Incomplete partial fractions for parallel evaluation of rational matrix functions

Frequently, one needs to evaluate expressions of the form [p(A)]−1q(A)b, where A ∈ RN × N, b ∈ RN, and p and q are polynomials with degree q ⩽ degree p, and such that no zero of p is an eigenvalue of A. Algorithms based on the partial fraction representation of qp when evaluating [p(A)]−1 q(A)b lend themselves well to implementation on a parallel computer, but might yield poor accuracy. We discuss how to determine an incomplete partial fraction representation of qp which allows parallel computation, while retaining high accuracy.