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Backward error analysis of polynomial approximations for computing the action of the matrix exponential

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Abstract

We describe how to perform the backward error analysis for the approximation of \(\exp (A)v\) by \(p(s^{-1}A)^sv\), for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja–Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples.

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Notes

  1. The excerpts of code reported here do work in SageMath. Furthermore, we feel they are short, easy to understand, and can be regarded as pseudo-codes.

  2. In [7], \(\theta _m\) was selected equal to \(\bar{c}_m\) and therefore the values reported in Table 3 are larger than those in [7, Table 1].

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Authors and Affiliations

  1. Department of Computer Science, University of Verona, Verona, Italy

    Marco Caliari

  2. Department of Mathematics, University of Innsbruck, Innsbruck, Austria

    Peter Kandolf

  3. Department of Mathematics, University of Trento, Trento, Italy

    Franco Zivcovich

Authors
  1. Marco Caliari
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  2. Peter Kandolf
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  3. Franco Zivcovich
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Correspondence to Marco Caliari.

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Communicated by Daniel Kressner.

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Caliari, M., Kandolf, P. & Zivcovich, F. Backward error analysis of polynomial approximations for computing the action of the matrix exponential. Bit Numer Math 58, 907–935 (2018). https://doi.org/10.1007/s10543-018-0718-9

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  • DOI: https://doi.org/10.1007/s10543-018-0718-9

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