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Open AccessArticle

Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

1
Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain
2
Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain
3
IMAC and Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1174; https://doi.org/10.3390/math7121174
Received: 11 November 2019 / Revised: 20 November 2019 / Accepted: 21 November 2019 / Published: 3 December 2019
(This article belongs to the Special Issue Geometric Numerical Integration)
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 Padé 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. View Full-Text
Keywords: exponential of a matrix; scaling and squaring; matrix polynomials exponential of a matrix; scaling and squaring; matrix polynomials
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MDPI and ACS Style

Bader, P.; Blanes, S.; Casas, F. Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation. Mathematics 2019, 7, 1174.

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