Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems
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Numer. Math. (2013) 123:333–362 DOI 10.1007/s00211-012-0489-1
Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems Daniel B. Szyld · Fei Xue
Received: 22 August 2011 / Revised: 10 July 2012 / Published online: 18 September 2012 © Springer-Verlag 2012
Abstract We study the local convergence of several inexact numerical algorithms closely related to Newton’s method for the solution of a simple eigenpair of the general nonlinear eigenvalue problem T (λ)v = 0. We investigate inverse iteration, Rayleigh quotient iteration, residual inverse iteration, and the single-vector Jacobi–Davidson method, analyzing the impact of the tolerances chosen for the approximate solution of the linear systems arising in these algorithms on the order of the local convergence rates. We show that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves. We discuss the connections and emphasize the differences between the standard inexact Newton’s method and these inexact algorithms. When the local symmetry of T (λ) is present, the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of convergence rates. The convergence results are illustrated by numerical experiments. Mathematics Subject Classification (2000)
65F15 · 15A18 · 15A22
This work was supported by the US Department of Energy under Grant DEFG0205ER25672, and the US National Science Foundation under Grant DMS-1115520. D. B. Szyld (B) Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122-6094, USA e-mail: [email protected] F. Xue Department of Mathematics, University of Louisiana at Lafayette, P.O. Box 41010, Lafayette, LA 70504-1010, USA e-mail: [email protected]
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1 Introduction Nonlinear eigenvalue problems (NEPs) of the form T (λ)v = 0 arise naturally in a variety of science and engineering applications, such as the dynamic analysis of structures, the optimization of the acoustic emissions of high speed trains, and the solution of optimal control problems; see, for example, the survey [30] and references therein, as well as, e.g., the recent paper [29]. Among all types of NEPs, the quadratic eigenvalue problem (QEP) is of particular interest and has been studied extensively, due to its wide range of applications; see [39] and some recent references [2,8,16,17, 19,21,27]. For quadratic, polynomial and rational eigenvalue problems, a commonly used approach is to transform the original problem to a linear generalized eigenvalue problem of larger size and with identical spectrum by a linearization [1,15,26,36], while preserving the same matrix structure as T (λ). For general NEPs, variants of Newton’s method constituent a most important class of algorithms for computing a single eigenpair, provided that a good initial eigenpair approximation is available; see [33] for a detailed discussion of a few exact algorithms of
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