Low-memory Krylov subspace methods for optimal rational matrix function approximation

TL;DR

This paper introduces Lanczos-OR, a low-memory, optimal rational matrix function approximation method based on Krylov subspaces, which improves accuracy over prior methods with minimal additional computational cost.

Contribution

The paper presents a novel Lanczos-based algorithm for optimal rational matrix function approximation that is memory-efficient and applicable to various matrix functions.

Findings

Lanczos-OR achieves better approximation quality than standard methods.

The low-memory implementation requires storing only a small number of vectors.

Lanczos-OR can be used for functions like the matrix sign function and rational quadrature approximations.

Abstract

We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.