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New algorithms for computing the matrix sine and cosine separately or simultaneously. (English) Zbl 1315.65045

65F60 Numerical computation of matrix exponential and similar matrix functions
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[1] A. H. Al-Mohy and N. J. Higham, A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970–989. · Zbl 1194.15021
[2] A. H. Al-Mohy and N. J. Higham, Improved inverse scaling and squaring algorithms for the matrix logarithm, SIAM J. Sci. Comput., 34 (2012), pp. C153–C169. · Zbl 1252.15027
[3] A. H. Al-Mohy, N. J. Higham, and S. D. Relton, Computing the Fréchet derivative of the matrix logarithm and estimating the condition number, SIAM J. Sci. Comput., 35 (2013), pp. C394–C410. · Zbl 1362.65051
[4] E. Anderson, Z. Bai, C. H. Bischof, S. Blackford, J. W. Demmel, J. J. Dongarra, J. J. Du Croz, A. Greenbaum, S. J. Hammarling, A. McKenney, and D. C. Sorensen, LAPACK Users’ Guide, 3rd ed., SIAM, Philadelphia, 1999. · Zbl 0934.65030
[5] G. A. Baker, Jr., Essentials of Padé Approximants, Academic Press, New York, 1975.
[6] J. P. Coleman, Rational approximations for the cosine function; P-acceptability and order, Numer. Algorithms, 3 (1992), pp. 143–158. · Zbl 0785.65093
[7] E. B. Davies, Approximate diagonalization, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 1051–1064. · Zbl 1157.65024
[8] E. Deadman and N. J. Higham, Testing matrix function algorithms using identities, MIMS EPrint 2014.13, Manchester Institute for Mathematical Sciences, The University of Manchester, Manchester, UK, 2014; ACM Trans. Math. Software, to appear.
[9] J. Diblík, D. Ya. Khusainov, J. Lukáčová, and M. R\ružičková, Control of oscillating systems with a single delay, Adv. Difference Equ., 2010 (2010), 108218.
[10] N. J. Dingle and N. J. Higham, Reducing the influence of tiny normwise relative errors on performance profiles, ACM Trans. Math. Software, 39 (2013), 24. · Zbl 1298.65086
[11] E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Programming, 91 (2002), pp. 201–213. · Zbl 1049.90004
[12] J. M. Franco, New methods for oscillatory systems based on ARKN methods, Appl. Numer. Math., 56 (2006), pp. 1040–1053. · Zbl 1096.65068
[13] F. R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York, 1959. · Zbl 0085.01001
[14] V. Grimm and M. Hochbruck, Rational approximation to trigonometric operators, BIT, 48 (2008), pp. 215–229. · Zbl 1148.65075
[15] G. I. Hargreaves and N. J. Higham, Efficient algorithms for the matrix cosine and sine, Numer. Algorithms, 40 (2005), pp. 383–400. · Zbl 1084.65039
[16] D. J. Higham and N. J. Higham, MATLAB Guide, 2nd ed., SIAM, Philadelphia, 2005.
[17] N. J. Higham, The Matrix Computation Toolbox, http://www.maths.manchester.ac.uk/ higham/mctoolbox/.
[18] N. J. Higham, The Matrix Function Toolbox, http://www.maths.manchester.ac.uk/ higham/mftoolbox/.
[19] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002. · Zbl 1011.65010
[20] N. J. Higham, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179–1193. · Zbl 1081.65037
[21] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, 2008. · Zbl 1167.15001
[22] N. J. Higham and L. Lin, A Schur–Padé algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl., 32 (2011), pp. 1056–1078. · Zbl 1242.65091
[23] N. J. Higham and L. Lin, An improved Schur–Padé algorithm for fractional powers of a matrix and their Fréchet derivatives, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 1341–1360. · Zbl 1279.65050
[24] N. J. Higham and S. D. Relton, Higher order Fréchet derivatives of matrix functions and the level-2 condition number, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1019–1037. · Zbl 1308.65066
[25] N. J. Higham and M. I. Smith, Computing the matrix cosine, Numer. Algorithms, 34 (2003), pp. 13–26. · Zbl 1033.65027
[26] N. J. Higham and F. Tisseur, A block algorithm for matrix \(1\)-norm estimation, with an application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1185–1201. · Zbl 0959.65061
[27] A. Iserles and S. P. Nørsett, Order Stars, Chapman and Hall, London, 1991.
[28] A. Magnus and J. Wynn, On the Padé table of \(\cos z\), Proc. Amer. Math. Soc., 47 (1975), pp. 361–367. · Zbl 0297.65016
[29] D. L. Michels, G. A. Sobottka, and A. G. Weber, Exponential integrators for stiff elastodynamic problems, ACM Trans. Graph., 33 (2014), 7. · Zbl 1288.68231
[30] M. S. Paterson and L. J. Stockmeyer, On the number of nonscalar multiplications necessary to evaluate polynomials, SIAM J. Comput., 2 (1973), pp. 60–66. · Zbl 0262.65033
[31] J. Sastre, J. Ibán͂ez, P. Ruiz, and E. Defez, Efficient computation of the matrix cosine, Appl. Math. Comput., 219 (2013), pp. 7575–7585. · Zbl 1288.65059
[32] S. M. Serbin, Rational approximations of trigonometric matrices with application to second-order systems of differential equations, Appl. Math. Comput., 5 (1979), pp. 75–92. · Zbl 0408.65047
[33] S. M. Serbin and S. A. Blalock, An algorithm for computing the matrix cosine, SIAM J. Sci. Statist. Comput., 1 (1980), pp. 198–204. · Zbl 0445.65023
[34] J. P. Sharma and R. K. George, Controllability of matrix second order systems: A trigonometric matrix approach, Electron. J. Differential Equations, 2007 (2007), pp. 1–14. · Zbl 1136.93311
[35] L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, 2013. · Zbl 1264.41001
[36] B. Wang, K. Liu, and X. Wu, A Filon-type asymptotic approach to solving highly oscillatory second-order initial value problems, J. Comput. Phys., 243 (2013), pp. 210–223. · Zbl 1349.65219
[37] D. Yuan and W. Kernan, Explicit solutions for exit-only radioactive decay chains, J. Appl. Phys., 101 (2007), 094907.
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