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Matrix inverse trigonometric and inverse hyperbolic functions: theory and algorithms. (English) Zbl 1388.15012

Summary: 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 the 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 Padé 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.

MSC:

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
65F60 Numerical computation of matrix exponential and similar matrix functions
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