Computing the Action of Trigonometric and Hyperbolic Matrix Functions

Higham, Nicholas J. and Kandolf, Peter (2017) Computing the Action of Trigonometric and Hyperbolic Matrix Functions. SIAM Journal on Scientific Computing, 39 (2). A613-A627. ISSN 1095-7197

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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: Article
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: 28 Apr 2017
Last Modified: 20 Oct 2017 14:13
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2545

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