BIT Numerical Mathematics

, Volume 58, Issue 4, pp 907–935 | Cite as

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

  • Marco CaliariEmail author
  • Peter Kandolf
  • Franco Zivcovich


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.


Backward error analysis Action of matrix exponential Leja–Hermite interpolation Taylor series 

Mathematics Subject Classification

65D05 65F30 65F60 


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of VeronaVeronaItaly
  2. 2.Department of MathematicsUniversity of InnsbruckInnsbruckAustria
  3. 3.Department of MathematicsUniversity of TrentoTrentoItaly

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