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MODIFIED BERNOULLI ITERATION METHODS FOR QUADRATIC MATRIX EQUATION

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Journal of Computational Mathematics 2007年第5期25卷 498-511页

作者： zhong-zhi bai Yong-Hua Gao LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 China

We construct a modified Bernoulli iteration method for solving the quadratic matrix equation AX^2 ＋ BX ＋ C = 0, where A, B and C are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the ShermanMorrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.

关键词： Quadratic matrix equation Quadratic eigenvalue problem Solvent Bernoulli's iteration Newton's method Local convergence.

RELAXED ASYNCHRONOUS ITERATIONS FOR THE LINEAR COMPLEMENTARITY PROBLEM

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Journal of Computational Mathematics 2002年第1期20卷 97-112页

作者： zhong-zhi bai Yu-guang Huang LSEC ICMSECAcademy of Mathematics and Systems SciencesChinese Academy of SciencesBeijing 100080China Oxford University Computing Laboratory Wolfson BuildingParks RoadOxford OX1 3QDU.K.

Presents a class of relaxed asynchronous parallel multisplitting iterative methods for solving the linear complementarity problem on multiprocessor systems. Establishment of the methods; Convergence theories; Numerica... 详细信息

关键词： linear complementarity problem matrix multisplitting relaxation method asynchronous iteration convergence theory

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

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Journal of Computational Mathematics 2010年第6期28卷 745-766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 study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

关键词： Matrix-valued function Smooth LU decomposition Pivoting Nonlinear eigenvalue problem Multiple eigenvalue Newton method.

ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS

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

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.

ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES

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Journal of Computational Mathematics 2010年第2期28卷 235-260页

作者： zhong-zhi bai Xue-Ping Guo LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China Department of Mathematics East China Normal University Shanghai 200062 China

The Hermitian and skew-Hermitian splitting （HSS） method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.

关键词： Systems of nonlinear equations HSS iteration method Newton method Local convergence.

A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES

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Journal of Computational Mathematics 2006年第4期24卷 539-552页

作者： zhong-zhi bai Jun-feng Yin Yang-feng Su LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China Department of Mathematics Fudan University Shanghai 200433 China

A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.

关键词： Non-Hermitian positive definite matrix Matrix splitting Preconditioning Krylov subspace method Convergence.

A REGULARIZED CONJUGATE GRADIENT METHOD FOR SYMMETRIC POSITIVE DEFINITE SYSTEM OF LINEAR EQUATIONS

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Journal of Computational Mathematics 2002年第4期20卷 437-448页

作者： zhong-zhi bai Shao-liang Zhang LSEC ICMSECAcademy of Mathematics and System SciencesChinese Academy of SciencesBeijing 100080China Department of Applied Physics University of Tokyo7-3-1 HongoBunkyo-kuTokyo 113-8656Japan

A class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.

关键词： conjugate gradient method symmetric positive definite matrix regularization ill-conditioned linear system

BLOCK-TRIANGULAR PRECONDITIONERS FOR SYSTEMS ARISING FROM EDGE-PRESERVING IMAGE RESTORATION

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Journal of Computational Mathematics 2010年第6期28卷 848-863页

作者： zhong-zhi bai Yu-Mei Huang Michael K. Ng LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China School of Mathematics and Statistics Lanzhou University Lanzhou 730000 China Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong

Signal and image restoration problems are often solved by minimizing a cost function consisting of an l2 data-fidelity term and a regularization term. We consider a class of convex and edge-preserving regularization functions. In specific, half-quadratic regularization as a fixed-point iteration method is usually employed to solve this problem. The main aim of this paper is to solve the above-described signal and image restoration problems with the half-quadratic regularization technique by making use of the Newton method. At each iteration of the Newton method, the Newton equation is a structured system of linear equations of a symmetric positive definite coefficient matrix, and may be efficiently solved by the preconditioned conjugate gradient method accelerated with the modified block SSOR preconditioner. Our experimental results show that the modified block-SSOR preconditioned conjugate gradient method is feasible and effective for further improving the numerical performance of the half-quadratic regularization approach.

关键词： Block system of equations Matrix preconditioner Edge-preserving Image restoration Half-quadratic regularization.

A CLASS OF NEW PARALLEL HYBRID ALGEBRAIC MULTILEVEL ITERATIONS

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Journal of Computational Mathematics 2001年第6期19卷 651-672页

作者： zhong-zhi bai (LSEC ICMSEC, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China) 中科院计算数学与科学工程计算所北京 100080

Presents preconditioning matrices having parallel computing function for the coefficient matrix and a class of parallel hybrid algebraic multilevel iteration methods for solving linear equations. Solution to elliptic ... 详细信息

关键词： elliptic boundary value problem system of linear equations symmetric positive definite matrix multilevel iteration parallel method