Nadukandi, Prashanth and Higham, Nicholas J. (2018) Computing the WaveKernel Matrix Functions. [MIMS Preprint] (Submitted)
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Abstract
We derive an algorithm for computing the wavekernel 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 AlMohy and Higham (\textit{SIAM J. Matrix Anal.\ Appl.}, 31(3):970989, 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 floatingpoint arithmetic and is superior in this respect to the general purpose SchurParlett algorithm applied to these functions.
Item Type:  MIMS Preprint 

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:  19 Aug 2018 13:11 
Last Modified:  19 Aug 2018 13:11 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2651 
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Computing the WaveKernel Matrix Functions. (deposited 10 Feb 2018 16:46)
 Computing the WaveKernel Matrix Functions. (deposited 19 Aug 2018 13:11) [Currently Displayed]
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