From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: matrix equation
Date: 27 Mar 2000 11:37:28 -0500
Newsgroups: sci.math,sci.math.num-analysis
Summary: [missing]

In article <8bnvmv$1u1j7$1@hades.rz.uni-sb.de>,
Uwe Schmitt  <schmitt@karcher.num.uni-sb.de> wrote:
:hi,
:
:i try to solve the matrix-equation  A X B + \lambda X = M
:for some given matrices A, B, M and \lambda \ge 0.
:i'm looking for an efficient algorithm to solve this equation.
:
:yours, uwe.

(Assuming A and B are square of the same dimensions; otherwise adjustments
must be made)

This is a variant of Sylvester equation  F*Y - Y*G = C (unknown Y), for
which an algorithm, together with further references, is offered in

Matrix Computations, by Gene H. Golub and Charles F. Van Loan
(3rd Ed.), Johns Hopkins 1996
ISBN 0-8018-5414-8

(Algorithm 7.6.2, on p. 367)

Their (Bartels's and Stewart's) trick can be adapted: first make sure by a
unitary (or not-too-ill-conditioned invertible, in general) transformation
that A and B are upper triangular: if

C = U^(-1)*A*U  and  D = V*B*V^(-1)

are upper triangular, change the variable to  

X = U*Y*V

and obtain as simpler-looking equation

C*Y*D + lambda*Y = U^(-1)*M*V^(-1)

The triangular structure will allow you to write down a step-by-step
elimination procedure.

Good luck, ZVK(Slavek).