On the Solution of Linear Equations in Infinitely Many Variables by Successive Approximations

On the Solution of Linear Equations in Infinitely Many Variables by SuccessiveApproximationsAuthor(s): J. L. WalshSource: American Journal of Mathematics, Vol. 42, No. 2 (Apr., 1920), pp. 91-96Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370317 .

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On the Solution of Linear Equations in Infinitely Many Variables by Successive Approximations.*

BY J. L. WALSH.

In this paper we shall consider systems of equations of the type

allox + al2X2 + a13X3 + C1 '1 a2lxl+ a22X2+ a23X3 + C2 L (1

a31xl + a82X2 + a33X3 + * C3 ,

where the aij and c, are given real or complex numbers and the xi are to be determined. Systems of type (1) have been solved by various means,t in- cluding the method of successive approximations, but this method has been used chiefly for Hilbert space [i. e., the space of points {xk} for which oo

I | Xk 12 converges] with corresponding restrictions on the aij and ci.t It is k=1

the object of the present paper to give a number of new conditions under which (1) can be solved by successive approximations; in particular, it is shown that if (1) has a non-vanishing normal determinant and if a simple transformation of the system is made, then the method of successive approximations can be used. The method of successive approximations i-s very convenient for numerical computation.

We shall u-se the, method of successive approximations to prove the

following theorem, which applies to a system of equations of type slightly less general than (1) :

Theorem I. If there exist positive constants C, M, and P such that the

coefficients of the system

x1 + a12X2 + a,13X3 + C1 1

x2+a23x3 + * 2 (2) X3 + * -C3,

satisfy the restrictions

* Presented to the American Mathematical Society (Chicago), Apr. 7, 1917.

t See, e. g., F. Riesz, lquaations Lingaires. t See E. Goldschmidt, Wiurzburg Dissertation (1912).

91



92 WALSH: On the Solution of Linear Equations in

oo 00 ICk I Ck, MCk akhI convergent (k-1, 2, Y), I ak, IP for every

J=k+1 J=k+l

k > K, C < (1/P), C < 1, then (2) has one solution and only one solution for which I xkI yk, y < (1/P), yC 1.

We first consider the special case K 0, and we take for approximations

k1)~ Ck. (k-~1, 2, . . .,l( Xk

(3) X('+" )Ck [akk+jX(I)+ akk+,2x(+)+ l] 2i ...) k kc k+2

From (3) it follows that

X (1) Ck,

X(2) X(1) - [ak k+lX'' + ak k+2X'1' + 1

(4) k k k. 1 k+2 4 x'8)_ X(2)= - [ak, k+1 (X(2)- X' ) ak-, k+2(X2) X(1)) +* k k L +lk+1 k+1 k/ +

and therefore

X-k = -(f) + (x(2) -X1.1) - + (x(3) - (x2) )+ ** (5) k k k k k

< < MCk + PMUk+1 + P2MCk+2 + * *-MCk/ (1 PC). (6)

The xk as defined by (5) are a solution of (2), for if we add all the equations of (4) and sum by columns the resulting absolutely convergent double series, we have

Xk Ck [akk+lXk+l + akk+2Xk+2 +

By (6) we see that for the Xk defined by (5) there exist ju and y such that I Xk | C yy, y < (1/P), y ? 1. Under this restrietion the solution is unique, for if Xrk' and Xk" denote two solutions satisfying this restriction, their difference Xk X- XI:" is a solution of the lhomogeneous system corresponding to (2):

T,1 + a12X2 + a33+ =* 0,

X2 + a23x3 + o= (7)

X3? + *=0,

and we have

Xk ATXk, X<(1/P), X?1. Place

x<1 Xk (k=~1., 25, * , k

x+1 - [akk+lX()+ akk+2X()+ ] (=1, 2,



Infinitely Many Variables by Successive Approximations. 93

from which it follows that j xk(W) I_ P4NXk+4. Hence

lim Xk() = 0. t=00

But we have Xk() Xk(= Xk by equations (7). Hence xk= 0, which proves the uniqueness of the solution and completes the proof of Theorem I for the case K R 0. The reader will easily complete the proof of Theorem I in its generality [K 0] by the use of mathematical induction. In this proof, it will appear that equations (3) and (5) will give xk; for every value of k.

