zbMATH — the first resource for mathematics

On-the-fly backward error estimate for matrix exponential approximation by Taylor algorithm. (English) Zbl 06980367
Summary: In this paper we show that it is possible to estimate the backward error for the approximation of the matrix exponential on-the-fly, without the need to precompute in high precision quantities related to specific accuracies. In this way, the scaling parameter \(s\) and the degree \(m\) of the truncated Taylor series (the underlying method) are adapted to the input matrix and not to a class of matrices sharing some spectral properties, such as with the classical backward error analysis. The result is a very flexible method which can approximate the matrix exponential at any desired accuracy. Moreover, the risk of overscaling, that is the possibility to select a value \(s\) larger than necessary, is mitigated by a new choice as the sum of two powers of two. Finally, several numerical experiments in MATLAB with data in double and variable precision and with different tolerances confirm that the method is accurate and often faster than available good alternatives.

65 Numerical analysis
62 Statistics
Full Text: DOI
[1] Moler, C. B.; Van Loan, C. F., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45, 1, 3-49, (2003) · Zbl 1030.65029
[2] 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
[3] 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
[4] Higham, N. J., Functions of matrices, (2008), SIAM Philadelphia
[5] Ruiz, R.; Sastre, J.; Ibáñez, J.; Defez, E., High performance computing of the matrix exponential, J. Comput. Appl. Math., 291, 1, 370-379, (2016) · Zbl 1329.65092
[6] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, R., Accurate matrix exponential computation to solve coupled differential models in engineering, Math. Comput. Modelling, 54, 1835-1840, (2011) · Zbl 1235.65042
[7] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, R., Efficient orthogonal matrix polynomial based method for computing matrix exponential, Appl. Math. Comput., 217, 6451-6463, (2011) · Zbl 1211.65052
[8] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, R., New scaling-squaring Taylor algorithms for computing the matrix exponential, SIAM J. Sci. Comput., 37, 1, A439-A455, (2015) · Zbl 1315.65046
[9] Sastre, J.; Ibáñez, J.; Ruiz, R.; Defez, E., Accurate and efficient matrix exponential computation, Int. J. Comput. Math., 91, 1, 97-112, (2014) · Zbl 1291.65139
[10] Al-Mohy, A. H.; Higham, N. J., Computing the action of the matrix exponential with an application to exponential integrators, SIAM J. Sci. Comput., 33, 2, 488-511, (2011) · Zbl 1234.65028
[11] Caliari, M.; Kandolf, P.; Ostermann, A.; Rainer, S., The Leja method revisited: backward error analysis for the matrix exponential, SIAM J. Sci. Comput., 38, 3, A1639-A1661, (2016) · Zbl 1339.65061
[12] Caliari, M.; Kandolf, P.; Zivcovich, F., Backward error analysis of polynomial approximations for computing the action of the matrix exponential, BIT Numer. Math., (2018)
[13] van Loan, C., A note on the evaluation of matrix polynomials, IEEE Trans. Automat. Control, 24, 2, 320-321, (1979) · Zbl 0401.65027
[14] Kenney, C. S.; Laub, A. J., A Schur-Fréchet algorithm for computing the logarithm and exponential of a matrix, SIAM J. Matrix Anal. Appl., 19, 640-663, (1998) · Zbl 0913.65036
[15] Fischer, T. M., On the algorithm by al-mohy and higham for computing the action of the matrix exponential: A posteriori roundoff error estimation, Linear Algebra Appl., 531, 141-168, (2017) · Zbl 1372.65137
[16] Higham, N. J.; Tisseur, F., A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra, SIAM J. Matrix Anal. Appl., 21, 4, 1185-1201, (2000) · Zbl 0959.65061
[17] Golub, G. H.; Van Loan, C. F., (Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, (2012), Johns Hopkins University Press)
[18] N.J. Higham, The Matrix Computation Toolbox, version 1.2, 2002. URL http://www.ma.man.ac.uk/ higham/mctoolbox.
[19] N.J. Higham, The Matrix Function Toolbox, version 1.0, 2008. URL http://www.ma.man.ac.uk/ higham/mftoolbox.
[20] McCurdy, A.; Ng, K. C.; Parlett, B. N., Accurate computation of divided differences of the exponential function, Math. Comp., 43, 168, 501-528, (1984) · Zbl 0561.65009
[21] Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29, 1, 209-228, (1992) · Zbl 0749.65030
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.