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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 oftraditional approaches for solving this problem depends highly on the spectralradius $\rho(X_+^{-1}A)$. If $\rho(X_+^{-1}A)$ is close to one or equal to one,the iterations of traditional approaches converges very slowly or does notconverge. Our goal is to come up with a shifting tactic to remove thesingularities embedded in $\rho(X_+^{-1}A)$. Finally, an example is used todemonstrate the capacity of our method.

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