Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms

Aprahamian, Mary and Higham, Nicholas J. (2016) Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms. [MIMS Preprint]

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Abstract

Theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and principal values $\acos$, $\asin$, $\acosh$, and $\asinh$ are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a ``round trip'' formula that relates $\mathrm{acos}(\cos A)$ to $A$ and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pad\'e approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas.

Item Type: MIMS Preprint
Uncontrolled Keywords: matrix function, inverse trigonometric functions, inverse hyperbolic functions, matrix inverse sine, matrix inverse cosine, matrix inverse hyperbolic sine, matrix inverse hyperbolic cosine, matrix exponential, matrix logarithm, matrix sign function, rational approximation, Pad\'{e} approximation, MATLAB, GNU Octave, Fr\'echet derivative, condition number
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: 20 Jan 2016
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2432

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