Computing the Action of Trigonometric and Hyperbolic Matrix Functions - Higham, Nicholas J; Kandolf, Peter (arXiv.org, 2017)
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 \(e^AV\) 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 expmv through formulas such as \(\cos(A)V = (e^{iA}V + e^{-iA}V)/2\).