On the squared unsymmetric Lanczos method
type: Item
instance of: scholarly article
title: On the squared unsymmetric Lanczos method
zbMATH Open document ID: 0836.65045
DOI: 10.1016/0377-0427(94)90395-6
published in: Journal of Computational and Applied Mathematics
publication date: 1996-04-29T00:00:00Z
review text: This paper deals with linear systems, (1) \(Ax = b\), where \(A\) is a large nonsymmetric \(N \times N\) matrix. The biorthogonal Lanczos method (briefly, BiL) can be used to approximate the eigenvalues of nonsymmetric matrices and to solve systems like (1). The purpose of the present paper is to improve this procedure with respect to its both branches, computing the eigenvalues and solving (1). To serve as such a starting point, BiL offers some advantages. For instance, as compared to the class of Krylov subspace methods, it has a quite modest main memory storage requirement. At the first stage, it turns out that the solution of (1), by the BiL method, can be obtained in two different ways, called briefly BiLQR and BiLMINRES. As his main step the author applies the squaring and restarting processes in terms of the matrix polynomial presentation. The first branch, computing the eigenvalues, constitutes the contents of section 4. This is in fact a review of an own work of the author, published only in a report series before, in 1989. Thus, section 5 entitled ``The restarted squared Lanczos method for linear systems'' contains the actual contribution to this article. The methods BiLQR and BiLMINRES have now got their squared and restarted forms, denoted SBiLQR(m) and SBiLMINRES(m), respectively. The corresponding algorithm falls into cycles, consisting of \(m\) iterations. In the so-called modified SBiLQR method the cycle \(m\) does not remain fixed during the iteration. Section 6 contains numerical tests in terms of two boundary value problems. In one of them the number of the equations corresponding to the system (1) is even 40 000. The author compares the number of the iteration steps needed in the three new methods to that needed in the known GMRES\((m)\) method. The comparison seems to be favorable, modified SBiLQR being the best. In section 7 there are some further conclusions.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F15
zbMATH DE Number: 739572
zbMATH Keywords: squared unsymmetric Lanczos method
zbMATH Keywords: linear systems
zbMATH Keywords: biorthogonal Lanczos method
zbMATH Keywords: eigenvalues
zbMATH Keywords: Krylov subspace methods
zbMATH Keywords: restarting processes
zbMATH Keywords: numerical tests
author: Anthony T. Chronopoulos
reviewed by: Veikko Seppälä
describes a project that uses: CGS
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90395-6
OpenAlex ID: W1964556214
cites work: A generalized conjugate gradient, least square method
cites work: s-Step Iterative Methods for (Non)Symmetric (In)Definite Linear Systems
cites work: Variational Iterative Methods for Nonsymmetric Systems of Linear Equations
cites work: Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method
cites work: Q4091368
cites work: A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems
cites work: QMR: A quasi-minimal residual method for non-Hermitian linear systems
cites work: A Completed Theory of the Unsymmetric Lanczos Process and Related Algorithms, Part I
cites work: An efficient nonsymmetric Lanczos method on parallel vector computers
cites work: Solution of Sparse Indefinite Systems of Linear Equations
cites work: A Look-Ahead Lanczos Algorithm for Unsymmetric Matrices
cites work: The Lanczos Biorthogonalization Algorithm and Other Oblique Projection Methods for Solving Large Unsymmetric Systems
cites work: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
cites work: CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems
cites work: Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems
cites work: A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations
cites work: Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods
Recommended article: Q3140302
Recommended article: Lanczos Methods for the Solution of Nonsymmetric Systems of Linear Equations
Recommended article: An Efficient Implementation of the Nonsymmetric Lanczos Algorithm
Recommended article: Q4356575
Recommended article: The Lanczos algorithm applied to unsymmetric generalized eigenvalue problem
Recommended article: On the Lanczos Method for Solving Symmetric Linear Systems with Several Right-Hand Sides
Recommended article: Extended nonsymmetric global Lanczos method for matrix function approximation
Recommended article: Rayleigh-Ritz and Lanczos methods for symmetric matrix pencils
Recommended article: A generalized nonsymmetric Lanczos procedure
Recommended article: Q3474830
description: scientific article; zbMATH DE number 739572