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.
Categories: math.NA cs.NA math.SP