Efficient algorithms for the matrix cosine and sine
 Gareth I. Hargreaves,
 Nicholas J. Higham
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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 ∞norm to 1 or less, evaluates the [8/8] Padé approximant, then uses the doubleangle formula cos (2A)=2cos ^{2} A−I to recover the cosine of the original matrix. The first improvement is to phrase truncation error bounds in terms of ‖A^{2}‖^{1/2} instead of the (no smaller and potentially much larger quantity) ‖A‖. The second is to choose the degree of the Padé 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 Padé 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 doubleangle 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.
AMS subject classification
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 Title
 Efficient algorithms for the matrix cosine and sine
 Journal

Numerical Algorithms
Volume 40, Issue 4 , pp 383400
 Cover Date
 20051201
 DOI
 10.1007/s1107500581410
 Print ISSN
 10171398
 Online ISSN
 15729265
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 matrix function
 matrix cosine
 matrix sine
 matrix exponential
 Taylor series
 Padé approximation
 Padé approximant
 doubleangle formula
 rounding error analysis
 Schur–Parlett method
 MATLAB
 Industry Sectors
 Authors

 Gareth I. Hargreaves ^{(001)}
 Nicholas J. Higham ^{(001)}
 Author Affiliations

 001. School of Mathematics, University of Manchester, Sackville Street, Manchester, M60 1QD, UK