Rational Iterative Methods for the Matrix Sign Function

In this paper an analysis of rational iterations for the matrix sign function is presented. This analysis is based on Padé approximations of a certain hypergeometric function and it is shown that local convergence results for “multiplication-rich” polynomial iterations also apply to these rational methods. Multiplication-rich methods are of particular interest for many parallel and vector computing environments. The main diagonal Padé recursions, which include Newton’s and Halley’s methods as special cases, are globally convergent and can be implemented in a multiplication-rich fashion which is computationally competitive with the polynomial recursions (which are not globally convergent). Other rational iteration schemes are also discussed, including Laurent approximations, Cayley power methods, and globally convergent eigenvalue assignment methods.