Newton's Method for Discrete Algebraic Riccati Equations when the Closed-Loop Matrix Has Eigenvalues on the Unit Circle

When Newton's method is applied to find the maximal symmetric solution of a discrete algebraic Riccati equation (DARE), convergence can be guaranteed under moderate conditions. In particular, the initial guess does not need to be close to the solution. The convergence is quadratic if the Fréchet derivative is invertible at the solution. When the closed-loop matrix has eigenvalues on the unit circle, the derivative at the solution is not invertible. The convergence of Newton's method is shown to be either quadratic or linear with the common ratio $\frac{1}{2}$, provided that the eigenvalues on the unit circle are all semisimple. The linear convergence appears to be dominant, and the efficiency of the Newton iteration can be improved significantly by applying a double Newton step at the right time.