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

Authors:
Kumar Dookhitram;Ravindra Boojhawon;Ashvin Gopaul;Muddun Bhuruth
Affiliations:
Department of Mathematics, Faculty of Science, University of Mauritius, Reduit, Mauritius;Department of Mathematics, Faculty of Science, University of Mauritius, Reduit, Mauritius;Department of Mathematics, Faculty of Science, University of Mauritius, Reduit, Mauritius;Department of Mathematics, Faculty of Science, University of Mauritius, Reduit, Mauritius
Venue:
Numerical Algorithms
Year:
2011

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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.