On iterative algorithms for the polar decomposition of a matrix and the matrix sign function

In this paper we consider relations between the principal iterations from the Padé family of Kenney and Laub for computing the matrix sign function and the principal iterations from the reciprocal Padé family of Greco, Iannazzo and Poloni, and the dual Padé family of Ziȩtak. We show global convergence of the principal reciprocal Padé iterations and the principal dual Padé iterations.We adopt the dual Padé family of iterations, to which the Newton method belongs, for computing the unitary polar factor of a nonsingular matrix. We present numerical experiments with the scaled Newton method of Higham for computing the unitary polar factor, which show how the quality of the computed inverses of matrices in the scaled Newton method affects the accuracy of the computed polar factorization of a nonsingular matrix. It indicates that assumptions, under which the backward stability of the scaled Newton method has been proved in the literature, cannot be weaker.