Fast calculation of Laurent expansions for matrix inverses

Previously described algorithms for calculating the Laurent expansion of the inverse of a matrix-valued analytic function become impractical already for singularity orders as low as around p=6, since they require over O(28p) matrix multiplications and correspondingly large amounts of memory. In place of using mathematically exact recursions, we show that, for floating point calculations, a rational approximation approach can avoid this cost barrier without any significant loss in accuracy.