Local convergence analysis of conjugate gradient methods for solving algebraic Riccati equations

Abstract

Necessary and sufficient conditions are given for local convergence of the conjugate gradient (CG) method for solving symmetric and nonsymmetric algebraic Riccati equations. For these problems, the Frobenius norm of the residual matrix is minimized via the CG method, and convergence in a neighborhood of the solution is predicated on the positive definiteness of the associated Hessian matrix. For the nonsymmetric case, the Hessian eigenvalues are determined by the squares of the singular values of the closed-loop Sylvester operator. In the symmetric case, the Hessian eigenvalues are closely related to the squares of the closed-loop Lyapunov singular values. In particular, the Hessian is positive definite if and only if the associated operator is nonsingular. The invertibility of these operators can be expressed as a noncancellation condition on the eigenvalues of the closed-loop matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>