@article{vanbeeumen-sisc-2013,
abstract = {This paper proposes a new rational Krylov method for solving the nonlinear eigenvalue problem: \$A(\backslash lambda)x = 0\$. The method approximates \$A(\backslash lambda)\$ 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(\backslash sigma)\$, where \$\backslash sigma\$ 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.},
author = {{Van Beeumen}, Roel and Meerbergen, Karl and Michiels, Wim},
doi = {10.1137/120877556},
journal = {SIAM Journal on Scientific Computing},
keywords = {rational Krylov,Newton polynomials,Hermite interpolation,nonlinear eigenvalue problem},
number = {1},
pages = {A327--A350},
title = {A Rational {K}rylov Method Based on {H}ermite Interpolation for Nonlinear Eigenvalue Problems},
url = {http://epubs.siam.org/doi/abs/10.1137/120877556},
volume = {35},
year = {2013}
}