Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

Recently, two general methods for evaluating matrix polynomials requiring one product less than the Paterson–Stockmeyer method were proposed, where cost of a polynomial is given asymptotically by total number evaluations. An analysis stability those was and have been applied to Taylor-based implementations computing exponential, cosine hyperbolic tangent functions. Moreover, particular example evaluation exponential Taylor approximation degree 15 four products given, whereas maximum available using with 9. Based on this example, new family more efficiently having potential achieve much higher efficiency, i.e., certain degree, or increasing same cost. However, difficulty these lies in calculation coefficients involved approximations. In paper, we provide Paterson-Stockmeyer degrees 30. approximations 21 five evaluations, respectively, are 9 12, respectively. Finally, practical examples logarithm accurately given.