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

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