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

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