Author:

Keywords:

Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, rational Krylov, Newton polynomials, Hermite interpolation, nonlinear eigenvalue problem, DAVIDSON ITERATION METHOD, MATRIX POLYNOMIALS, ALGORITHM, LINEARIZATION, KUL-CoE-Optec, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, 0802 Computation Theory and Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics, 4903 Numerical and computational mathematics

Abstract:

This paper proposes a new rational Krylov method for solving the nonlinear eigenvalue problem: A(λ)x = 0. The method approximates A(λ) by Hermite interpolation where the degree of the interpolating polynomial and the interpolation points are not fixed in advance. It uses a companion-type reformulation to obtain a linear generalized eigenvalue problem (GEP). To this GEP we apply a rational Krylov method that preserves the structure. The companion form grows in each iteration and the interpolation points are dynamically chosen. Each iteration requires a linear system solve with A(σ), where σ is the last interpolation point. The method is illustrated by small- and large-scale numerical examples. In particular, we illustrate that the method is fully dynamic and can be used as a global search method as well as a local refinement method. In the last case, we compare the method to Newton's method and illustrate that we can achieve an even faster convergence rate. © 2013 Society for Industrial and Applied Mathematics.