Higham, Nicholas J. and Kandolf, Peter (2017) Computing the Action of Trigonometric and Hyperbolic Matrix Functions. SIAM Journal on Scientific Computing, 39 (2). A613A627. ISSN 10957197
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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 AlMohy 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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Computing the Action of Trigonometric and Hyperbolic Matrix Functions. (deposited 26 Jul 2016)

Computing the Action of Trigonometric and Hyperbolic Matrix Functions. (deposited 03 Feb 2017)
 Computing the Action of Trigonometric and Hyperbolic Matrix Functions. (deposited 28 Apr 2017) [Currently Displayed]

Computing the Action of Trigonometric and Hyperbolic Matrix Functions. (deposited 03 Feb 2017)
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