Extracting partial canonical structure for large scale eigenvalue problems
type: Item
instance of: scholarly article
title: Extracting partial canonical structure for large scale eigenvalue problems
zbMATH Open document ID: 0957.65025
DOI: 10.1023/A:1019153528915
author: Bo Kågström
author: Petter Wiberg
published in: Numerical Algorithms
publication date: 2000-11-22T00:00:00Z
review text: The paper is motivated by the current large scale computations of eigenvalues which only pay a marginal attention to possible degeneracy of computed eigenvalues. The problem is extremely challenging since the corresponding Jordan canonical form of the matrix \(A\) (or the Weierstrass canonical form of the matrix pencil \(A - \lambda B\) in general) is unstable with respect to the perturbations and/or influence of the input errors. A convenient regularization is needed, using some suitable deflation criteria. The use of the so called Jordan- or Weierstrass-Schur forms for nearby matrices is also recommended in place of the bundles of the strict canonical representations. The paper describes the eligible numerical machinery which, in essence, combines the implicitly restarted Arnoldi method (giving the partial Schur forms within invariant subspace(s)) and a subsequent (final) staircase algorithm. Emphasis is given to a new Gershgorin clustering heuristics (with a reliable control of errors). The reliability and robustness are demonstrated via numerical experiments. The reading of the paper is easy (presenting the encountered difficulties with ill-conditioning etc at an appropriate speed) and rewarding (due to its steady progress towards the clarity and completeness of the discussion).
reviewed by: Miloslav Znojil
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 65F30
zbMATH DE Number: 1534771
zbMATH Keywords: large scale eigenvalue problems
zbMATH Keywords: degenerate eigenvalues
zbMATH Keywords: eigenvalue clustering
zbMATH Keywords: canonical structures
zbMATH Keywords: Jordan form
zbMATH Keywords: Weierstrass form
zbMATH Keywords: implicitly restarted Arnoldi method
zbMATH Keywords: staircase algorithm
zbMATH Keywords: Schur versions
zbMATH Keywords: Gershgorin clustering
zbMATH Keywords: numerical experiments
describes a project that uses: SRRIT
describes a project that uses: JDQR
MaRDI profile type: Publication
full work available at URL: a:1019153528915
OpenAlex ID: W1644302294
description: scientific article; zbMATH DE number 1534771