TY - JOUR
T1 - On Eigenvalue Bounds and Iteration Methods for Discrete Algebraic Riccati Equations
AU - Hua Dai & Zhong-Zhi Bai
JO - Journal of Computational Mathematics
VL - 3
SP - 341
EP - 366
PY - 2011
DA - 2011/06
SN - 29
DO - http://doi.org/10.4208/jcm.1010-m3258
UR - https://global-sci.org/intro/article_detail/jcm/8482.html
KW - Discrete algebraic Riccati equation, Symmetric positive definite solution,
Eigenvalue bound, Fixed-point iteration, Convergence theory.
AB - <p style="text-align: justify;">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<span style="text-align: justify;">&nbsp;not</span> been 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 computing 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.</p>