Journal ArticleDOI
Solution of the matrix equation AX + XB = C [F4]
R. H. Bartels,G. W. Stewart +1 more
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The algorithm is supplied as one file of BCD 80 character card images at 556 B.P.I., even parity, on seven ~rack tape, and the user sends a small tape (wt. less than 1 lb.) the algorithm will be copied on it and returned to him at a charge of $10.O0 (U.S.and Canada) or $18.00 (elsewhere).Abstract:
and Canada) or $18.00 (elsewhere). If the user sends a small tape (wt. less than 1 lb.) the algorithm will be copied on it and returned to him at a charge of $10.O0 (U.S. only). All orders are to be prepaid with checks payable to ACM Algorithms. The algorithm is re corded as one file of BCD 80 character card images at 556 B.P.I., even parity, on seven ~rack tape. We will supply the algorithm at a density of 800 B.P.I. if requested. The cards for the algorithm are sequenced starting at 10 and incremented by 10. The sequence number is right justified in colums 80. Although we will make every attempt to insure that the algorithm conforms to the description printed here, we cannot guarantee it, nor can we guarantee that the algorithm is correct.-L.D.F. Descdption The following programs are a collection of Fortran IV sub-routines to solve the matrix equation AX-.}-XB = C (1) where A, B, and C are real matrices of dimensions m X m, n X n, and m X n, respectively. Additional subroutines permit the efficient solution of the equation ArX + xa = C, (2) where C is symmetric. Equation (1) has applications to the direct solution of discrete Poisson equations [2]. It is well known that (1) has a unique solution if and only if the One proof of the result amounts to constructing the solution from complete systems of eigenvalues and eigenvectors of A and B, when they exist. This technique has been proposed as a computational method (e.g. see [1 ]); however, it is unstable when the eigensystem is ill conditioned. The method proposed here is based on the Schur reduction to triangular form by orthogonal similarity transformations. Equation (1) is solved as follows. The matrix A is reduced to lower real Schur form A' by an orthogonal similarity transformation U; that is A is reduced to the real, block lower triangular form.read more
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References
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Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
The Direct Solution of the Discrete Poisson Equation on a Rectangle
TL;DR: In this article, the authors provide a survey of direct methods for solving finite difference equations with rectilinear domains. But the authors do not discuss whether the methods are easily adaptable to more general regions, and to general elliptic partial differential equations.
Journal ArticleDOI
TheQ R algorithm for real hessenberg matrices
TL;DR: The volume of work involved in a QR step is far less if the matrix is of Hessenberg form, and since there are several stable ways of reducing a general matrix to this form, the QR algorithm is invariably used after such a reduction.
Journal ArticleDOI
Matrix and other direct methods for the solution of systems of linear difference equations
W. G. Bickley,J. McNamee +1 more
TL;DR: In this paper, the problem of boundary problems arising in the approximate solutions of linear PDEs was investigated. But the work was conducted on desk machines, and the operations involved are, however, capable of being handled efficiently and simply by modern high-speed digital computers.