Computing the Action of Trigonometric and Hyperbolic Matrix Functions

TL;DR

This paper introduces a new, efficient algorithm for computing the action of trigonometric and hyperbolic matrix functions on matrices without explicitly calculating the functions, leveraging existing exponential algorithms for improved stability and speed.

Contribution

The authors develop a novel algorithm that computes multiple matrix functions simultaneously using real arithmetic, enhancing efficiency and stability over previous methods.

Findings

The new algorithm is forward stable.

It outperforms methods based on multiple invocations of expmv.

It can compute cosine, sine, cosh, sinh simultaneously.

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 $\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V + \mathrm{e}^{\mathrm{-i}A}V)/2$.