Poèsie : émotion ou rigueur?

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2002; 9:163–179 (DOI: 10.1002/nla.266)

The convergence of Jacobi–Davidson iterations for Hermitian eigenproblems
Jasper van den Eshof ∗
Department of Mathematics; Utrecht University; P.O. Box 80.010; NL-3508 TA Utrecht; The Netherlands

SUMMARY Rayleigh quotient iteration is an iterative method with some attractive convergence properties for ÿnding (interior) eigenvalues of large sparse Hermitian matrices. However, the method requires the accurate (and, hence, often expensive) solution of a linear system in every iteration step. Unfortunately, replacing the exact solution with a cheaper approximation may destroy the convergence. The (Jacobi-)Davidson correction equation can be seen as a solution for this problem. In this paper we deduce quantitative results to support this viewpoint and we relate it to other methods. This should make some of the experimental observations in practice more quantitative in the Hermitian case. Asymptotic convergence bounds are given for ÿxed preconditioners and for the special case if the correction equation is solved with some ÿxed relative residual precision. A dynamic tolerance is proposed and some numerical illustration is presented. Copyright ? 2002 John Wiley & Sons, Ltd.
KEY WORDS:

Hermitian matrices; eigenvalue problem; Jacobi–Davidson; Davidson’s method; inexact inverse iteration; convergence rate

0. INTRODUCTION We are interested in methods to calculate an eigenvalue and its associated eigenvector x, possibly in the interior of the spectrum, of a large sparse Hermitian matrix A: Ax = x Iterative methods that employ a shift-and-invert approach, often are successful methods for ÿnding such eigenpairs ( ; x), examples of these methods include the shift-and-invert power method (inverse iteration) and Rayleigh quotient iteration (RQI) (see, for example [1, 2]). This latter method has very nice local convergence properties, but practical

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