The Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method: Theory and Algorithms

Details The Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method: Theory and Algorithms The Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method: Theory and Algorithms

2011-09-26

arxiv.org/abs/1109.5455v2

Mathematics

Numerical Analysis

30 pages, 12 figures

Scientific paper

Using a new analysis approach, we establish a general convergence theory of the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple eigenvalue nearest to a target $\sigma$ and/or the associated eigenvector. In SIRA, a subspace expansion vector at each step is obtained by solving a certain inner linear system. We prove that the inexact SIRA method mimic the exact SIRA well provided that the inner linear systems are iteratively solved with {\em low} or {\em modest} accuracy. The results demonstrate that SIRA is superior to the inexact Shift-Invert Arnoldi (SIA) method, where the inner linear system involved must be solved with very high accuracy whenever the approximate eigenpair is of poor accuracy and are only solved with decreasing accuracy after the approximate eigenpair starts converging. Based on the theory, we design a practical stopping criterion for inner solvers. For practical purpose, we propose a restarted SIRA algorithm. Our analysis approach applies to the JD method as well, a general convergence theory is obtained similarly for the standard Jacobi--Davidson (JD) method with a fixed target, and a practical JD algorithm is developed. Numerical experiments confirm our theory and the considerable superiority of the (non-restarted and restarted) inexact SIRA to the inexact SIA, and demonstrate that the inexact SIRA and JD are similarly effective and mimic the exact SIRA very well.

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