## 2014.31: New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously

2014.31:
Awad H. Al-Mohy, Nicholas J. Higham and Samuel D. Relton
(2015)
*New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously.*
SIAM J. Sci. Comput., 37 (1).
A456-A487.
ISSN 1095-7197

*This is the latest version of this eprint.*

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DOI: 10.1137/140973979

## Abstract

Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic and behave in a forward stable manner in floating point arithmetic. Our new algorithms employ both Pad\'e approximants of $\sin x$ and new rational approximants to $\cos x$ and $\sin x$ obtained from Pad\'e approximants to $e^x$. The amount of scaling and the degree of the approximants are chosen to minimize the computational cost subject to backward stability in exact arithmetic. Numerical experiments show that the new algorithms have backward and forward errors that rival or surpass those of existing algorithms and are particularly favorable for triangular matrices.

Item Type: | Article |
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Uncontrolled Keywords: | matrix sine, matrix cosine, matrix exponential, matrix function, backward error, forward error, rational approximation, Pad\'{e} approximation, MATLAB, double angle formula, triple angle formula |

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

MIMS number: | 2014.31 |

Deposited By: | Nick Higham |

Deposited On: | 13 February 2015 |

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- New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously (deposited 13 February 2015)
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