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

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.