A Rational Krylov Method Based on Hermite Interpolation for Nonlinear Eigenvalue Problems

This paper proposes a new rational Krylov method for solving the nonlinear eigenvalue problem: $A(\lambda)x = 0$. The method approximates $A(\lambda)$ 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 $A(\sigma)$, where $\sigma$ 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.