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ON eigenvalue boundS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS

Journal of Computational Mathematics 2011年第3期29卷 341-366页

作者： Hua Dai Zhong-Zhi Bai Department of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing 210016 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

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.

关键词： Discrete algebraic Riccati equation Symmetric positive definite solution eigenvalue bound Fixed-point iteration Convergence theory.

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The Relation Between a Tensor and Its Associated Semi-Symmetric Form

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Numerical Mathematics(Theory,Methods and Applications) 2022年第2期15卷 530-564页

作者： Hassan Bozorgmanesh Masoud Hajarian Anthony Theodore Chronopoulos Department of Applied Mathematics Faculty of Mathematical SciencesShahid Beheshti UniversityTehranIran Department of Computer Science University of TexasSan AntonioTexas 78249USA (Visiting Faculty)Department of Computer Engineering&Informatics University of PatrasRio 26500Greece

It is known that every tensor has an associated semi-symmetric *** purpose of this paper is to investigate the shared properties of a tensor and its semi-symmetric *** particular,a corresponding semi-symmetric tensor has smaller Frobenius norm under some conditions and can be used to get smaller bounds for eigenvalues and solutions of dynamical systems and tensor complementarity *** addition,every tensor has the same eigenvalues as its corresponding semi-symmetric form,also a corresponding semi-symmetric tensor inherits properties like being circulant,Toeplitz,Z-tensor,M-tensor,H-tensor and some ***,there are a two-way connection for properties like being positive definite,P-tensor,semi-positive,primitive and several others.

关键词： Tensor eigenvalue semi-symmetric tensor eigenvalue bound logarithmic norm tensor complementarity problem

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