一类自适应参数牛顿迭代法的构造与收敛性分析

Construction and Convergence Analysis of a Class of Adaptive Parameter Newton Iteration Methods

摘要: 本文提出了一类自适应参数的牛顿迭代法，利用所研究的函数构造自适应参数。首先，从$\left|f\left(x_n\right)\right|$ 的角度分析了其与步长的关系，从而构造出可调整步长的自适应参数${\lambda }_n=\frac{2}{1+\sqrt{1+\alpha {\left|f\left(x_n\right)\right|}^p}}$ ；其次，构造出自适应参数牛顿迭代法，通过证明$\left|f\left(x_{n+1}\right)\right|<\left|f\left(x_n\right)\right|$ 的成立，说明了每次迭代的有效性，通过分型图探讨了$\alpha$ 和$p$ 取不同值时，对收敛域的影响，并给出了自适应参数牛顿法算法流程；最后，通过收敛性分析数值实验说明了该算法的局部二阶收敛性和满足一定条件下的全局收敛性，同时利用Matlab验证了该算法相较于牛顿迭代法较广的应用性。

Abstract: This paper proposes a class of Newton iterative methods with adaptive parameters, where the adaptive parameters are constructed using the function under study. Firstly, from the perspective of $\left|f\left(x_n\right)\right|$ , the relationship between it and the step size is analyzed, thereby constructing an adaptive parameter ${\lambda }_n=\frac{2}{1+\sqrt{1+\alpha {\left|f\left(x_n\right)\right|}^p}}$ that can adjust the step size. Secondly, the Newton iterative method with adaptive parameters is constructed. The validity of each iteration is illustrated by proving the establishment of $\left|f\left(x_{n+1}\right)\right|<\left|f\left(x_n\right)\right|$ . The influence of different values of $\alpha$ and $p$ on the convergence domain is discussed through fractal diagrams, and the algorithm flow of the adaptive parameter Newton method is given. Finally, convergence analysis and numerical experiments show that the algorithm has local second-order convergence and global convergence under certain conditions. Meanwhile, Matlab is used to verify that the algorithm has wider applicability compared with the Newton iterative method.