On Scaling Newton´s Method for Polar Decomposition and the Matrix Sign Function

A precise charcterization is given of the speed of convergence of the optimally scaled Newton method for the polar decomposition of a nonsingular complex matrix. The results are readily and practically implementable and give a sile way of bounding the number of steps required for a given degree of accuracy. For the matrix sign problem, optimal scaling requires complete knowledge of the eigenvalues of the original matrix. Because this is impractical, we examine spectral scaling, which is asymptotically optimal but slow when the eigenvalues are near the imaginary axis, and the more common determinantal scaling, which does well near the imaginary axis but is not asymptotically optimal. The complementary strengths of these two methods are combined into a new hybrid scaling strategy which is practical and nearly optimal.