O. Gomilko, F. Greco, K. Zietak, 
A Pade family of iterations for the matrix sign  function and related problem,
 Numerical Linear Algebra with Applications (2011)

In this paper we consider the Pade family of iterations for computing the matrix sign function
and the Pade family of iterations for computing the matrix p-sector function. We prove that all
the iterations of the Pade family for the matrix sign function have a common convergence region.
It completes a similar result of Kenney and Laub for half of the Pade family. 
We show that the iterations of the Pade family for the matrix p-sector function are well defined
in an analogous common region, depending on p. For this purpose we proved the Pade approximants
to the function $(1-z)^{-\sigma}, 0<\sigma<1$, are a quotient of hypergeometric functions
whose poles we have localized. Furthemore we proved that the coefficients of the power expansion
of a certain analytic function form a positive sequence and in a special case this sequence
has the log-concavity property.



key words: matrix sign function; matrix sector function; Pade family of iterations;
local convergence; Pade approximant; hypergeometric function; 
root of hypergeometric polynomial; reciprocal of power series;
log-concavity of sequence