Computing the Action of Trigonometric and Hyperbolic Matrix Functions

Higham, Nicholas J. and Kandolf, Peter (2016) Computing the Action of Trigonometric and Hyperbolic Matrix Functions. [MIMS Preprint]

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Abstract

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 $\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V + \mathrm{e}^{\mathrm{-i}A}V)/2$.

Item Type: MIMS Preprint
Additional Information: To appear in SIAM J. Sci. Comput.
Uncontrolled Keywords: matrix function, action of matrix function, trigonometric function, hyperbolic function, matrix exponential, Taylor series, backward error analysis, exponential integrator, splitting methods
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 03 Feb 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2524

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