The dual Padé families of iterations for the matrix pth root and the matrix p-sector function

摘要

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 pth root, Numer. Linear Algebra Appl. 16 (2009) 951–970] and a new dual Padé family of iterations for computing the principal pth 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 p-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.We show a connection of the series expansion with respect to B of the iterates, generated by iterations of the dual Padé family for computing the matrix pth root (I−B)1/p, with binomial scalar expansion of (1−b)1/p.