Papers › Computing the Action of Trigonometric and Hyperbolic Matrix Functions
Computing the Action of Trigonometric and Hyperbolic Matrix Functions
Nicholas J. Higham, Peter Kandolf
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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 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 \texttt{expmv} through formulas such as cos(A)V = (eⁱᴬV + e⁻ⁱᴬV)/2.
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