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