AlMohy, Awad H. and Higham, Nicholas J. and Relton, Samuel D. (2014) New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously. [MIMS Preprint]
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Abstract
Several existing algorithms for computing the matrix cosine employ polynomial or rational approximations combined with scaling and use of a double angle formula. Their derivations are based on forward error bounds. We derive new algorithms for computing the matrix cosine, the matrix sine, and both simultaneously, that are backward stable in exact arithmetic and behave in a forward stable manner in floating point arithmetic. Our new algorithms employ both Pad\'e approximants of $\sin x$ and new rational approximants to $\cos x$ and $\sin x$ obtained from Pad\'e approximants to $e^x$. The amount of scaling and the degree of the approximants are chosen to minimize the computational cost subject to backward stability in exact arithmetic. Numerical experiments show that the new algorithms have backward and forward errors that rival or surpass those of existing algorithms and are particularly favorable for triangular matrices.
Item Type:  MIMS Preprint 

Additional Information:  To appear in SIAM Journal on Scientific Computing 
Uncontrolled Keywords:  matrix sine, matrix cosine, matrix exponential, matrix function, backward error, forward error, rational approximation, Pad\'{e} approximation, MATLAB, double angle formula, triple angle formula 
Subjects:  MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis 
Depositing User:  Nick Higham 
Date Deposited:  09 Jan 2015 
Last Modified:  08 Nov 2017 18:18 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2235 
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New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously. (deposited 20 Jun 2014)
 New Algorithms for Computing the Matrix Sine and Cosine Separately or Simultaneously. (deposited 09 Jan 2015) [Currently Displayed]
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