Jorge Sastre, Javier Ibáñez, Journal of Computational and Applied Mathematics, Volume 419, 2003, 114706, https://doi.org/10.1016/j.cam.2022.114706
A new formula to write the forward error of Taylor approximations of analytical functions in terms of the backward error of those approximations is given, overcoming problems of the backward error analysis that use inverse functions. Examples for the backward error analysis of functions such as the matrix cosine cos(A) or cos(sqrt(A)) are given.
Boosting the computation of the matrix exponential, J. Sastre, J. Ibáñez, E. Defez, Appl. Math. Comput. in press, 2018, doi:10.1016/j.amc.2018.08.017, Preprint, Matlab code expmpol.m.
This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson–Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Padé algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances.
First article with an application of the new matrix polynomial evaluation methods from J. Sastre, Efficient evaluation of matrix polynomials, Linear Algebra Appl. 539, (2018) 229-250. With the new matrix polynomial evaluation methods, Taylor approximation methods are more efficient than Padé approximant based methods.