zbMATH — the first resource for mathematics

Two algorithms for computing the matrix cosine function. (English) Zbl 1426.65059
Summary: The computation of matrix trigonometric functions has received remarkable attention in the last decades due to its usefulness in the solution of systems of second order linear differential equations. Several state-of-the-art algorithms have been provided recently for computing these matrix functions. In this work, we present two efficient algorithms based on Taylor series with forward and backward error analysis for computing the matrix cosine. A MATLAB implementation of the algorithms is compared to state-of-the-art algorithms, with excellent performance in both accuracy and cost.

65F60 Numerical computation of matrix exponential and similar matrix functions
15A16 Matrix exponential and similar functions of matrices
Full Text: DOI
[1] Serbin, S., Rational approximations of trigonometric matrices with application to second-order systems of differential equations, Appl. Math. Comput., 5, 1, 75-92, (1979) · Zbl 0408.65047
[2] Serbin, S. M.; Blalock, S. A., An algorithm for computing the matrix cosine, SIAM J. Sci. Stat. Comput., 1, 2, 198-204, (1980) · Zbl 0445.65023
[3] Defez, E.; Sastre, J.; Ibáñez, J. J.; Ruiz, P. A., Computing matrix functions arising in engineering models with orthogonal matrix polynomials, Math. Comput. Model., 57, 7-8, 1738-1743, (2013) · Zbl 1305.65137
[4] Sastre, J.; Ibáñez, J.; Ruiz, P.; Defez, E., Efficient computation of the matrix cosine, Appl. Math. Comput., 219, 7575-7585, (2013) · Zbl 1288.65059
[5] Al-Mohy, A. H.; Higham, N. J.; Relton, S. D., New algorithms for computing the matrix sine and cosine separately or simultaneously, SIAM J. Sci. Comput., 37, 1, A456-A487, (2015) · Zbl 1315.65045
[6] Alonso, P.; Ibáñez, J.; Sastre, J.; Peinado, J.; Defez, E., Efficient and accurate algorithms for computing matrix trigonometric functions, J. Comput. Appl. Math., 309, 325-332, (2017) · Zbl 1416.65126
[7] 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
[8] Tsitouras, C.; Katsikis, V. N., Bounds for variable degree rational L_∞ approximations to the matrix cosine, Comput. Phys. Commun., 185, 11, 2834-2840, (2014) · Zbl 1348.65083
[9] Higham, N. J., Functions of matrices: theory and computation, (2008), SIAM Philadelphia, PA, USA · Zbl 1167.15001
[10] Golub, G. H.; Loan, C. V., Matrix computations, Johns Hopkins Studies in Mathematical Sciences, (1996), The Johns Hopkins University Press
[11] Sastre, J.; Ibáñez, J. J.; Defez, E.; Ruiz, P. A., Efficient scaling-squaring Taylor method for computing matrix exponential, SIAM J. Sci. Comput., 37, 1, A439-A455, (2015) · Zbl 1315.65046
[12] 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
[13] Higham, J.; Tisseur, F., A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., 21, 1185-1201, (2000) · Zbl 0959.65061
[14] T.G. Wright, Eigtool, Version 2.1, 16, March 2009. Available online at: http://www.comlab.ox.ac.uk/pseudospectra/eigtool/.
[15] Higham, N. J., The test matrix toolbox for MATLAB, Numerical Analysis Report No. 237, (1993), The University of Manchester, England
[16] Franco, J. M., New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math., 56, 8, 1040-1053, (2006) · Zbl 1096.65068
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.