Nadukandi, Prashanth and Higham, Nicholas J. (2018) Computing the Wave-Kernel Matrix Functions. [MIMS Preprint]
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Abstract
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.
Item Type: | MIMS Preprint |
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Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis |
Divisions: | Manchester Institute for the Mathematical Sciences |
Depositing User: | Dr Prashanth Nadukandi |
Date Deposited: | 10 Feb 2018 16:46 |
Last Modified: | 10 Feb 2018 16:51 |
URI: | http://eprints.maths.manchester.ac.uk/id/eprint/2621 |
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