Volume 29, Issue 3
On Eigenvalue Bounds and Iteration Methods for Discrete Algebraic Riccati Equations

Hua Dai & Zhong-Zhi Bai

DOI: 10.4208/jcm.1010-m3258

J. Comp. Math., 29 (2011), pp. 341-366

Published online: 2011-06

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  • Abstract

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 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.

  • Keywords

Discrete algebraic Riccati equation Symmetric positive definite solution Eigenvalue bound Fixed-point iteration Convergence theory

  • AMS Subject Headings

15A15 15A18 15A24 15A48 65F30.

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COPYRIGHT: © Global Science Press

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@Article{JCM-29-341, author = {Hua Dai and Zhong-Zhi Bai}, title = {On Eigenvalue Bounds and Iteration Methods for Discrete Algebraic Riccati Equations}, journal = {Journal of Computational Mathematics}, year = {2011}, volume = {29}, number = {3}, pages = {341--366}, abstract = { 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 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.}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1010-m3258}, url = {http://global-sci.org/intro/article_detail/jcm/8482.html} }
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 KW - Symmetric positive definite solution KW - Eigenvalue bound KW - Fixed-point iteration KW - Convergence theory AB - 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 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.
Hua Dai & Zhong-Zhi Bai. (1970). On Eigenvalue Bounds and Iteration Methods for Discrete Algebraic Riccati Equations. Journal of Computational Mathematics. 29 (3). 341-366. doi:10.4208/jcm.1010-m3258
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