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 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 important, 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 discretization points. It is mathematically equivalent to an Arnoldi method with an infinite matrix, corresponding to the limit where we have an infinite number of discretization points. We also show an equivalence with the Arnoldi method in an operator setting. It turns out that with an appropriately defined operator over a space equipped with scalar product with respect to which Chebyshev polynomials are orthonormal, the vectors in the Arnoldi iteration can be interpreted as the coefficients in a Chebyshev expansion of a function. The presented method yields the same Hessenberg matrix as the Arnoldi method applied to the operator.