Optimality of the Paterson-Stockmeyer method for evaluating matrix polynomials and rational matrix functions

Many state-of-the-art algorithms reduce the computation of transcendental matrix functions to the evaluation of polynomial or rational approximants at a matrix argument. This task can be accomplished efficiently by resorting to the Paterson–Stockmeyer method, an evaluation scheme originally developed for matrix polynomials that extends quite naturally to rational functions. An important feature of this technique is that the number of matrix multiplications required to evaluate an approximant of order n grows slower than n itself, with the result that different approximants yield the same asymptotic computational cost. We analyze the number of matrix multiplications required by the Paterson–Stockmeyer method and by two widely used generalizations, one for evaluating diagonal Padé approximants of general functions and one specifically tailored to those of the exponential. In all three cases, we identify the approximants of maximum order for any given computational cost