When the Xk defined by (3) and (5) are computed in terms of the as} and ci, it is found that

Xk=Ck- akjcj + 1 Y akJajic- i (8) j=k+l j=k+l i=j+1

which is the so-called Neumann series. The following special case of Theorem I will be used in the sequel:

Theorem II. If for the system (2) we have E I akj I convergent for 00 4. j=k+1

every 7c, Y I ak <P < 1 for every k > K, .f=k+1

I Ck C for every k,

then (2) has one solution and only one solution for which the Xk are bounded. Moreover, this solution is given by the Neumann series (8) .*

We now return to the system of general type (1), and shall proceed to

show that if (1) has a non-vanishing normal determinant and if the Ck are

bounded, then (1) can be transformed into an equivalent system of type (2). This latter system will be shown to satisfy the hypotheses of Theorem II, and therefore the method of successive approximations can be used. To show the possibility of making this transformation we need the

Lemma. In any non-vanishing determinant

all a12 . . . alk

a2l a22 * a2k

A (k) . . . . . X

aki Uk2 . . . aA-k

* Theorem II is similar to a theorem given by von Koch, Jahresbericht, 1913, p. 289.

2



94 WALSH: On the Solution of Linear Equations in

the r-ows can be arranged so that no minor

all a12 . . . alt

a21 a22 . . . a2i

(i=-1, 2, . . k)

ail ai2 . . . ait

will vanish.

This lemma is evidently true (although trivial) for k = 1 and k- = 2; the reader can readily complete the proof by induction.

We have supposed (1) to have a non-vanishing normal determinant; 00

that is, we suppose Y I ai- di, I to be convergent [dij is the Kronecker 4' J=1

symbol whose value is zero or unity according as i #7 j or i j] and

lim (n) = A# 0. n=oo

Then there exists k such that A(n) 7& 0 for n =- k, kc + 1, Hence, by the Lemma, the order of the equations can be changed (if necessary) so that A(n) = 0 for n = 1, 2, . Such rearrangement will not affect the con-

vergence of the double series Y j Ia- j nor the value of A. i, j=1

We suppose, now, that this arrangement has been made, and we proceed to transform (1) into an equivalent system of the type

b11x, + bI2x2 + b13X3 + 1, 1 b22X2+b23X3 + * P2 (9)

b33X3 + * 3 j

This transformation is made by placing * blk alk, 81 cl, and for n >1,

a,, a.2 . . . al, n-I a1k

bn, k . . . . . . . , (k 1, 2, . ) anl an2 . . . an, "-I an, k

all a12 . . . al, n-l C1

an, aCn2 . . . an, n-I Cn

The 8,n as thus defined are bounded for if we set

* Riesz, 1. c., p. 11.



Infinitely Many Variables by Successive Approximations. 95

00 00

AL Y I aik-dik |, II 1 T(1 + Ak), 4=1 k=1

we shall have

f IBn | absolute value of . C, (1) an. an2 * an itl c.

where ck < C (kl , 2, . * ).

Similarly, we slhall prove that E I bi, 1 converges for every i. For by 1=4+1

using a method similar to that used in (10), it follows that for n < k,

I bnk ?lI[ | alk |+ I a2k I + + I ank ]

Hence we have

bn, ?+1|+| bn. ni+2 +*

< II + | bl, a+ , I + 2 + +f | a2 + I +I2, n+2I1 +. + | an, , + I a, n+2 1 +

+ an, .R+1I+Iafl,fl+2I? Then for every value of n, the series of absolute values of the coefficients b,k, k > n, converges; and its sum approaches zero as n becomes infinite.

None of the bn,n is zero and they approach a limit different from zero, so when we wIrite system (9) in the form (2) the hypothesis of Theorem II is satisfied. Therefore we have shown that if (1) has a non-vanishing normal determinant and the ci are bounded, and if (1) is transformed into an equivalent system of type (2), then the method of successive approximations can be used. Incidentally we have proved the theorem of von Koch that if (1) has a non-vanishing normal determinant and if the ci are bounded, then (1) has one solution and only one solution such that the xi are bounded.

We shall now state two theorems, each of which is proved precisely as Theorem I was proved. The Xk of each theorem are given by the Neumann series (8).

Theorem III. If system (2) is such that E I akj VP (p> 1) converges =ki+1

00

for every k, > I ak,j V QP for every k > K, j=k+1

1Ck I -MCk) C <Ti+ 1

then (2) has one solution and only one solution for which



96 WALSH: On the Solution of Linear Equations.

I Xk I _< 9ykj, -Y < (1 + QpI(P1)) (r1)/P

Theorem IV. Under the restrictions J aik I < NTk-4 for every k > i,

I Ck Il MCkX, TC < 1/(1 + N),

system (2) has one solution and only one solution for which

I xk | C5 pyk, T-Y < I/ (1 + N).

MADISON, Wis.

June, 1917.

* A slightly less general theorem for the case p = 2 was proved by von Koch using infinite determinants. See Proc. Camb. Cong. Math. (1912) I, p. 354.

In proving Theorem III there will be found useful the following inequality due to H6lder:

I . akbk |P? (E I ak |P) (Y I bk Jv1)t1. k=1 k=1 k=1

See Riesz, 1. c., p. 45. t Cf. von Koch, 1. c., p. 355.



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On the Solution of Linear Equations in Infinitely Many Variables by Successive Approximations