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