, Volume 34, Issue 1, pp 1326
First online:
Computing the Matrix Cosine
 Nicholas J. HighamAffiliated withDepartment of Mathematics, University of Manchester
 , Matthew I. SmithAffiliated withDepartment of Mathematics, University of Manchester
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An algorithm is developed for computing the matrix cosine, building on a proposal of Serbin and Blalock. The algorithm scales the matrix by a power of 2 to make the ∞norm less than or equal to 1, evaluates a Padé approximant, and then uses the double angle formula cos (2A)=2cos (A)^{2}−I to recover the cosine of the original matrix. In addition, argument reduction and balancing is used initially to decrease the norm. We give truncation and rounding error analyses to show that an [8,8] Padé approximant produces the cosine of the scaled matrix correct to machine accuracy in IEEE double precision arithmetic, and we show that this Padé approximant can be more efficiently evaluated than a corresponding Taylor series approximation. We also provide error analysis to bound the propagation of errors in the double angle recurrence. Numerical experiments show that our algorithm is competitive in accuracy with the Schur–Parlett method of Davies and Higham, which is designed for general matrix functions, and it is substantially less expensive than that method for matrices of ∞norm of order 1. The dominant computational kernels in the algorithm are matrix multiplication and solution of a linear system with multiple righthand sides, so the algorithm is well suited to modern computer architectures.
 Title
 Computing the Matrix Cosine
 Journal

Numerical Algorithms
Volume 34, Issue 1 , pp 1326
 Cover Date
 200309
 DOI
 10.1023/A:1026152731904
 Print ISSN
 10171398
 Online ISSN
 15729265
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

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

 Nicholas J. Higham ^{(1)}
 Matthew I. Smith ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of Manchester, Manchester, M13 9PL, England e