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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Veröffentlicht in:	arXiv.org
Hauptverfasser:	Higham, Nicholas J, Kandolf, Peter
Format:	Paper
Sprache:	Englisch
Veröffentlicht:	Ithaca Cornell University Library, arXiv.org 29.04.2017
Schlagworte:	Algorithms Computation Error analysis Trigonometric functions
ISSN:	2331-8422
Online-Zugang:	Volltext
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Beschreibung
Zusammenfassung:	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 \(\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V + \mathrm{e}^{\mathrm{-i}A}V)/2\).
Bibliographie:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.1607.04012