On the sensitivity of Lanczos recursions to the spectrum
type: Item
instance of: scholarly article
title: On the sensitivity of Lanczos recursions to the spectrum
zbMATH Open document ID: 1079.65045
author: V. Druskin
author: L. Borcea
author: Leonid Knizhnerman
published in: Linear Algebra and its Applications
publication date: 2005-02-22T00:00:00Z
review text: This paper is concerned with the inverse eigenvalue problem of finding a symmetric tridiagonal matrix \(T\) having a prespecified set \(\Lambda\) of eigenvalues and a prespecified first row \(e^T S\) of the orthogonal eigenvector matrix \(S\). In particular, the individual sensitivities of the diagonal and (positive) off-diagonal entries of \(T\) with respect to perturbations in \(\Lambda\) and \(S\) are investigated. For this purpose, the matrices \(T\) and \(S\) along with their perturbations are related to each other via a projected Sylvester equation. Linearizing this equation yields a first order perturbation expansion, which is employed to derive relative error bounds for the off-diagonal entries and absolute error bounds for the diagonal entries of \(T\). As an important application of the obtained results, stability estimates for the coefficients produced by the Lanczos process applied to a symmetric matrix are given. The relevance of these estimates is discussed in detail for discrete Sturm-Liouville problems.
reviewed by: Daniel Kressner
Mathematics Subject Classification ID: 65F18
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65L15
zbMATH DE Number: 2136674
zbMATH Keywords: Lanczos process
zbMATH Keywords: tridiagonal matrix
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: discrete Gel'fand-Levitan method
zbMATH Keywords: Sylvester equation
zbMATH Keywords: error bounds
zbMATH Keywords: stability estimates
zbMATH Keywords: discrete Sturm-Liouville problems
MaRDI profile type: Publication
full work available at URL: j.laa.2004.08.031
OpenAlex ID: W2150902005
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DOI: 10.1016/J.LAA.2004.08.031
description: scientific article; zbMATH DE number 2136674