Computing the Wave-Kernel Matrix Functions

Revision as of 15:30, 5 May 2020

Abstract

We derive an algorithm for computing the wave-kernel functions $\cos h{\sqrt {A}}$ and $\sin h{\sqrt {A}}$ for an arbitrary square matrix $A$ , where $\sin hcz=\sin h{\frac {(z)}{z}}$ . The algorithm is based on Padé approximation and the use of double angle formulas. We show that the backward error of any approximation to $\cos h{\sqrt {A}}$ 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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 ==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 <math>\cos h\sqrt{A}</math> 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 \sqrt A</math> and <math>\sin h \sqrt 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 <math>\cos h\sqrt{A}</math> can be explicitly expressed in terms of a hypergeometric function. To bound the backward error we derive and exploit a new bound for <math>\|A^k\|^{1/k}</math> 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 <math>\cosh\sqrt{A}</math> 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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