A comparison of limited-memory Krylov methods for Stieltjes functions of Hermitian matrices

TL;DR

This paper compares various limited-memory Krylov methods for approximating Stieltjes functions of large Hermitian matrices, providing new error bounds and performance predictions to guide method selection.

Contribution

It introduces a systematic comparison of polynomial Krylov methods for matrix functions, including new error bounds and performance estimates, and extends analysis to inexact Krylov iterations.

Findings

Performance ranking of Krylov methods based on new error bounds

Derivation of new results on inexact Krylov iterations

Guidelines for method selection based on accuracy and memory constraints

Abstract

Given a limited amount of memory and a target accuracy, we propose and compare several polynomial Krylov methods for the approximation of f(A)b, the action of a Stieltjes matrix function of a large Hermitian matrix on a vector. Using new error bounds and estimates, as well as existing results, we derive predictions of the practical performance of the methods, and rank them accordingly. As by-products, we derive new results on inexact Krylov iterations for matrix functions in order to allow for a fair comparison of rational Krylov methods with polynomial inner solves.