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

Al-Mohy, Awad H. and Higham, Nicholas J. and Relton, Samuel D. (2014) New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously. [MIMS Preprint]

## 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: MIMS Preprint matrix sine, matrix cosine, matrix exponential, matrix function, backward error, forward error, rational approximation, Pad\'{e} approximation, MATLAB, double angle formula, triple angle formula 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 20 Jun 2014 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2149