Parallel approximation of the exponential of Hermitian matrices

Approximation parallèle de l'exponentielle de matrices Hermitiennes

Frédéric Hecht (1) , Sidi-Mahmoud Kaber (2) , Lucas Perrin (3) , Alain Plagne (4) , Julien Salomon (3)

1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 ANGE - Numerical Analysis, Geophysics and Ecology
4 CMLS - Centre de Mathématiques Laurent Schwartz

We are interested in the approximation of the exponential function on a real interval by a particular rational function (inverse of a polynomial). Using the partial fraction decomposition of this rational function, we build an algorithm for computing the exponential of a matrix that is adapted to parallel computing. The performances of this parallel algorithm, both in accuracy and in computing time, are quantitatively analyzed.

Keywords: Matrix exponential; Parallel computing; Truncation error; Taylor series; Partial fraction decomposition; Pad\'e approximation; MATLAB; Octave; expm; Roundoff error.

Domains: Numerical Analysis [math.NA]; Distributed, Parallel, and Cluster Computing [cs.DC]

Frédéric Hecht, Sidi-Mahmoud Kaber, Lucas Perrin, Alain Plagne, Julien Salomon. Parallel approximation of the exponential of Hermitian matrices. 2023. ⟨hal-03948509⟩