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.
Highlights
In recent years research into the computation of matrix functions has primarily focused on the matrix exponential, the logarithm, and matrix powers
In this work we develop three new algorithms for the sine and cosine that are based on backward error analysis and are backward stable in exact arithmetic
We begin by analyzing the backward error of Pade approximants to the matrix sine and cosine
Summary
By contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK. NEW ALGORITHMS FOR COMPUTING THE MATRIX SINE AND COSINE SEPARATELY OR SIMULTANEOUSLY∗
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