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An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation

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  • Bader, Philipp
  • Blanes, Sergio
  • Casas, Fernando
  • Seydaoğlu, Muaz

Abstract

We present a practical algorithm to approximate the exponential of skew-Hermitian matrices up to round-off error based on an efficient computation of Chebyshev polynomials of matrices and the corresponding error analysis. It is based on Chebyshev polynomials of degrees 2, 4, 8, 12 and 18 which are computed with only 1, 2, 3, 4 and 5 matrix–matrix products, respectively. For problems of the form exp(−iA), with A a real and symmetric matrix, an improved version is presented that computes the sine and cosine of A with a reduced computational cost. The theoretical analysis, supported by numerical experiments, indicates that the new methods are more efficient than schemes based on rational Padé approximants and Taylor polynomials for all tolerances and time interval lengths. The new procedure is particularly recommended to be used in conjunction with exponential integrators for the numerical time integration of the Schrödinger equation.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:194:y:2022:i:c:p:383-400
    DOI: 10.1016/j.matcom.2021.12.002
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    References listed on IDEAS

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    1. 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.
    2. Sastre, J. & Ibáñez, J. & Defez, E., 2019. "Boosting the computation of the matrix exponential," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 206-220.
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