An estimate of approximation of an analytic function of a matrix by a rational function
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Let A be a square complex matrix; z_1, ..., z_N∈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 σ(A) of the matrix A and the points z_1, ..., z_N; and the rational function r=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 ‖ f(A)-r(A)‖≤max_t∈[0,1];μ∈convex hull{z_1,z_2,…,z_N}‖Ω(A)[v(A)]^-1(vf)^(N)((1-t)μ1+tA)/N!‖, where Ω(z)=∏_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.
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