一般条件下牛顿下降法的收敛性

Convergence of Newton- Decline Method Under Common Conditions

对于求解非线性方程f(x)=0,牛顿下降法xn+1=xn-ωnf′-1(xn)f(xn)是一种经典的迭代法,具有大范围收敛等优点,有必要研究其收敛条件。为了使其能够适应更多环境的需要,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},证明了此情形下牛顿下降法的收敛性。该条件可以表示为‖f′-1(x0)·L(u+‖x-x0‖)dx,而此条件f(x0)‖≤β,‖f′-1(x0)f″(x0)‖≤γ,‖f′-1(x0)(f″(x)-f″(y))‖≤∫‖x-y‖0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 x_(n+1)=x_n-ω_nf′^(-1)(x_n)f(x_n) is a traditional iterative method for solving nonlinear equation f(x)=0,and has big range of convergence.It is necessary to research its convergent conditions. To make it more meaningful in general, by choosing a common decline factor sequence｛ω_n｝ under a more common condition, the convergence of Newton-decline method was proved. This condition can be expressed as ‖f′^(-1)(x_0)f(x_0)‖≤β,‖f′^(-1)(x_0)f″(x_0)‖≤γ,‖f′^(-1)(x_0)(f″(x)-f″(y))‖≤∫^(‖x-y‖)_0L(u+‖x-x_0‖)dx, 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.