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COMPUTING THE WAVE-KERNEL MATRIX FUNCTIONS
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DOI：10.1137/18M1170352.png)
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Abstract
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We derive an algorithm for computing the wave-kernel functions cosh root A and sinhc root A for an arbitrary square matrix A, where sinhcz = sinh(z)/z. The algorithm is based on Pade approximation and the use of double angle formulas. We show that the backward error of any approximation to cosh root A can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for parallel to A(k)parallel to(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 Pade approximant are chosen to minimize the computational cost subject to achieving backward stability for cosh root 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.
Keywords:
wave kernel
matrix function
Pade approximation
backward stability
hypergeometric function
matrix norm estimation
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