The Shifting Technique for Computing the Extreme Solutions of $X + A^\top X^{-1} A = Q$

We propose a new way for speeding up the search of the maximal solution $X_+$ of $X + A^\top X^{-1} A = Q$. It is known that the speed of convergence of traditional approaches for solving this problem depends highly on the spectral radius $ρ(X_+^{-1}A)$. If $ρ(X_+^{-1}A)$ is close to one or equal to one, the iterations of traditional approaches converges very slowly or does not converge. Our goal is to come up with a shifting tactic to remove the singularities embedded in $ρ(X_+^{-1}A)$. Finally, an example is used to demonstrate the capacity of our method.

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