P. Nadukandi, N. Higham
We derive an algorithm for computing the wave-kernel functions cosh \surd A and sinhc\surd A for an arbitrary square matrix A, where sinhcz = sinh(z)/z. The algorithm is based on Pad\'e approximation and the use of double angle formulas. We show that the backward error of any approximation to cosh \surd A can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for \| Ak\| 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 Pad\'e approximant are chosen to minimize the computational cost subject to achieving backward stability for cosh \surd 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.
Published on 01/01/2018
DOI: 10.1137/18M1170352Licence: CC BY-NC-SA license
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