Computing the Action of Trigonometric and Hyperbolic Matrix Functions
We derive a new algorithm for computing the action \(f(A)V\) of the cosine, sine, hyperbolic cosine, and hyperbolic sine of a matrix \(A\) on a matrix \(V\), without first computing \(f(A)\). The algorithm can compute \(\cos(A)V\) and \(\sin(A)V\) simultaneously, and likewise for \(\cosh(A)V\) and \...
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| Vydané v: | arXiv.org |
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| Hlavní autori: | , |
| Médium: | Paper |
| Jazyk: | English |
| Vydavateľské údaje: |
Ithaca
Cornell University Library, arXiv.org
29.04.2017
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| Predmet: | |
| ISSN: | 2331-8422 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | We derive a new algorithm for computing the action \(f(A)V\) of the cosine, sine, hyperbolic cosine, and hyperbolic sine of a matrix \(A\) on a matrix \(V\), without first computing \(f(A)\). The algorithm can compute \(\cos(A)V\) and \(\sin(A)V\) simultaneously, and likewise for \(\cosh(A)V\) and \(\sinh(A)V\), and it uses only real arithmetic when \(A\) is real. The algorithm exploits an existing algorithm \texttt{expmv} of Al-Mohy and Higham for \(\mathrm{e}^AV\) and its underlying backward error analysis. Our experiments show that the new algorithm performs in a forward stable manner and is generally significantly faster than alternatives based on multiple invocations of \texttt{expmv} through formulas such as \(\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V + \mathrm{e}^{\mathrm{-i}A}V)/2\). |
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| Bibliografia: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.1607.04012 |