An estimate of approximation of an analytic function of a matrix by a rational function

TL;DR

This paper provides estimates for approximating an analytic matrix function by a rational function interpolant, with applications demonstrated through impulse response approximation in dynamic systems.

Contribution

It introduces new bounds for the approximation error of analytic functions of matrices by rational interpolants, extending previous results with explicit estimates.

Findings

Derived explicit error bounds for rational approximation of matrix functions.

Applied the estimates to impulse response approximation in reduced-order systems.

Validated the bounds through numerical examples comparing actual and estimated errors.

Abstract

Let $A$ be a square complex matrix; $z_1$, ..., $z_{N}\in\mathbb C$ be arbitrary (possibly repetitive) points of interpolation; $f$ be an analytic function defined on a neighborhood of the convex hull of the union of the spectrum $\sigma(A)$ of the matrix $A$ and the points $z_1$, ..., $z_{N}$; and the rational function $r=\frac uv$ (with the degree of the numerator $u$ less than $N$) interpolates $f$ at these points (counted according to their multiplicities). Under these assumptions estimates of the kind $$ \bigl\Vert f(A)-r(A)\bigr\Vert\le \max_{t\in[0,1];\mu\in\text{convex hull}\{z_1,z_{2},\dots,z_{N}\}}\biggl\Vert\Omega(A)[v(A)]^{-1} \frac{\bigl(vf\bigr)^{{(N)}} \bigl((1-t)\mu\mathbf1+tA\bigr)}{N!}\biggr\Vert, $$ where $\Omega(z)=\prod_{k=1}^N(z-z_k)$, are proposed. As an example illustrating the accuracy of such estimates, an approximation of the impulse response of a dynamic system obtained using the reduced-order Arnoldi method is considered, the actual accuracy of the approximation is compared with the estimate based on this paper.