Computing the Wave-Kernel Matrix Functions

We derive an algorithm for computing the wave-kernel functions $\cosh{\sqrt{A}}$ and $\mathrm{sinhc}{\sqrt{A}}$ for an arbitrary square matrix $A$, where $\mathrm{sinhc}(z) = \sinh(z)/z$. 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 $\cosh{\sqrt{A}}$ can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for $\|A^k\|^{1/k}$ that is sharper than one previously obtained by Al-Mohy and Higham (\textit{SIAM J. Matrix Anal.\ Appl.}, 31(3):970--989, 2009). The amount of scaling and the degree of the Pade approximant are chosen to minimize the computational cost subject to achieving backward stability for $\cosh{\sqrt{A}}$ 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