The method called Arnoldi is currently a very popular method to solve large-scale eigenvalue problems. The general purpose of this paper is to generalize Arnoldi to the characteristic equation of a delay-differential equation (DDE), here called a delay eigenvalue problem.
The DDE can equivalently be expressed with a linear infinite dimensional operator which eigenvalues are the solutions to the delay eigenvalue problem. A common approach to solve the delay eigenvalue problem is to discretize the operator and compute the eigenvalues of a standard or generalized eigenvalue problem.
We derive a new method by applying Arnoldi to the generalized eigenvalue problem associated with a spectral discretization of the operator. It turns out that the structure of the problem can be heavily exploited and we derive an efficient matrix vector product. More importantly, the structure is such that if the Arnoldi scheme is started in an appropriate way, it is mathematically equivalent to an Arnoldi scheme with an infinite matrix, corresponding to the limit where we have an infinite number of discretization points. The resulting method is a scheme where we expand a subspace, not only in the traditional way done in Arnoldi, but the subspace vectors are also expanded with one block of rows in each iteration. In this way, the number of discretization points increases with the number of Arnoldi iterations such that the number of discretization points does not have to be fixed before the iteration starts.
It turns out that there is a complete equivalence between the presented method and Arnoldi applied in the setting of the infinite dimensional operator. More precisely, we show that if the functions space on which the operator acts is equipped with an appropriate scalar product, the vectors in the Arnoldi iteration can be interpreted as the coefficients in a Chebyshev expansion of a function, and the Hessenberg matrix produced by the presented method and the standard Arnoldi scheme applied to the linear infinite dimensional operator are equal.