The Short-term Rational Lanczos Method and Applications

Abstract

Rational Krylov subspaces have become a reference tool in dimension reduction procedures for several application problems. When data matrices are symmetric, a short-term recurrence can be used to generate an associated orthonormal basis. In the past this procedure was abandoned because it requires twice the number of linear system solves per iteration than with the classical long-term method. We propose an implementation that allows one to obtain key rational subspace matrices without explicitly storing the whole orthonormal basis, with a moderate computational overhead associated with sparse system solves. Several applications are discussed to illustrate the advantages of the proposed procedure.

Computation of generalized matrix functions with rational Krylov methods

“…This phenomenon has been studied for the polynomial Lanczos case in [22]. A brief study of the problem for the rational Lanczos case can be found in [23].…”

State-dependent Riccati equation feedback stabilization for nonlinear PDEs

“…The latter feature is particularly important for large scale problems, for which dealing with the orthogonal approximation basis represents one of the major computational and memory costs. To the best of our knowledge, this variant of the Riccati solver is new, while it is currently explored in [40] for related control problems and the rational Krylov space.…”