Evaluating Non-Analytic Functions of Matrices

TL;DR

This paper analyzes how the smoothness of a function and the matrix's properties affect the convergence of Chebyshev polynomial expansions used to evaluate matrix functions.

Contribution

It provides new bounds on convergence rates linking function smoothness and matrix diagonalizability, with numerical illustrations.

Findings

Convergence rates depend on function smoothness and matrix diagonalizability.

Derived bounds clarify the relation between function properties and expansion convergence.

Numerical examples validate the theoretical analysis.

Abstract

The paper revisits the classical problem of evaluating $f(A)$ for a real function $f$ and a matrix $A$ with real spectrum. The evaluation is based on expanding $f$ in Chebyshev polynomials, and the focus of the paper is to study the convergence rates of these expansions. In particular, we derive bounds on the convergence rates which reveal the relation between the smoothness of $f$ and the diagonalizability of the matrix A. We present several numerical examples to illustrate our analysis.