Krystyna Zietak, 
The dual Pade families of iterations for the matrix p-th root and the matrix p-sector function



In the paper we consider the Pade family of iterations 
[B. Laszkiewicz, K. Zietak, A Pade family of iterations for the matrix sector function and the matrix p-th root, 
Numer. Linear Algebra Appl. 16 (2009), 951--970] and a new dual Pade family of iterations
for computing the principal p-th 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 Pade 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 Pade approximants to the function $(1-z)^{-1/p}$. 
 On this occasion we eliminate  a gap in the  proof of Theorem 2.6 in
[O. Gomilko, D.B.  Karp, M. Lin, K. Zietak, Regions of convergence of a Pade family of iterations
for the matrix sector function and the matrix p-th root, J. Comput. Appl. Math. 236 (2012), 4410--4420]. 

We show  a connection of the series expansion with respect to B of the iterates,
generated by iterations of the dual Pade family for computing
the matrix p-th root $(I-B)^{1/p}$, with binomial scalar expansion of $(1-b)^{1/p}$.
Guo   has shown it for  iterates of the Newton  method  and the Halley method 
in [Ch.-H. Guo, On Newton's method and Halley's method for the principal p-th root of a matrix,
Linear Algebra Appl. 432 (2010), 1905--1922].  


Keywords: matrix root; matrix sector function; matrix sign function;  Pade approximant; 
hypergeometric function;  rational matrix iteration;