A rational Krylov method based on Newton and/or Hermite interpolation for the nonlinear eigenvalue problem
Roel Van Beeumen, Karl Meerbergen, Wim Michiels · Lirias (KU Leuven) · 2012
In this talk we present a new rational Krylov method for solving the nonlinear eigen value problem (NLEP): A(λ)x = 0. The method approximates A(λ) by polynomial Newton and/or Hermite interpolation. It uses a companion-type reformulation to obtain a linear generalized eigenvalue problem (GEP). This GEP is solved by a rational Krylov method, where the number of iteration points is not fixed in advance. As a result, the companion form grows in each iteration. The number of interpolation points is dynamically chosen. Each iteration requires a linear system solve with A(σ) where $\sigma$ is the last interpolation point. We illustrate the method by numerical examples and compare with residual inverse iteration. We also give a number of scenarios where the method performs very well.