Inversion of analytic matrix functions that are singular at the origin

In this paper we study the inversion of an analytic matrix valued function A (z). This problem can also be viewed as an analytic perturbation of the matrix A(0) = A(0). We are mainly interested in the case where A(0) is singular but A (z) has an inverse in some punctured disc around z = 0. It is known that A(-1) (z) can be expanded as a Laurent series at the origin. The main purpose of this paper is to provide efficient computational procedures for the coefficients of this series. We demonstrate that the proposed algorithms are computationally superior to symbolic algebra when the order of the pole is small.