Computing the Action of Trigonometric and Hyperbolic Matrix Functions

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 $\mathtt{\text{expmv}}$ of Al-Mohy and Higham for ${\mathrm{e}}^{A}V$ 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 $\mathtt{\text{expmv}}$ through formulas such as $\cos (A)V=({\mathrm{e}}^{\mathrm{i}A}V+{\mathrm{e}}^{-\mathrm{i}A}V)\mathrm{/}2$.