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 $r=\frac{u}{v}$ (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 $$\Vert f(A)-r(A)\Vert \le \underset{t\in [0,1];\mu \in \text{convex hull}\{z_1,z_2,\dots ,z_N\}}{\max }\Vert \Omega (A)[v(A){]}^{-1}\frac{\left(vf{)}^{(N)}\right((1-t)\mu 1+tA)}{N!}\Vert ,$$ 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.

21 pages, 1 figure