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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 ›› 2016, Vol. 33 ›› Issue (4): 382-390.doi: 10.3969/j.issn.1005-3085.2016.04.005

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非线性方程组自反解的非精确Newton-MCG算法

梁志艳,   张凯院,   宁倩芝   

  1. 西北工业大学应用数学系，西安 710072
  • 收稿日期:2015-03-18 接受日期:2015-11-30 出版日期:2016-08-15 发布日期:2016-10-15
  • 基金资助:
    国家自然科学基金 (11471262).

Inexact Newton-MCG Algorithm for Reflexive Solution of Nonlinear Algebraic Equations

LIANG Zhi-yan,  ZHANG Kai-yuan,  NING Qian-zhi   

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072
  • Received:2015-03-18 Accepted:2015-11-30 Online:2016-08-15 Published:2016-10-15
  • Supported by:
    The National Natural Science Foundation of China (11471262).

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摘要： 针对源于科学计算和工程应用领域的非线性代数方程组，本文应用Newton算法求其自反解，并采用修正共轭梯度法(MCG算法)求由Newton算法每一步迭代计算导出的线性代数方程组的近似自反解或其近似自反最小二乘解，建立了求其自反解的非精确Newton-MCG算法．基于MCG算法适用面宽和有限步收敛的特点，建立的非精确Newton-MCG算法仅要求非线性代数方程组有自反解，而不要求它的自反解唯一．数值算例表明，非精确Newton-MCG算法是有效的．

关键词: 非线性代数方程组, 自反解, Newton算法, MCG算法, 非精确Newton-MCG算法

Abstract:

Nonlinear algebraic equations have wide applications in scientific computation and engineering application. In this paper, the inexact Newton-MCG algorithm for computing the reflexive solution of the nonlinear algebraic equation is proposed. The algorithm is constructed based on the Newton method for calculating the reflexive solution of the nonlinear algebraic equations and the modified conjugate gradient method for the approximate reflexive solution or the approximate reflexive least-square solution of the linear algebraic equation derived from each Newton step. Moreover, the proposed algorithm only requires the nonlinear algebraic equation to have the reflexive solution and the solution may not be unique, owing to the wide scope of applications and the finite-step convergent property of the MCG method. Finally, some numerical experiments illustrate the efficiency of the new algorithm.

Key words: nonlinear algebraic equations, reflexive solution, Newton method, MCG method, inexact Newton-MCG algorithm

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