Latest revision as of 15:31, 5 May 2020

Abstract

We derive an algorithm for computing the wave-kernel functions ${\displaystyle \cos h{\sqrt {A}}}$ and ${\displaystyle \sin h{\sqrt {A}}}$ for an arbitrary square matrix ${\displaystyle A}$, where ${\displaystyle \sin hcz=\sin h{\frac {(z)}{z}}}$. The algorithm is based on Padé approximation and the use of double angle formulas. We show that the backward error of any approximation to ${\displaystyle \cos h{\sqrt {A}}}$ can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for ${\displaystyle \|A^{k}\|^{1/k}}$ that is sharper than one previously obtained by Al-Mohy and Higham [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970--989]. The amount of scaling and the degree of the Padé approximant are chosen to minimize the computational cost subject to achieving backward stability for ${\displaystyle \cosh {\sqrt {A}}}$ in exact arithmetic. Numerical experiments show that the algorithm behaves in a forward stable manner in floating-point arithmetic and is superior in this respect to the general purpose Schur--Parlett algorithm applied to these functions.

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