Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices

TL;DR

This paper develops bounds for the Arnoldi method's residuals when approximating functions of non-Hermitian matrices, leveraging decay properties of matrix functions, with numerical validation of the bounds.

Contribution

It introduces a priori residual bounds and iteration accuracy strategies for inexact Arnoldi approximations based on decay behavior of matrix functions.

Findings

Derived a priori residual bounds for inexact Arnoldi approximations.

Established decay bounds for functions of banded non-Hermitian matrices.

Numerical experiments confirm the effectiveness of the bounds.

Abstract

We derive a priori residual-type bounds for the Arnoldi approximation of a matrix function and a strategy for setting the iteration accuracies in the inexact Arnoldi approximation of matrix functions. Such results are based on the decay behavior of the entries of functions of banded matrices. Specifically, we will use a priori decay bounds for the entries of functions of banded non-Hermitian matrices by using Faber polynomial series. Numerical experiments illustrate the quality of the results.