Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

[EN] A new way to compute the Taylor polynomial of a matrix exponential is presented which reduces the number of matrix multiplications in comparison with the de-facto standard Paterson-Stockmeyer method for polynomial evaluation. Combined with the scaling and squaring procedure, this reduction is sufficient to make the Taylor method superior in performance to Pade approximants over a range of values of the matrix norms. An efficient adjustment to make the method robust against overscaling is also introduced. Numerical experiments show the superior performance of our method to have a similar accuracy in comparison with state-of-the-art implementations, and thus, it is especially recommended to be used in conjunction with Lie-group and exponential integrators where preservation of geometric properties is at issue.This work was funded by Ministerio de Economia, Industria y Competitividad (Spain) through project MTM2016-77660-P (AEI/FEDER, UE). P.B. was additionally supported by a contract within the Program Juan de la Cierva Formacion (Spain).Bader, P.; Blanes Zamora, S.; Casas, F. (2019). Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation. Mathematics. 7(12):1-19. https://doi.org/10.3390/math7121174S119712Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P., & Zanna, A. (2000). Lie-group methods. Acta Numerica, 9, 215-365. doi:10.1017/s0962492900002154Blanes, S., Casas, F., Oteo, J. A., & Ros, J. (2009). The Magnus expansion and some of its applications. Physics Reports, 470(5-6), 151-238. doi:10.1016/j.physrep.2008.11.001Casas, F., & Iserles, A. (2006). Explicit Magnus expansions for nonlinear equations. Journal of Physics A: Mathematical and General, 39(19), 5445-5461. doi:10.1088/0305-4470/39/19/s07Celledoni, E., Marthinsen, A., & Owren, B. (2003). Commutator-free Lie group methods. Future Generation Computer Systems, 19(3), 341-352. doi:10.1016/s0167-739x(02)00161-9Crouch, P. E., & Grossman, R. (1993). Numerical integration of ordinary differential equations on manifolds. Journal of Nonlinear Science, 3(1), 1-33. doi:10.1007/bf02429858Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286. doi:10.1017/s0962492910000048Najfeld, I., & Havel, T. F. (1995). Derivatives of the Matrix Exponential and Their Computation. Advances in Applied Mathematics, 16(3), 321-375. doi:10.1006/aama.1995.1017Sidje, R. B. (1998). Expokit. ACM Transactions on Mathematical Software, 24(1), 130-156. doi:10.1145/285861.285868Higham, N. J., & Al-Mohy, A. H. (2010). Computing matrix functions. Acta Numerica, 19, 159-208. doi:10.1017/s0962492910000036Paterson, M. S., & Stockmeyer, L. J. (1973). On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials. SIAM Journal on Computing, 2(1), 60-66. doi:10.1137/0202007Ruiz, P., Sastre, J., Ibáñez, J., & Defez, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics, 291, 370-379. doi:10.1016/j.cam.2015.04.001Sastre, J., Ibán͂ez, J., Defez, E., & Ruiz, P. (2015). New Scaling-Squaring Taylor Algorithms for Computing the Matrix Exponential. SIAM Journal on Scientific Computing, 37(1), A439-A455. doi:10.1137/090763202Sastre, J. (2018). Efficient evaluation of matrix polynomials. Linear Algebra and its Applications, 539, 229-250. doi:10.1016/j.laa.2017.11.010Westreich, D. (1989). Evaluating the matrix polynomial I+A+. . .+A/sup N-1/. IEEE Transactions on Circuits and Systems, 36(1), 162-164. doi:10.1109/31.16591An Efficient Alternative to the Function Expm of Matlab for the Computation of the Exponential of a Matrix http://www.gicas.uji.es/Research/MatrixExp.htmlKenney, C. S., & Laub, A. J. (1998). A Schur--Fréchet Algorithm for Computing the Logarithm and Exponential of a Matrix. SIAM Journal on Matrix Analysis and Applications, 19(3), 640-663. doi:10.1137/s0895479896300334Dieci, L., & Papini, A. (2000). Padé approximation for the exponential of a block triangular matrix. Linear Algebra and its Applications, 308(1-3), 183-202. doi:10.1016/s0024-3795(00)00042-2Higham, N. J., & Tisseur, F. (2000). A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIAM Journal on Matrix Analysis and Applications, 21(4), 1185-1201. doi:10.1137/s0895479899356080Celledoni, E., & Iserles, A. (2000). Approximating the exponential from a Lie algebra to a Lie group. Mathematics of Computation, 69(232), 1457-1481. doi:10.1090/s0025-5718-00-01223-

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