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2017.29: A New Algorithm for Computing the Actions of Trigonometric and Hyperbolic Matrix Functions

2017.29: Awad H. Al-Mohy (2017) A New Algorithm for Computing the Actions of Trigonometric and Hyperbolic Matrix Functions.

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Abstract

A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with $n_0 \ll n$. $A^{1/2}$ denotes any matrix square root of $A$ and it is never required to be computed. The algorithm offers six independent output options given $t$, $A$, $B$, and a tolerance. For each option, actions of a pair of trigonometric or hyperbolic matrix functions are simultaneously computed. The algorithm scales the matrix $A$ down by a positive integer $s$, approximates $f(s^{-1}tA)B$ by a truncated Taylor series, and finally uses the recurrences of the Chebyshev polynomials of the first and second kind to recover $f(tA)B$. The selection of the scaling parameter and the degree of Taylor polynomial are based on a forward error analysis and a sequence of the form $\|A^k\|^{1/k}$ in such a way the overall computational cost of the algorithm is optimized. Shifting is used where applicable as a preprocessing step to reduce the scaling parameter. The algorithm works for any matrix $A$ and its computational cost is dominated by the formation of products of $A$ with $n\times n_0$ matrices that could take advantage of the implementation of level-3 BLAS. Our numerical experiments show that the new algorithm behaves in a forward stable fashion and in most problems outperforms the existing algorithms in terms of CPU time, computational cost, and accuracy.

Item Type:MIMS Preprint
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2017.29
Deposited By:Awad Al-Mohy
Deposited On:28 August 2017

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