Roel Van Beeumen

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:

NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

A(lambda)x = 0

COMPACT RATIONAL KRYLOV METHODS FOR NONLINEAR EIGENVALUE PROBLEMS

. The method approximates

A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems

A(lambda)

Model Reduction by Balanced Truncation of Linear Systems with a Quadratic Output

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

Linearization of Lagrange and Hermite interpolating matrix polynomials

A(sigma)

Determining bound states in a semiconductor device with contacts using a nonlinear eigenvalue solver

, where

Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems

sigma

Compact rational Krylov methods for solving nonlinear eigenvalue problems

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 Newtons method and illustrate that we can achieve an even faster convergence rate.

Fast algorithms for computing the distance to instability of nonlinear eigenvalue problems, with application to time-delay systems

A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear operator and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear operator has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independent of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included.