## 2016.40: Computing the Action of Trigonometric and Hyperbolic Matrix Functions

2016.40:
Nicholas J. Higham and Peter Kandolf
(2017)
*Computing the Action of Trigonometric and Hyperbolic Matrix Functions.*
SIAM Journal on Scientific Computing, 39 (2).
A613-A627.
ISSN 1095-7197

*This is the latest version of this eprint.*

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DOI: 10.1137/16M1084225

## 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 |
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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 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2016.40 |

Deposited By: | Nick Higham |

Deposited On: | 28 April 2017 |

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