Computing the Action of Trigonometric and Hyperbolic Matrix Functions
Nicholas John Higham, Peter Kandolf · MIMS EPrints (University of Southampton) · 2016
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 $\\cos(A)V = (\\mathrm{e}^{\\mathrm{i}A}V + \\mathrm{e}^{\\mathrm{-i}A}V)/2$.