利用优序列方法证明牛顿下降法在一般条件下的收敛性

Proving convergence of Newton-Decline method under common conditions by using dominating sequence method

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,利用优序列的方法,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)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 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,has big range of convergence. It is necessary to research its convergent conditions.To make it more meaningful in general,by using dominating sequence method and choosing a common decline factor sequence {ω_n} under a more common condition,this paper proves the convergence of Newton-Decline method.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‖)_(0)L(u+‖x-x_0‖)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.