## Efficient mixed rational and polynomial approximation of matrix functions.(English)Zbl 1283.65047

Summary: 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.

### MSC:

 65F60 Numerical computation of matrix exponential and similar matrix functions 41A10 Approximation by polynomials 41A20 Approximation by rational functions

### Software:

mftoolbox; MATLAB expm; LAPACK
Full Text:

### References:

 [2] Golub, G. H.; Loan, C. V., Matrix Computations. Matrix Computations, Johns Hopkins Studies in Math. Sci. (1996), The Johns Hopkins University Press · Zbl 0865.65009 [3] Paterson, M. S.; Stockmeyer, L. J., On the number of nonscalar multiplications necessary to evaluate polynomials, SIAM J. Comput., 2, 1, 60-66 (1973) · Zbl 0262.65033 [4] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, P., Efficient orthogonal matrix polynomial based method for computing matrix exponential, Appl. Math. Comput., 217, 14, 6451-6463 (2011) · Zbl 1211.65052 [5] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, P., Accurate matrix exponential computation to solve coupled differential models in engineering, Math. Comput. Model., 54, 1835-1840 (2011) · Zbl 1235.65042 [6] Al-Mohy, A. H.; Higham, N. J., A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl., 31, 3, 970-989 (2009) · Zbl 1194.15021 [7] Defez, E.; Sastre, J.; Ibáñez, J.; Ruiz, P., Computing matrix functions solving coupled differential models, Math. Comput. Model., 50, 5-6, 831-839 (2009) · Zbl 1185.65078 [8] Defez, E.; Jódar, L., Some applications of Hermite matrix polynomials series expansions, J. Comput. Appl. Math., 99, 105-117 (1998) · Zbl 0929.33006 [9] Moler, C. B.; Loan, C. V., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45, 3-49 (2003) · Zbl 1030.65029 [11] Higham, N. J., The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26, 4, 1179-1193 (2005) · Zbl 1081.65037 [12] Calvetti, D.; Gallopoulos, E.; Reichel, L., Incomplete partial fractions for parallel evaluation of rational matrix functions, J. Comput. Appl. Math., 59, 349-380 (1995) · Zbl 0839.65054 [13] Calvetti, D.; Reichel, L., On the evaluation of polynomial coefficients, Numer. Alg., 33, 153-161 (2003) · Zbl 1035.65156
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.