Fixed-Point Iteration Schemes to Solve Symmetric Algebraic Riccati Equation \(XBX-XA-A^{T}X-C=0\)

Nonlinear matrix equations have important applications in optimal control problems. This study introduces parametric iterative methods aimed at determining solutions for the symmetric algebraic Riccati equation (SARE) expressed as: \(\begin{aligned} S(X)=XBX-XA-A^{T}X-C=0. \end{aligned}\) $$\begin{array}{c}\hfill S(X)=XBX-XA-A^{T}X-C=0.\end{array}$$ These methods leverage fixed-point and weight-splitting schemes for computation. The study demonstrates the convergence of the proposed schemes to nonnegative solutions of the SARE in specific situations. Additionally, various numerical examples illustrate the effectiveness of these schemes in solving several instances of SAREs.