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Boosting the computation of the matrix exponential. (English) Zbl 1429.65093
Summary: This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson-Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Padé algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances.

##### MSC:
 65F60 Numerical computation of matrix exponential and similar matrix functions 15A16 Matrix exponential and similar functions of matrices
##### Software:
CONEST; LAPACK; Matlab; MATLAB expm; mftoolbox; SONEST
Full Text:
##### References:
 [1] Higham, N. J., Functions of matrices: Theory and computation, Society for Industrial and Applied Mathematics (2008), Philadelphia, PA, USA · Zbl 1167.15001 [2] 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 [3] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, P. A., Efficient scaling-squaring Taylor method for computing matrix exponential, SIAM J. Sci. Comput., 37, 1 (2015) · Zbl 1315.65046 [4] 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 [5] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, P., Accurate and efficient matrix exponential computation, Int. J. Comput. Math., 91, 1, 97-112 (2014) · Zbl 1291.65139 [6] Ruiz, P.; Sastre, J.; Ibáñez, J.; Defez, E., High performance computing of the matrix exponential, J. Comput. Appl. Math., 291, 370-379 (2016) · Zbl 1329.65092 [7] Sastre, J.; Ibáñez, J.; Alonso, P.; Peinado, J.; Defez, E., Two algorithms for computing the matrix cosine function, Appl. Math. Comput., 312, 66-77 (2017) · Zbl 1426.65059 [8] Defez, E.; Ibáñez, J.; Sastre, J.; Peinado, J.; Alonso, P., A new efficient and accurate spline algorithm for the matrix exponential computation, J. Comput. Appl. Math., 373, 354-365 (2018) · Zbl 1398.65091 [9] Sastre, J., Efficient evaluation of matrix polynomials, Linear Algebra Appl., 539, 229-250 (2018) · Zbl 1432.65029 [10] Blackford, S.; Dongarra, J., Installation Guide for LAPACK, LAPACK, Working Note 41 (1999), Department of Computer Science University of Tennessee [11] 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 [13] Higham, N. J.; Tisseur, F., Fortran codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation, ACM Trans. Math. Softw, 14, 4, 381-396 (2000) [14] 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 [15] 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 [16] Dolan, E. D.; Moré, J. J., Benchmarking optimization software with performance profiles, Math. Program., 91, 201-213 (2002) · Zbl 1049.90004
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