Treffer: Schur Decomposition Methods for the Computation of Rational Matrix Functions.

In this work we consider the problem to compute the vector $\vec{y}=\Phi_{m,n}(A)\vec{x}$ where Φm,n(z) is a rational function, $\vec{x}$ is a vector and A is a matrix of order N, usually nonsymmetric. The problem arises when we need to compute the matrix function f(A), being f(z) a complex analytic function and Φm,n(z) a rational approximation of f. Hence Φm,n(A) is a approximation for f(A) cheaper to compute. We consider the problem to compute first the Schur decomposition of A then the matrix rational function exploting the partial fractions expansion. In this case it is necessary to solve a sequence of linear systems with the shifted coefficient matrix $(A-z_jI)\vec{y}=\vec{b}$. [ABSTRACT FROM AUTHOR]