NDLI: A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Content Provider	Springer Nature Link
Author	Gopaul, Ashvin Bhuruth, Muddun Dookhitram, Kumar Boojhawon, Ravindra
Copyright Year	2010
Abstract	Convergence of the implicitly restarted Arnoldi (IRA) method for nonsymmetric eigenvalue problems has often been studied by deriving bounds for the angle between a desired eigenvector and the Krylov projection subspace. Bounds for residual norms of approximate eigenvectors have been less studied and this paper derives a new a-posteriori residual bound for nonsymmetric matrices with simple eigenvalues. The residual vector is shown to be a linear combination of exact eigenvectors and a residual bound is obtained as the sum of the magnitudes of the coefficients of the eigenvectors. We numerically illustrate that the convergence of the residual norm to zero is governed by a scalar term, namely the last element of the wanted eigenvector of the projected matrix. Both cases of convergence and non-convergence are illustrated and this validates our theoretical results. We derive an analogous result for implicitly restarted refined Arnoldi (IRRA) and for this algorithm, we numerically illustrate that convergence is governed by two scalar terms appearing in the linear combination which drives the residual norm to zero. We provide a set of numerical results that validate the residual bounds for both variants of Arnoldi methods.
Ending Page	495
Page Count	15
Starting Page	481
File Format	PDF
ISSN	10171398
e-ISSN	15729265
Journal	Numerical Algorithms
Issue Number	4
Volume Number	56
Language	English
Publisher	Springer US
Publisher Date	2010-06-26
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Eigenvalues, singular values, and eigenvectors Nonsymmetric eigenvalue problems Algorithms Algebra Approximation by polynomials Numeric Computing Implicitly restarted Arnoldi Eigenvalues, eigenvectors Residual bounds Theory of Computation Mathematics Refined Arnoldi
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Convergence of Arnoldi’s method for generalized eigenvalue problems

Implicitly Restarted Generalized Second-order Arnoldi Type Algorithms for the Quadratic Eigenvalue Problem

Restarted block Lanczos bidiagonalization methods

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

The Use of Refined Harmonic Shifts in the Implicitly Restarted Refined Harmonic Arnoldi Method

H-matrix theory vs. eigenvalue localization

Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

Fast Inexact Implicitly Restarted Arnoldi Method for Generalized Eigenvalue Problems with Spectral Transformation

Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems

Similar Documents

Convergence of Arnoldi’s method for generalized eigenvalue problems

Implicitly Restarted Generalized Second-order Arnoldi Type Algorithms for the Quadratic Eigenvalue Problem

Restarted block Lanczos bidiagonalization methods

Restarted generalized Krylov subspace methods for solving large-scale polynomial eigenvalue problems

The Use of Refined Harmonic Shifts in the Implicitly Restarted Refined Harmonic Arnoldi Method

H-matrix theory vs. eigenvalue localization

Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems

Fast Inexact Implicitly Restarted Arnoldi Method for Generalized Eigenvalue Problems with Spectral Transformation

Error bounds of Rayleigh–Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems

A-posteriori residual bounds for Arnoldi’s methods for nonsymmetric eigenvalue problems