Return to the Middle Ages: A Half-Angle Iteration for the Logarithm of a Unitary Matrix

If A is unitary then A = eiH = cos H + i sin H where H is Hermitian. We examine the problem of approximating H . The standard approach to approximating logarithms is to take successive square roots until A1/2 n is close to the identity and then apply a Padé approximation to log A = 2n log A1/2 n . For the unitary problem we have A1/2 n = cos(H/2n) + i sin(H/2n) which suggests the use of halfangle formulas in computing the square roots. In this paper we consider problems associated with incomplete DenmanBeavers square root approximations as applied to the halfangle formulation. Square roots of unitary matrices are again unitary and the desire to have approximate square roots retain this property leads us to a tangent formulation of the halfangle iteration. Numerical tests illustrate this new procedure on a variety of examples.