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==Abstract== | ==Abstract== | ||
| − | We derive an algorithm for computing the wave-kernel functions<math>\cos h \surd A</math> and <math>\sin h \surd A</math> for an arbitrary square matrix <math>A</math>, where <math>\sin h c z = \sin h\frac{(z)}z</math>. The algorithm is based on Padé 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 [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970--989]. The amount of scaling and the degree of the Padé 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. | + | We derive an algorithm for computing the wave-kernel functions <math>\cos h \surd A</math> and <math>\sin h \surd A</math> for an arbitrary square matrix <math>A</math>, where <math>\sin h c z = \sin h\frac{(z)}z</math>. The algorithm is based on Padé 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 [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970--989]. The amount of scaling and the degree of the Padé 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. |
==Full Document== | ==Full Document== | ||
Revision as of 15:26, 5 May 2020
Abstract
We derive an algorithm for computing the wave-kernel functions and for an arbitrary square matrix , where . The algorithm is based on Padé 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 [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 970--989]. The amount of scaling and the degree of the Padé 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.
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2 The main text
3 Bibliography
4 Acknowledgments
5 References
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Published on 01/01/2018
DOI: 10.1137/18M1170352
Licence: CC BY-NC-SA license
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