# Efficient Algorithms for the Matrix Cosine and Sine

Hargreaves, Gareth I. and Higham, Nicholas J. (2005) Efficient Algorithms for the Matrix Cosine and Sine. Numerical Algorithms, 40 (4). pp. 383-400. ISSN 1572-9265

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## Abstract

Several improvements are made to an algorithm of Higham and Smith for computing the matrix cosine. The original algorithm scales the matrix by a power of 2 to bring the $\infty$-norm to 1 or less, evaluates the [8/8] Pad\'e approximant, then uses the double-angle formula $\cos(2A) = 2\cos^2A - I$ to recover the cosine of the original matrix. The first improvement is to phrase truncation error bounds in terms of $\norm{A^2}^{1/2}$ instead of the (no smaller and potentially much larger quantity) $\norm{A}$. The second is to choose the degree of the Pad\'e approximant to minimize the computational cost subject to achieving a desired truncation error. A third improvement is to use an absolute, rather than relative, error criterion in the choice of Pad\'e approximant; this allows the use of higher degree approximants without worsening an a priori error bound. Our theory and experiments show that each of these modifications brings a reduction in computational cost. Moreover, because the modifications tend to reduce the number of double-angle steps they usually result in a more accurate computed cosine in floating point arithmetic. We also derive an algorithm for computing both $\cos(A)$ and $\sin(A)$, by adapting the ideas developed for the cosine and intertwining the cosine and sine double angle recurrences.

Item Type: Article The original publication is available at www.springerlink.com matrix function, matrix cosine, matrix sine, matrix exponential, Taylor series, Pad\'e approximation, Pad\'e approximant, double-angle formula, rounding error analysis, Schur--Parlett method, MATLAB MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Nick Higham 15 Dec 2005 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/124

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