Schur Decomposition Methods for the Computation of Rational Matrix Functions

In this work we consider the problem to compute the vector y=Φm,n(A)x where Φm,n(z) is a rational function, 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 j I)y = b.