ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS

• Published in: Journal of computational mathematics 2011-05, Vol.29 (3), p.341-366
• Main Authors: Dai, Hua, Bai, Zhong-Zhi
• Format: Article
• Language: English
• Subjects: Algebra; Automats; Eigenvalues; Internships; Iterative solutions; Linear algebra; Mathematical inequalities; Matrix equations; Partial sums; Riccati equation; 代数Riccati方程; 单调收敛; 对称积; 特征值; 矩阵反演; 离散; 边界; 迭代方法
• ISSN: 0254-9409; 1991-7139
• DOI: 10.4208/jcm.1010-m3258

We derive new and tight bounds about the eigenvalues and certain sums of the eigenvalues for the unique symmetric positive definite solutions of the discrete algebraic Riccati equations. These bounds considerably improve the existing ones and treat the cases that have been not discussed in the literature. Besides, they also result in completions for the available bounds about the extremal eigenvalues and the traces of the solutions of the discrete algebraic Riccati equations. We study the fixed-point iteration methods for com- puting the symmetric positive definite solutions of the discrete algebraic Riccati equations and establish their general convergence theory. By making use of the Schulz iteration to partially avoid computing the matrix inversions, we present effective variants of the fixed-point iterations, prove their monotone convergence and estimate their asymptotic convergence rates. Numerical results show that the modified fixed-point iteration methods are feasible and effective solvers for computing the symmetric positive definite solutions of the discrete algebraic Riccati equations.