The Short-term Rational Lanczos Method and Applications

TL;DR

This paper introduces a computationally efficient short-term recurrence method for rational Krylov subspaces applicable to symmetric data matrices, reducing storage and computational costs in dimension reduction tasks.

Contribution

It proposes a novel implementation that combines two system solves to lower overall costs and enables basis storage reduction, improving upon previous methods.

Findings

Lower computational costs compared to existing methods

Effective basis storage reduction in applications

Demonstrated advantages through multiple examples

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 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.