摘要
牛顿下降法xn+1=xn-ωnf′-1(xn)f(xn)是求解非线性方程f(x)=0的一种经典的迭代法,有必要研究其收敛条件,使其保持大范围收敛等优点.为了使其能够适应更多环境的需要,利用优序列方法,在一个更一般的条件下,选取了一个较为一般的下降因子序列{ωn},证明牛顿下降法的收敛性.该条件可表示为‖f′-1(x0)f(x0)‖≤β,‖f′-1(x0)f″(x0)‖≤γ,‖f′-1(x0)(f″(x)-f″(y))‖≤‖∫x-y‖0L(u+‖x-x0‖)du.而此条件比传统的Kantorovich型条件更具有一般的代表性,主要表现为不减的正的有界函数L(u)取值的灵活性,能够适应更多的环境.
Newton - Decline method xn＋1=xn-ωnf′^-1（xn）f（xn） is a traditional iterative method for solving nonlinear equation f（x） = 0 . It is necessary to research it＇s convergent conditions to keep it＇s big range of convergence. To make it more meaningful in general, by using dominating sequence method and choosing a common decline factor sequence {ωn} under a more common condition. This paper proves the convergence of Newton - Decline method. The condition can be expressed as ‖f′^-1（x0）f（x0）‖≤β,‖f′^-1（x0）f″（x0）‖≤γ,‖f′^-1（x0）（f″（x）-f″（y）‖≤∫^‖x-y‖ 0 L（u＋‖x-x0‖）du. While the condition has more common quality than traditional Kantorovich- kind conditions, mainly lying on the flexibility of the no reducible and positive function L（ u ） , and it can adapt to much more environments.
出处
《江西师范大学学报（自然科学版）》
CAS
北大核心
2005年第4期294-297,318,共5页
Journal of Jiangxi Normal University(Natural Science Edition)
基金
辽宁省自然科学基金资助项目(001084).
