Efficient mixed rational and polynomial approximation of matrix functions

This paper presents an efficient method for computing approximations for general matrix functions based on mixed rational and polynomial approximations. A method to obtain this kind of approximation from rational approximations is given, reaching the highest efficiency when transforming nondiagonal rational approximations with a higher numerator degree than the denominator degree. Then, the proposed mixed rational and polynomial approximation can be successfully applied for matrix functions which have any type of rational approximation, such as Padé, Chebyshev, etc., with maximum efficiency for higher numerator degrees than the denominator degrees. The efficiency of the mixed rational and polynomial approximation is compared with the best existing evaluating schemes for general polynomial and rational approximations, providing greater theoretical accuracy with the same cost in terms of matrix multiplications. It is well known that diagonal rational approximants are generally more accurate than the corresponding nondiagonal rational approximants which have the same computational cost. Using the proposed mixed approximation we show that the above statement is no longer true, and nondiagonal rational approximants are in fact generally more accurate than the corresponding diagonal rational approximants with the same cost.