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\centerline {\Large\bf  Positive Definite Solutions }
\centerline {\Large\bf of the Equation $ X - A^* X^{-2} A = I $}

\vspace{0.4cm}
\centerline {\Large\bf Ivan G. Ivanov, Borislav V. Minchev, Vezhdi I.
Hasanov }
\vspace{0.4cm}

\rm
In this paper we study the positive solutions of the matrix equation
$ X - A^* X^{-2} A = I $. Necessary and
sufficient conditions for existence of positive definite solutions of this
equation are derived. Algorithms  for numerical computation of such
solutions are given.

Sufficient
conditions for convergence of the iterative methods
\begin{eqnarray}
 X_0 = \alpha I, \ \
 X_{k+1} &=& \sqrt {A (X_k - I)^{-1} A^*}, \ \ \ k=0,1,2, \ldots \ \ \ (
\alpha
> 2 )   \\
 X_0 = I,  \ \
 X_{k+1} &=& I + A^* X_k^{-2} A, \ \ \ k=0,1,2, \ldots
\end{eqnarray}
are presented.
If there exsists a real number  $\alpha$ for wich  the following
conditions are satisfied $$ A A^* <  \alpha^2(\alpha-1) I, \ \ \
 \sqrt {\frac{A A^*}{\alpha-1}} - \frac{1}{\alpha^2} A^* A > I $$
 or
$$ A A^* >  \alpha^2(\alpha-1) I, \ \ \
 \sqrt {\frac{A A^*}{\alpha-1}} - \frac{1}{\alpha^2} A^* A < I $$
then the matrix sequence obtained by formula
$(1)$ is convergent to a positive definite solution of the equation.
If the matrix $A$ is normal
$ ( A A^* =  A^* A )$ and the spectral norm  satisfies
$$ \|A\| < \sqrt {\frac{\sqrt{3}-1}{2}}, $$ then the matrix sequence
obtained by formula $(2)$ is convergent to a  positive definite solution of
the
equation.

Numerical experiments for computing of positive definite solutions
of the considered equation are described. The results from the
experiments show the effectiveness of the iterative methods for the described
examples.

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