We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation $X = Q + LX^{-1}L^T$, where Q is symmetric positive definite and L is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly SZ algorithm to solve the DARE. This approach is compared to several fixed point type iterative methods suggested in the literature.

We study numerical methods for finding the maximal symmetric positive definite solution of the nonlinear matrix equation $X = Q + LX^{-1}L^T$, where Q is symmetric positive definite and L is nonsingular. Such equations arise for instance in the analysis of stationary Gaussian reciprocal processes over a finite interval. Its unique largest positive definite solution coincides with the unique positive definite solution of a related discrete-time algebraic Riccati equation (DARE). We discuss how to use the butterfly SZ algorithm to solve the DARE. This approach is compared to several fixed point type iterative methods suggested in the literature.