A Krylov Method for the Delay Eigenvalue Problem

Abstract. The Arnoldi method is currently a very popular algorithm to solve large-scale eigenvalue problems. The main goal of this paper is to generalize the Arnoldi method to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem (DEP). The DDE can equivalently be expressed with a linear infinite dimensional operator whose eigenvalues are the solutions to the DEP. We derive a new method by applying the Arnoldi method to the generalized eigenvalue problem (GEP) associated with a spectral discretization of the operator and by exploiting the structure. The result is a scheme where we expand a subspace not only in the traditional way done in the Arnoldi method. The subspace vectors are also expanded with one block of rows in each iteration. More importantly, the structure is such that if the Arnoldi method is started in an appropriate way, it has the (somewhat remarkable) property that it is in a sense independent of the number of discre...

• Type: Journal
• Year: 2010
• Where: SIAMSC
• Authors: Elias Jarlebring, Karl Meerbergen, Wim Michiels
• Keywords: Arnoldi Method, Eigenvalue Problem, Operator, SIAMSC 2010