• Original Articles •

### ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS

Hua Dai1, Zhong-Zhi Bai2

1. 1. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China;
2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
• Received:2009-11-11 Revised:2010-09-14 Online:2011-05-15 Published:2011-05-15
• Supported by:

This work was started when the first author was visiting State Key Lab-oratory of Scientific/Engineering Computing, Chinese Academy of Sciences, during March-May in 2008. The support and hospitality from LSEC are very much appreciated. Supported by The National Basic Research Program (No. 2005CB321702), The China Outstanding Young Scien-tist Foundation (No. 10525102), and The National Natural Science Foundation for Innovative Research Groups (No. 11021101), P.R. China.

Hua Dai, Zhong-Zhi Bai. ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS[J]. Journal of Computational Mathematics, 2011, 29(3): 341-366.

We derive new and tight bounds about the eigenvalues and certain sums of the eigen-values 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.

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