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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (1): 29-38.doi: 10.3969/j.issn.1005-3085.2015.01.004

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一类非线性代数方程组的Newton-Triangle Splitting迭代法

胡纪洋1,  王川龙2,  温瑞萍2   

  1. 1- 太原理工大学数学学院，太原  030024
    2- 太原师范学院工程科学计算山西省高等学校重点实验室，太原 030012
  • 收稿日期:2013-07-01 接受日期:2014-01-03 出版日期:2015-02-15 发布日期:2015-04-15
  • 基金资助:
    国家自然科学基金 (11371275; 11071184)；山西省自然科学基金 (2010011006; 2012011015-6).

On Newton-Triangle Splitting Methods for the Systems of Nonlinear Algebraic Equations

HU Ji-yang1,  WANG Chuan-long2,  WEN Rui-ping2   

  1. 1- Institution of Mathematics, Taiyuan University of Technology, Taiyuan 030024
    2- Higher Education Key Laboratory of Engineering Science Computing in Shanxi Province, Taiyuan Normal University, Taiyuan 030012
  • Received:2013-07-01 Accepted:2014-01-03 Online:2015-02-15 Published:2015-04-15
  • Supported by:
    The National Natural Science Foundation of China (11371275; 11071184); the Natural Science Foundation of Shanxi Province (2010011006; 2012011015-6).

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摘要： Triangle Splitting迭代方法是求解大型稀疏非Hermitian正定线性代数方程组的一种有效迭代算法．为了有效求解大型稀疏且Jacobi矩阵为非Hermitian正定的非线性代数方程组，本文将Triangle Splitting迭代方法作为不精确Newton方法的内迭代求解器，构造了不精确Newton-Triangle Splitting迭代方法．在适当的约束条件下，给出了该方法的两类局部收敛性定理．通过数值实验结果验证了该方法的可行性和有效性，并说明了该方法在计算时间和迭代次数方面比Newton-BTSS迭代方法更有优势.

关键词: Triangle Splitting迭代方法, 非线性代数方程组, 不精确Newton方法, 局部收敛性

Abstract:

The Triangle Splitting iteration method is an effective iteration method for solving large-scale sparse non-Hermitian positive definite system of linear algebraic equations. By making use of the Triangle Splitting iteration method on non-Hermitian positive definite matrices as the inner solver of the inexact Newton method, we establish a class of inexact Newton-Triangle Splitting iteration methods for solving the large-scale sparse system of nonlinear algebraic equa-tions with positive definite Jacobian matrices in the paper. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions. The numerical results are given to examine their feasibility and effectiveness. The numerical implementations also show that the Newton-Triangle Splitting methods have advantages over Newton-BTSS methods with less computation time and iteration steps.

Key words: Triangle Splitting iteration method, nonlinear algebraic equations, inexact Newton method, local convergence

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