The Short-Term Rational Lanczos Method and Applications

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 compared with the classical long-term method. We propose an implementation that allows one to obtain the rational subspace reduced matrices at lower overall computational costs than proposed in the literature by also conveniently combining the two system solves. Several applications are discussed where the short-term recurrence feature can be exploited to avoid storing the whole orthonormal basis. We illustrate the advantages of the proposed procedure with several examples.