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非线性方程组的Newton流线法

陈传淼, 胡宏伶, 雷蕾, 曾星星   

  1. 高性能计算与随机信息处理省部共建教育部重点实验室 湖南师范大学数学与计算机科学学院, 长沙 410081
  • 收稿日期:2011-06-26 出版日期:2012-08-15 发布日期:2012-08-16
  • 基金资助:

    国家自然科学基金(No.11071067)资助项目.

陈传淼, 胡宏伶, 雷蕾, 曾星星. 非线性方程组的Newton流线法[J]. 计算数学, 2012, 34(3): 235-258.

Chen Chuanmiao, Hu Hongling, Lei Lei, Zeng Xingxing. NEWTON FLOW METHOD FOR NONLINEAR SYSTEMS OF EQUATIONS[J]. Mathematica Numerica Sinica, 2012, 34(3): 235-258.

NEWTON FLOW METHOD FOR NONLINEAR SYSTEMS OF EQUATIONS

Chen Chuanmiao, Hu Hongling, Lei Lei, Zeng Xingxing   

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China
  • Received:2011-06-26 Online:2012-08-15 Published:2012-08-16
为求解非线性方程组F(x)=0, 研究了Newton流方程xt=V(x)=-(DF(x))-1F(x),x(0)=x0,及数值Newton流xj+1=xj+hV(xj),h∈(0,1].导出了减幅指标gj(h)=||F(xj+1)||/||F(xj)||=1-h+h2djh<1和m重根x*附近的表示gj(h)=(1-h/m)m+h2O(||xj-x*||).最后基于4个可计算量gj,dj,Kj,qj,提出了新的Newton流线法,如果投入大量的随机初始点, 能找到所有实根、重根和复根.
To solve nonlinear systems of equations F(x)=0, Newton’s flow equation xt=V(x)=-(DF(x))-1F(x),x(0)=x0, and its numerical flow xj+1=xj+hV(xj),h∈(0,1]. are studied. The damped index gj(h)=||F(xj+1)||/||F(xj)||=1-h+h2djh<1 and refine expression gj(h)=(1-h/m)m+h2O(||xj-x*||). near the m-ple root x? are derived. Finally based on fourth computable quantities gj,dj,Kj,qj, a new Newton flow algorithm is proposed, which can find all real, multiple and complex roots, if put into a large number of stochastic initial points.

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