Efficient computation of the extreme solutions of $X+A^*X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$

Abstract

We propose a new quadratically convergent algorithm, having a low computational cost per step and good numerical stability properties, which allows the simultaneous approximation of the extreme solutions of the matrix equations $X+A^* X^{-1}A=Q$ and $X-A^*X^{-1}A=Q$. The algorithm is based on the cyclic reduction method.