对非线性方程(组)简单迭代法的一种新改进

A New Method of Improvement on the Simple Iterative for Nonlinear Equation(s)

在使用简单迭代法解非线性方程(组)时,要求迭代函数f(x)(F(x))必须满足q=supx∈D|f′(x)|<1(q′=supx∈D‖F′(x)‖<1)。如将迭代函数f(x)导数的最大模(F(x)的Jacobi矩阵最大范数)超出上述取值区间情况下的迭代函数f(x)(F(x))进行一系列恒等变形,建立一个新的迭代函数,让其导数的最大模(Jacobi矩阵最大范数)落在上述取值区间内,再运用压缩映射原理逐步逼近求出非线性方程(组)的近似解。这是一种新的改进,有更广的应用范围。两个数值计算实例表明,恒等变形得到这种新的迭代序列收敛,该方法可行。

In the application of simple iterative method to the solution of nonlinear equation（s）, the iterative function off（x）（F（x）） is required to meet the condition of, q = sup x∈D[ f′（x） ] 〈 1 （q′ = sup ｜｜ F′（x） ｜｜ 〈 1 ） This paper suggests an improved method for simple iterative method, that is, to carry out a sequence of identity transformation in iterative function f（x）（F（x）） on the condition that the maximum modulus of derivative of iterative function f（x） （maximum norms of Jacobi matrix of F（x）） exceeds the above range of values, and thus set up a new iterative function and its maximum modulus of derivative（maximum norms of Jacobi matrix） is to be fallen into the above range of values. Then, the approximate solution of nonlinear equation（s） is to be iterative approximated by means of the compression mapping principle convergence . As a result, this simple iterative method will be put into wider use. These two numberical computation experiments will ultimately proved that this kind of new iterative sequence via identity transformation is convergent and this new method is therefore feasible.