COMPUTING THE ACTION OF TRIGONOMETRIC AND HYPERBOLIC MATRIX FUNCTIONS

Research output: Contribution to journalArticle

Abstract

We derive a new algorithm for computing the action $f(A)V$ of the cosine, sine, hyperbolic cosine, and hyperbolic sine of a matrix $A$ on a matrix $V$, without first computing $f(A)$. The algorithm can compute $\cos(A)V$ and $\sin(A)V$ simultaneously, and likewise for $\cosh(A)V$ and $\sinh(A)V$, and it uses only real arithmetic when $A$ is real. The algorithm exploits an existing algorithm expmv of Al-Mohy and Higham for $\mathrm{e}^AV$ and its underlying backward error analysis. Our experiments show that the new algorithm performs in a forward stable manner and is generally significantly faster than alternatives based on multiple invocations of expmv through formulas such as $\cos(A)V = (\mathrm{e}^{\mathrm{i}A}V + \mathrm{e}^{\mathrm{-i}A}V)/2$.

Bibliographical metadata

Original languageEnglish
Pages (from-to)A613–A627
Number of pages15
JournalSIAM Journal on Scientific Computing
Volume39
Issue number2
Early online date20 Apr 2017
DOIs
StatePublished - 2017