-j-)

values

(0.2) of the fifth and sixth

degree

by the formulae

(k=i,...,

-T(j+-2j0+j-)

r), (2.8)

f+=

f (to + h, 20 + hj( lo +

j-=

f(t.-h,zo-hfjt,--,zo--f~)),

$ ,zo

+ Sfo

of the method

Proof.

(2.9) (2.10)

so that the order of the error of the approximate degree

)) >

solution

may be equal to the

for all n >/r.

We find the expansion

of formula

(2.9)

f+=fo+hfo’+;f,“+;[G+3&jo+3&jo2+

+-a~fos+3&fo~+3+fOfo’+ O3

0')

,-~;~O(~2+2~fo+~fo)]

+L’(h4). 0

of ordinary

Numerical integration

The expansion expansions

0n

of (2.11)

4.

3.

it is not difficult

Theorem

Formulae

5 is a particular

(2.8) - (2.10)

case

=

20 - Wo + -

4

k Jk P(fe+ - fe-) - 6 (k =

of Coroliary

2-Q,

z. -

to be very large because

of Theorem

9.

7.

as follows:

(8 = 0,1,2,)

1, 2, . . . . r);

2-ehf (to - 2-e-q

Hence if 8 = iI formulae (2.8) - (2.10) be chosen

these

that the initial

22e(fe- So + fe_)

fe+ = f (to+ 2,-%, z. + 2-ehf (to+ 2-e-a, fe_ = f (to -

ourselves

the conditions

can be generalized

k% z-k,e,e

Then, on substituting

to convince

f_k = f(tO - kh, Z-k, 6) satisfy

vector functions Votes.

similarly.

33

equations

in (2.8) we determine

basis

the

is obtained

of (2.10)

differential

are obtained.

z. + 2-e--*hfO)), z. - 2-e-*hfo) j. It is obvious

of the discrepancies

caused

that 0 cannot by the round-off

errors. Ry similar for methods

methods

we can construct

formulae for calculating

initial

values

of higher degrees. Translated 6y FT. F. Cleaves

REFERENCES

1.

BEREZIN, 1. S. and ZHIDKOV, Vol. 2, M., Fizmatgiz, 1%0.

F;. P.

2.

TOKMALAEVA,

formulae

differential

S. S.

Ordinate

equations,

Computational

Sb. ‘S~chisl.

methods

for the numerical

(Wetody vychislenii).

integration

Mat.” 5, U., fxu. Rkod.

of first-order

Nauk SSSR

Zi -

1959. 3 I

.

RAKITSK’II, Yu. V. Asymptotic problem

by difference

formulae

methods.

Prik.

for approximate

solutions

Mat. Me&h. 2, 7, 130 -

of the Cauchy 134, 1966.

57,

Yu. V. Rakitskii

4.

MYSOVSKIKA,

I. P.

vychislenii). 5.

RAKITSKII, equations

Fis. 6.

on computational

Yu. V.

Some properties

by one-step

(Lektsii

methods

po metodam

1962.

methods

of solutions of numerical

of sets

of ordinary

integration.

differential

Zh. vy’chisl.

Ma:. mat.

1,6,947- 962.1%1.

GEL’FOND, raznostei).

7.

Lectures

M., Fizmatgiz,

SALIKHOV, problem

A. 0.

N. P. for sets

of finite

The calculus

M., Fizmatgiz,

On polar difference of ordinary

differences

(Ischislenie

konechnykh

1959. methods

differential

for the solution

equations.

of the Cauchy

Zh. vy’chisl. Mat. mat. Fiz.

2, 4, 515 - 528. 1962. 8.

TIKHONOV,

A. N. and GORBUNOV,

of a difference ordinary

1%2.

method

differential

A. D.

for the solution

equations.

Asymptotic

expansions

of the Cauchy

problem

Zh. qchisl.

Mat. mat. Fiz.

of the errors for

sets

of

2, 4, 537 - 548,