On inexact Newton methods based on doubling iteration scheme for symmetric algebraic Riccati equations

Numerical methods for solving the symmetric algebraic Riccati equation by using Newton’s method are considered in this paper. Instead of direct methods, a fast doubling iteration scheme is applied to inexactly solve the Lyapunov equations arising in each Newton iteration. Then, a new inexact Newton method is proposed by using the Newton iteration as the outer iteration and the doubling iteration as the inner iteration. By controlling the inner iteration for each Newton iteration step, we prove the monotonicity and global convergence of the inexact Newton method. The efficiency of these methods are illustrated by several numerical examples.