Krylov Subspace Recycling For Matrix Functions

TL;DR

This paper introduces an augmented Krylov subspace method with subspace recycling for efficiently computing sequences of matrix functions on vectors, especially when matrices are related or change slightly.

Contribution

It develops three practical versions of the method and demonstrates their effectiveness through numerical experiments, focusing on the sign function in lattice QCD.

Findings

Effective for sequences of related matrices and vectors

Improves computational efficiency in matrix function applications

Demonstrated success with various functions and matrices

Abstract

We derive an augmented Krylov subspace method with subspace recycling for computing a sequence of matrix function applications on a set of vectors. The matrix is either fixed or changes as the sequence progresses. We assume consecutive matrices are closely related, but make no assumptions on the relationship between the vectors. We present three versions of the method with different practical implementations. We demonstrate the effectiveness of the method using a range of numerical experiments with a selection of functions and matrices. We primarily focus our attention on the sign function arising in the overlap formalism of lattice QCD.