Accurate and efficient matrix exponential computation

[EN] This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Pad´e state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost.This work has been supported by the Programa de Apoyo a la Investigacion y el Desarrollo of the Universitat Politecnica de Valencia grant PAID-06-11-2020Sastre, J.; Ibáñez González, JJ.; Ruiz Martínez, PA.; Defez Candel, E. (2014). Accurate and efficient matrix exponential computation. International Journal of Computer Mathematics. 91(1):97-112. https://doi.org/10.1080/00207160.2013.791392S97112911Al-Mohy, A. H., & Higham, N. J. (2010). A New Scaling and Squaring Algorithm for the Matrix Exponential. SIAM Journal on Matrix Analysis and Applications, 31(3), 970-989. doi:10.1137/09074721xArioli, M., Codenotti, B., & Fassino, C. (1996). The Padé method for computing the matrix exponential. Linear Algebra and its Applications, 240, 111-130. doi:10.1016/0024-3795(94)00190-1S. Blackford and J. Dongarra,Installation guide for LAPACK, LAPACK Working Note 411, Department of Computer Science, University of Tenessee, 1999.Dieci, 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)00042-2Dieci, L., & Papini, A. (2001). Numerical Algorithms, 28(1/4), 137-150. doi:10.1023/a:1014071202885Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2), 201-213. doi:10.1007/s101070100263C. Fassino,Computation of matrix functions, Ph.D. thesis TD-7/93, Università di Pisa, Genova, 1993.Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms. doi:10.1137/1.9780898718027Higham, N. J. (2005). The Scaling and Squaring Method for the Matrix Exponential Revisited. SIAM Journal on Matrix Analysis and Applications, 26(4), 1179-1193. doi:10.1137/04061101xHigham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Higham, 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/s0895479899356080Moler, C., & Van Loan, C. (2003). Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, 45(1), 3-49. doi:10.1137/s00361445024180Paterson, 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:10.1137/0202007Sastre, J., Ibáñez, J., Defez, E., & Ruiz, P. (2011). Accurate matrix exponential computation to solve coupled differential models in engineering. Mathematical and Computer Modelling, 54(7-8), 1835-1840. doi:10.1016/j.mcm.2010.12.04

Fast computation of the matrix exponential for a Toeplitz matrix

The computation of the matrix exponential is a ubiquitous operation in numerical mathematics, and for a general, unstructured $n\times n$ matrix it can be computed in $\mathcal{O}(n^3)$ operations. An interesting problem arises if the input matrix is a Toeplitz matrix, for example as the result of discretizing integral equations with a time invariant kernel. In this case it is not obvious how to take advantage of the Toeplitz structure, as the exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself. The main contribution of this work are fast algorithms for the computation of the Toeplitz matrix exponential. The algorithms have provable quadratic complexity if the spectrum is real, or sectorial, or more generally, if the imaginary parts of the rightmost eigenvalues do not vary too much. They may be efficient even outside these spectral constraints. They are based on the scaling and squaring framework, and their analysis connects classical results from rational approximation theory to matrices of low displacement rank. As an example, the developed methods are applied to Merton's jump-diffusion model for option pricing

A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called phi-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the Matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.Comment: 20 pages, 3 colour figures, code available from http://www.maths.leeds.ac.uk/~jitse/software.html . v2: Various changes to improve presentation as suggested by the refere