The group you are posting to is a Usenet group. Messages posted to this group will make your email address visible to anyone on the Internet.
Your reply message has not been sent.
Your post was successful
 |
Newsgroups: sci.math
From: Han de Bruijn <Han.deBru...@DTO.TUDelft.NL>
Date: Mon, 13 Nov 2006 10:09:48 +0100
Local: Mon, Nov 13 2006 8:09 pm
Subject: Re: Infinite system of linear equations
Jules wrote: Special cases of such infinite systems of linear equations have been > Suppose we have a doubly-indexed sequence {b_(i, j)} of real numbers, > with i, j positive integers. Suppose also that we have a sequence > {c_i} of reals. We wish to find a sequence {a_j} of reals so that the > sum as j goes from 1 to infinity of b_(i, j) * a_j = c_i for each i. > This is, in some sense, a collection of countably-many linear equations > in countably-many variables. Are there any conditions on the > coefficients b_(i, j) that would guarantee existence and/or uniqueness > of a solution {a_j}? If there were only finitely-many equations and > variables (the same number of each), then one could simply check that > the determinant of the coefficient matrix is non-zero. Is there any > analog of determinant for an "omega-by-omega" matrix? investigated. In "Multigrid Calculus" it is proved that the infinite tri-diagonal system of linear equations: ........ -b 1 -a -b 1 -a -b 1 -a ......... is equivalent with a second order ordinary differential equation: d^2u/dx^2 - P.du/dx + (P^2-Q^2)/4.u = 0 where the matrix coefficients - with infinitesimal grid-spacings (dx) - a = exp(-(P-Q)/2.dx) ; b = exp(+(P+Q)/2.dx) And the general solution is of the form (with A,B arbitrary): u(x) = A.exp((P-Q)/2.x) + B.exp((P+Q)/2.x) The key reference is: http://hdebruijn.soo.dto.tudelft.nl/hdb_spul/calculus.pdf Han de Bruijn You must Sign in before you can post messages.
To post a message you must first join this group.
Please update your nickname on the subscription settings page before posting.
You do not have the permission required to post.
| |||||||||||||