Towards backward perturbation bounds for approximate dual Krylov subspaces
type: Item
instance of: scholarly article
title: Towards backward perturbation bounds for approximate dual Krylov subspaces
zbMATH Open document ID: 1278.65034
author: Gang Wu
author: Lu Zhang
author: Yimin Wei
author: Zhigang Jia
author: Sitao Ling
published in: BIT
publication date: 2013-04-03T00:00:00Z
review text: The authors recall Arnoldi, Lanczos and Krylov decompositions. Under the assumption that for two given subspaces \(\mathcal{K}\) and \(\mathcal{L}\) a biorthonormal pair of bases \(V_m\) and \(W_m\) always exists, they define the dual Krylov subspaces with respect to the given matrix \(A\) and the dual Krylov decomposition with respect to \(A\) as a generalization of the Krylov decomposition. If \(A\) is an extended tridiagonal matrix, the dual Krylov decomposition is called ``the two-sided Lanczos decomposition'' -- a generalization of the Lanczos decomposition. The authors consider a pair of biorthonormal bases \(V_m\) and \(W_m\) and determine a backward perturbation matrix \(E\) whose norm is as small as possible, such that \(\mathcal{K}\) and \(\mathcal{L}\) are Krylov subspaces of \(A+E\) and its adjoint, respectively. A framework for determining the perturbation matrix \(E\) is given in Algorithm 2.1. The numerical experiments demonstrate the efficiency of the presented strategy for determining the backward perturbation.
reviewed by: Drahoslava Janovská
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 6150059
zbMATH Keywords: perturbation matrix
zbMATH Keywords: Krylov subspace method
zbMATH Keywords: dual Krylov subspaces
zbMATH Keywords: dual Krylov decomposition
zbMATH Keywords: two-sided Lanczos decomposition
zbMATH Keywords: Arnolid decomposition
zbMATH Keywords: numerical experiment
zbMATH Keywords: backward perturbation
describes a project that uses: JDQR
describes a project that uses: JDQZ
MaRDI profile type: Publication
full work available at URL: s10543-012-0402-4
OpenAlex ID: W1987535943
cites work: Interpolatory Model Reduction of Large-Scale Dynamical Systems
cites work: Error Analysis of the Lanczos Algorithm for the Nonsymmetric Eigenvalue Problem
cites work: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems
cites work: Templates for the Solution of Algebraic Eigenvalue Problems
cites work: On the QMR approach for iterative methods including coupled three-term recurrences for solving nonsymmetric linear systems
cites work: Q4091368
cites work: QMR: A quasi-minimal residual method for non-Hermitian linear systems
cites work: Q5689624
cites work: Residual Bounds on Approximate Eigensystems of Nonnormal Matrices
cites work: The Lanczos and Conjugate Gradient Algorithms
cites work: Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix
cites work: Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem
cites work: Q4414854
cites work: Numerical Methods for Large Eigenvalue Problems
cites work: Recent computational developments in Krylov subspace methods for linear systems
cites work: A Krylov--Schur algorithm for large eigenproblems
cites work: Matrix algorithms. Vol. 2: Eigensystems
cites work: Backword error bounds for approximate Krylov subspaces
cites work: Model reduction using the Vorobyev moment problem
cites work: On Efficient Numerical Approximation of the Bilinear Form $c^*A^{-1}b$
DOI: 10.1007/S10543-012-0402-4
description: scientific article; zbMATH DE number 6150059