一类边值问题的有限差分牛顿型方法

A Finite-Difference Newton-Like Method for a Class of Boundary Value Problems

本文利用微分方程的非线性差分格式的特殊结构,提出了一种新的牛顿型方法求解非线性差分方程。若新方法每步不附加计算非线性方程组的函数值,那么新算法收敛速度可达到R-((1+5^(1/2))/2)阶;若新方法每步附加计算一个非线性方程组的向量函数值,那么新算法收敛速度可达到Q-平方阶。

In this paper, we present a Newton-like method (modified Newton method and modified secant method), which explores the special structure of the finite-difference approximation(yi+1 -2yi + yi-1)/(h2) = f(ti,yi), i = 1,…,N,y0 =α, yN+1 =β, to the boundary value problemsy' = f(t,y), t∈[a,b],y(a) =α, y(b) =β.At each iteration, modified secant method only computes one function vector (i.e., no additional cast in function evaluations), and it has a R-(1+5^(1/2))/2convergence rate; and modifiedNewton method only calls two function vectors, and it has a Q-quadratic convergence rate. At last, our numerical results show the new methods are very effective.