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BIT Numerical Mathematics

Erschienen in:

15.05.2020

Approximation of the matrix exponential for matrices with a skinny field of values

verfasst von: Marco Caliari, Fabio Cassini, Franco Zivcovich

Erschienen in: BIT Numerical Mathematics | Ausgabe 4/2020

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Abstract

The backward error analysis is a great tool which allows selecting in an effective way the scaling parameter s and the polynomial degree of approximation m when the action of the matrix exponential \(\exp (A)v\) has to be approximated by \(\left( p_m(s^{-1}A)\right) ^sv=\exp (A+\varDelta A)v\). We propose here a rigorous bound for the relative backward error \(\left\Vert \varDelta A\right\Vert _{2}/\left\Vert A\right\Vert _{2}\), which is of particular interest for matrices whose field of values is skinny, such as the discretization of the advection–diffusion or the Schrödinger operators. The numerical results confirm the superiority of the new approach with respect to methods based on the classical power series expansion of the backward error for the matrices of our interest, both in terms of computational cost and achieved accuracy.

Vorheriger Artikel Numerical method with fractional splines for a subdiffusion problem

Nächster Artikel Semi-implicit Euler–Maruyama method for non-linear time-changed stochastic differential equations

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Titel: Approximation of the matrix exponential for matrices with a skinny field of values
verfasst von: Marco Caliari
Fabio Cassini
Franco Zivcovich
Publikationsdatum: 15.05.2020
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 4/2020
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-020-00809-0

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