# 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]

## 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 To appear in SIAM J. Sci. Comput. matrix function, action of matrix function, trigonometric function, hyperbolic function, matrix exponential, Taylor series, backward error analysis, exponential integrator, splitting methods MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 03 Feb 2017 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2524