Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups

For any matrix automorphism group G associated with a bilinear or sesquilinear form, Mackey, Mackey, and Tisseur have recently shown that the matrix sign decomposition factors of A ∈ G also lie in G; moreover, the polar factors of A lie in G if the matrix of the underlying form is unitary. Groups satisfying the latter condition include the complex orthogonal, real and complex symplectic, and pseudo-orthogonal groups. This work is concerned with exploiting the structure of G when computing the polar and matrix sign decompositions of matrices in G. We give sufficient conditions for a matrix iteration to preserve the group structure and show that a family of globally convergent rational Pade-based iterations of Kenney and Laub satisfy these conditions. The well-known scaled Newton iteration for computing the unitary polar factor does not preserve group structure, but we show that the approach of the iterates to the group is precisely tethered to the approach to unitarity, and that this forces a different and exploitable structure in the iterates. A similar relation holds for the Newton iteration for the matrix sign function. We also prove that the number of iterations needed for convergence of the structure-preserving methods can be precisely predicted by running an associated scalar iteration. Numerical experiments are given to compare the cubically and quintically converging iterations with Newton's method and to test stopping criteria. The overall conclusion is that the structure-preserving iterations and the scaled Newton iteration are all of practical interest, and which iteration is to be preferred is problem-dependent.