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Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation

Author

Listed:
  • Philipp Bader

    (Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain)

  • Sergio Blanes

    (Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain)

  • Fernando Casas

    (IMAC and Departament de Matemàtiques, Universitat Jaume I, 12071 Castellón, Spain)

Abstract

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.

Suggested Citation

  • Philipp Bader & Sergio Blanes & Fernando Casas, 2019. "Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation," Mathematics, MDPI, vol. 7(12), pages 1-19, December.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1174-:d:293584
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    Citations

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    Cited by:

    1. Bader, Philipp & Blanes, Sergio & Casas, Fernando & Seydaoğlu, Muaz, 2022. "An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 383-400.
    2. Bogdan Mocan & Claudiu Schonstein & Mircea Murar & Calin Neamtu & Mircea Fulea & Mihaela Mocan & Simona Dragan & Horea Feier, 2023. "Upper-Limb Robotic Exoskeleton for Early Cardiac Rehabilitation Following an Open-Heart Surgery—Mathematical Modelling and Empirical Validation," Mathematics, MDPI, vol. 11(7), pages 1-43, March.

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