POLAR DECOMPOSITION AND MATRIX SIGN FUNCTION CONDITION ESTIMATES

This paper presents reliable condition estimation procedures, based on Frechet derivatives, for polar decomposition and the matrix sign function. For polar decomposition, the condition number for complex matrices is equal to the reciprocal of the smallest singular value, and rather surprisingly, for real matrices it is equal to the reciprocal of the average of the two smallest singular values. By using inverse power methods, both of these condition numbers can be evaluated at a fraction of the cost of finding the polar decomposition. Except for special cases, such as for normal matrices, the condition number of the matrix sign function does not have such a precise characterization. However, accurate condition estimates can be obtained by using explicit forms of the Frechet derivative, or its finite-difference approximation, with a matricial inverse power method. These methods typically require two extra sign function evaluations, and it is an open problem whether accurate estimates can be obtained for a fraction of a function evaluation, as is the case for the polar decomposition. Related results for the stable Lyapunov equation and Newton's method for the matrix square root problem are discussed.