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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.

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

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