# zbMATH — the first resource for mathematics

A new efficient and accurate spline algorithm for the matrix exponential computation. (English) Zbl 1398.65091
Summary: In this work an accurate and efficient method based on matrix splines for computing matrix exponential is given. An algorithm and a MATLAB implementation have been developed and compared with the state-of-the-art algorithms for computing the matrix exponential. We also developed a parallel implementation for large scale problems. This implementation allowed us to get a much better performance when working with this kind of problems.

##### MSC:
 65F60 Numerical computation of matrix exponential and similar matrix functions 15A16 Matrix exponential and similar functions of matrices 65D07 Numerical computation using splines 65Y10 Numerical algorithms for specific classes of architectures
##### Software:
CUBLAS; CUDA; Matlab; MATLAB expm; mctoolbox; mftoolbox; testmatrix
Full Text:
##### References:
 [1] Bader, P.; Blanes, S.; Seydaoglu, M., The scaling, splitting, and squaring method for the exponential of perturbed matrices, SIAM J. Matrix Anal. Appl., 36, 2, 594-614, (2015) · Zbl 1328.65097 [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] Higham, N. J., Functions of Matrices: Theory and Computation, xx+425, (2008), SIAM Philadelphia, PA, USA · Zbl 1167.15001 [4] Sastre, J.; Ibáñez, J. J.; Defez, E.; Ruiz, P. A., Accurate matrix exponential computation to solve coupled differential models in engineering, Math. Comput. Modelling, 54, 1835-1840, (2011) · Zbl 1235.65042 [5] Sastre, J.; Ibáñez, J.; Defez, E.; Ruiz, P., New scaling-squaring Taylor algorithms for computing the matrix exponential, SIAM J. Sci. Comput., 37, 1, A439-A455, (2015) · Zbl 1315.65046 [6] 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 [7] Defez, E.; Tung, M.; Ibáñez, J. J.; Sastre, J., Approximating and computing nonlinear matrix differential models, Math. Comput. Modelling, 55, 7, (2012) · Zbl 1255.65148 [8] 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 [9] Golub, G. H.; Loan, C. V., (Matrix Computations, Johns Hopkins Studies in Mathematical Sciences, (1996), The Johns Hopkins University Press) [10] Sastre, J.; Ibáñez, J. J.; Defez, E.; Ruiz, P. A., Efficient orthogonal matrix polynomial based method for computing matrix exponential, Appl. Math. Comput., 217, 6451-6463, (2011) · Zbl 1211.65052 [11] Higham, N. J., Functions of Matrices: Theory and Computation, xx+425, (2008), Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 1167.15001 [12] Ruiz, P.; Sastre, J.; Ibáñez, J.; Defez, E., High perfomance computing of the matrix exponential, J. Comput. Appl. Math., 291, 370-379, (2016) · Zbl 1329.65092 [13] 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, 1185-1201, (2000) · Zbl 0959.65061 [14] N.J. Higham, The Test Matrix Toolbox for MATLAB, Numerical Analysis Report No. 237, Manchester, England, Dec. 1993. [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 [17] NVIDIA, CUDA. CUBLAS library, 2009. [18] NVIDIA, NVIDIA Tesla K20-K20X GPU Accelerators - Benchmarks, 2012. [19] E. D’Azevedo, A. Huang, K. Wong, W. Wu, Out-of-core algorithms for dense matrix factorization on GPGPU. [20] Mathworks, MATLAB MEX Files, http://www.mathworks.com/support/tech-notes/1600/1605.shtml#intro. [21] Alonso, P.; Ibáñez, J.; Sastre, J.; Peinado, J.; Defez, E., Efficient and accurate algorithms for computing matrix trigonometric functions, J. Comput. Appl. Math., 309, 1, 325-332, (2017) · Zbl 1416.65126 [22] Defez, E.; Sastre, J.; Ibáñez, J.; Peinado, J., Solving engineering models using hyperbolic matrix functions, Appl. Math. Model., 40, 4, 2837-2844, (2016)
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.