Computing the Action of Trigonometric and Hyperbolic Matrix Functions
This provides a more efficient and stable method for computing matrix trigonometric actions, benefiting applications in numerical linear algebra and differential equations.
The authors derive a new algorithm for computing the action of trigonometric and hyperbolic matrix functions on a matrix without computing the full function, achieving forward stability and significant speedup over alternatives based on multiple exponential evaluations.
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 $\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V + \mathrm{e}^{\mathrm{-i}A}V)/2$.