On the rational approximation of Markov functions,with applications to the computation of Markovfunctions of Toeplitz matrices

TL;DR

This paper develops new bounds and methods for accurately approximating Markov functions of symmetric Toeplitz matrices using rational interpolants, with practical strategies for high-precision computation.

Contribution

It introduces a new sharp a priori bound for rational interpolation error and analyzes efficient evaluation methods for matrix functions with finite precision considerations.

Findings

New upper bounds for interpolation error on spectral intervals.

Effective rational interpolant representations for high-precision scalar computations.

A novel stopping criterion for optimal rational degree selection in finite precision arithmetic.

Abstract

We investigate the problem of approximating the matrix function $f(A)$ by $r(A)$, with $f$ a Markov function, $r$ a rational interpolant of $f$, and $A$ a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error $1-r/f$ on the spectral interval of $A$. By minimizing this upper bound over all interpolation points, we obtain a new, simple and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant $r$. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating $r(A)$, where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for $r$ following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of $r$ leading to a small error, even in presence of finite precision arithmetic.