Beata Laszkiewicz, Krystyna Zietak, 
"Algorithms for the matrix sector function"

In this paper we consider   algorithms for the matrix sector function, which is 
a generalization of the matrix sign function. We develop    algorithms for computing
 the matrix sector function  based on  the (real)  Schur decompositions 
with and without reordering  and  the Parlett recurrence. We prove some results 
on convergence regions for the specialized  versions of the Newton and Halley methods 
applied to the matrix sector function, using  recent results of   Iannazzo 
for the principal matrix $p$th root.

Numerical experiments comparing the properties of  algorithms developed in this paper 
illustrate the differences in the  behaviour of the algorithms. We consider 
the conditioning of the matrix sector function  and the stability of the Newton 
and Halley methods.  We also prove a characterization of the Frechet derivative 
of  the matrix sector function,  which is  a generalization  of   the result 
of Kenney and Laub  for the Frechet derivative of the matrix sign function, 
and we provide  a way of computing it by the  Newton iteration.