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 expmv of Al-Mohy and Higham for eAV 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 expmv through formulas such as cos(A)V = (eiAV + e−iAV )/2.