Inversion of analytic matrix functions that are singular at the origin (Q2719192)

Let \(z\mapsto A(z)= A_0+z A_1+\cdots\) be an analytic \(n\times n\) matrix valued function, defined in a neighbourhood of the origin. It is supposed that \(A_0\) is singular but \(A(z)\) is not for small \(z\neq 0\). Then there exists an integer \(s\geq 1\) such that \(A^{-1}(z)=z^{-s} (X_0+zX_1+\cdots)\). The authors discuss three computational procedures for determining the matrix coefficients \(X_k\), \(k=0,1, \dots\). A comparison is made with algorithms based on symbolic algebra.