infinite equation system, solution convergence

[Henryk Trappmann]

Given an equation system that extends infinitely to the right and to
the bottom.
a_{1,1} x_1 + a_{1,2} x_2 + ... = b_1
a_{2,1} x_1 + a_{2,2} x_2 + ... = b_2
....

To obtain a solution (x_1,x_2,...) we truncate the equation system to
the first N variables and equations:

a_{1,1} x_{1,N} + ... + a_{1,N} x_{N,N} = b_1
...
a_{N,1} x_{1,N} + ....+ a_{N,N} x_{N,N} = b_N

Suppose each of the truncated equation systems is non-degenerate with
the solution (x_{1,N},...,x_{N,N}).

My question is whether there are established conditions on the
infinite equation system such that the limits
x_n = lim_{N->oo} x_{n,N}
exist and (x_1,x_2,...) is a solution of the original infinite
equation system.

In my special problem the equation system is upper triangular with
positive coefficients a_{m,n}.

[wonj...@yahoo.com.cn]

Is it a system of integral equation if u take that limit?

[Peter Spellucci]

In article <28f57e1a-569b-4995...@t39g2000prh.googlegroups.com>,

you are in the space of infinite sequences and your matrix is a linear operator
working on its elements. take some suitable norm. (this is the problem, practically)
||A|| is the operator norm defined by
sup_{x not=0} ||Ax||/||x||

now, consider your finite system in this infinite dim space., i.e.
you have operators

A(n) = {a{i,j} if 1<=i,j<=n, 0 otherwise }

and elements
x(n) = {x(i) 1<=i<=n , 0 otherwise} and similar for b(n)

then your question is
||x(n)-x||-> 0 for n-> inf in the sense of this norm
a sufficient condition:
if
||A(n)^{-1}|| is uniformly bounded (operator norm)
and
||A^{-1}|| exists
and
||A-A(n)|| ->0 ||b(n)-b||->0 ("the neglected elements become arbitrarily small")
then clearly
||x(n)->x||
poor
||x(n)-x||=||A(n)^{-1}b(n) - A^{-1}b|| =
|| A(n)^{-1}(b(n)-b)+(A(n)^{-1}-A^{-1})b ||<=
C||b(n)-b||+ ||A(n)^{-1}(A-A(n))A^{-1}b|| <=
C||B(n)-b||+ C||A-A(n)||||x||
(in the case of your triangular system the first question is to consider
the size of the diagonal elements and their relative size to the outer
diagonal elements..)

hth
peter

[Henryk Trappmann]

> Is it a system of integral equation if u take that limit?

The infinite equation system can perhaps be considered as the discrete
version of an integral equation. If a(s,t), b(s) are given functions
and x(t) is the searched function then the corresponding integral
equation would be:

integral from 0 to infinity over a(s,t)x(t) dt = b(s)

However I have no knowledge about solving integral equations, but it
looks as if it could help with the original infinite equation system.
Do you have any ideas?

[chrys...@googlemail.com]

On Feb 2, 4:43 pm, spellu...@fb04373.mathematik.tu-darmstadt.de (Peter
Spellucci) wrote:
> In article <28f57e1a-569b-4995-af45-3a72a5e5c...@t39g2000prh.googlegroups.com>,

> peter- Hide quoted text -
>
> - Show quoted text -

One must be more careful. If A(n)->A then A must be what's called a
compact operator. But compact operators do not have continuous
inverses and ||A^(-1)|| is not finite.

[Henryk Trappmann]

> One must be more careful. If A(n)->A then A must be what's called a
> compact operator. But compact operators do not have continuous
> inverses and ||A^(-1)|| is not finite.

Yes, in my particular case there is an infinity of solutions for the
infinite equation system. And I thought I can single out a special
solution by this limit approach. However to be sure I need to know
whether the limit of the solution really exists. Isnt there literature
specifically dealing with infinite matrices, where I could pick up
some criterions for the matrix coefficients?

[Henryk Trappmann]

Maybe some others are also interested so I give a summary what I found
so far.
Basically one can approach the problem via infinite determinants.
Investigations of infinite determinants reach far back to before
1900.
Basically the condition for an infinite determinant of a matrix to
exist is that

the sum over the non-diagonal elements and the product over the
diagonal elements converge absolutely.

There are several modifications of this with using p,1/(p-1)-
exponents. And also other methods and conditions for solving an
infinite equation system. A good summary is given in:
Harold T. Davis, The Theory of Linear Operators, 1936

Selected articles:

Helge von Koch, Sur la convergence des d?eterminants infinis, Palermo
Rend. 28, 265-266 (1909)

St. B?obr, Eine Verallgemeinerung des v. Kochschen Satzes ?uber die
absolute Konvergenz
der unendlichen Determinanten, Math. Zeitschr. 10, 1-11 (1921)

L. W. Cohen, A note on a system of equations with infinitely many
unknowns, Bulletin A. M. S. 36, 563-572 (1930)

T. Le?za?nski, The Fredholm theory of linear equations in Banach
spaces, Stud. Math. 13, 244-276 (1953)