The dual Padé families of iterations for the matrix ppth root and the matrix pp-sector function

In this paper we consider the Padé family of iterations [B. Laszkiewicz, K. Ziȩtak, A Padé family of iterations for the matrix sector function and the matrix ppth root, Numer. Linear Algebra Appl. 16 (2009) 951–970] and a new dual Padé family of iterations for computing the principal ppth root of a matrix, including the Newton and Halley methods as particular cases. We prove convergence of iterations of these families in certain regions. We also propose a new dual Padé family of iterations for computing the matrix pp-sector function and we determine a certain region of convergence. For this purpose we study properties of the Padé approximants to the function (1−z)−1/p(1−z)−1/p.We show a connection of the series expansion with respect to BB of the iterates, generated by iterations of the dual Padé family for computing the matrix ppth root (I−B)1/p(I−B)1/p, with binomial scalar expansion of (1−b)1/p(1−b)1/p.