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MODIFIED BERNOULLI ITERATION METHODS FOR QUADRATIC MATRIX EQUATION

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.

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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.

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维普期刊数据库

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Brown-Broyden修正算法

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高等学校计算数学学报 2001年第4期23卷 368-377页

作者：王德人王新力上海大学数学系上海大学数学系上海201800

In this paper, we proposed Brown-Broyden update algorithm for solving systems of nonlinear equations. We prove the local superlinear convergence.of the proposed algorithm. Numerical experiments show the superiority of... 详细信息

关键词： systems of nonlinear equations Brown method Broyden method local convergence.

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