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

TL;DR

This paper introduces a shifting technique to accelerate the computation of the maximal solution to a nonlinear matrix equation, especially when traditional methods converge slowly due to spectral radius issues.

Contribution

A novel shifting approach is proposed to improve convergence speed for solving the matrix equation by addressing spectral radius limitations.

Findings

The method effectively speeds up convergence in challenging cases.

The approach removes singularities related to spectral radius.

An example demonstrates the method's practical capacity.

Abstract

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 $\rho(X_+^{-1}A)$. 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.