Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative

Al-Mohy, Awad and Higham, Nicholas J. and Liu, Xiaobo (2021) Arbitrary Precision Algorithms for Computing the Matrix Cosine and its Fréchet Derivative. [MIMS Preprint]

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Abstract

Existing algorithms for computing the matrix cosine are tightly coupled to a specific precision of floating-point arithmetic for optimal efficiency so they do not conveniently extend to an arbitrary precision environment. We develop an algorithm for computing the matrix cosine that takes the unit roundoff of the working precision as input, and so works in an arbitrary precision. The algorithm employs a Taylor approximation with scaling and recovering and it can be used with a Schur decomposition or in a decomposition-free manner. We also derive a framework for computing the \fd, construct an efficient evaluation scheme for computing the cosine and its Fr\'echet derivative simultaneously in arbitrary precision, and show how this scheme can be extended to compute the matrix sine, cosine, and their \fd s all together. Numerical experiments show that the new algorithms behave in a forward stable way over a wide range of precisions. The transformation-free version of the algorithm for computing the cosine is competitive in accuracy with the state-of-the-art algorithms in double precision and surpasses existing alternatives in both speed and accuracy in working precisions higher than double.

Item Type: MIMS Preprint
Uncontrolled Keywords: multiprecision algorithm, multiprecision arithmetic, matrix cosine, matrix exponential, matrix function, Fréchet derivative double angle formula, Taylor approximation, forward error analysis, MATLAB
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: 16 Aug 2021 18:46
Last Modified: 16 Aug 2021 18:46
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2832

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