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{\Large ON THE ITERATIVE POSITIVE DEFINITE}\\
{\Large SOLUTION OF THE NONLINEAR MATRIX EQUATIONS}
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{\large Salah M. El-Sayed}\\\
{Department of Mathematics, Faculty of Science,}

{Benha University, Benha 13518, Egypt.}\\
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{\bf Abstract.} This paper is concerned with iterative positive definite
solution of the matrix $X+A^{\star}{\cal F}(X)A =I$, where
${\cal F}(X)=\sqrt[2^{m}]X$.
Theorems for the necessary and sufficient conditions of the existence a
positive solution are formulated and proved. The rate of convergence of
the iterative sequence of approximate
solution and stop criteria are obtained. Some numerical results are given
to illustrate the performance of the algorithm.



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