郭巧 杨兵 吴昌广

【摘   要】   利用定积分几何意义，推导出经典牛顿法、算术平均牛顿法和调和平均牛顿法，结合Lagrange插值定义，提出了一类新的六阶收敛的平均值牛顿迭代法。该算法每次迭代只需要计算两个函数值和两个一阶导数值，有效避免对函数进行高阶求导。收敛性分析和数值实例进一步验证该算法在求解非线性方程迭代时比牛顿迭代法、算术平均牛顿法和调和平均牛顿法效率更高、速度更快。

【关键词】   Lagrange插值；非线性方程；定积分；牛顿迭代

An Improved Mean Newton Iteration Method with Sixth-order

Convergence Based on Lagrange Interpolation

Guo Qiao1， Yang Bing1*， Wu Changguang2

（1. Anhui Vocational and Technical College， Hefei 230611， China;

2. Nanjing University of Science And Technology， Nanjing 210000， China）

【Abstract】    Using the geometric meaning of definite integral， the classical Newton method， arithmetic mean Newton method and harmonic mean Newton method are deduced. Combined with the definition of Lagrange interpolation， a new mean Newton iteration method with sixth-order convergence is proposed. Each iteration of the algorithm only needs to calculate two function values and two first-order derivative values， which effectively avoids high-order derivatives of functions. Convergence analysis and numerical examples further verify that the algorithm is more efficient and faster than Newton iteration method， arithmetic mean Newton method and harmonic mean Newton method in solving nonlinear equations interactively.

【Key words】     Lagrange interpolation; nonlinear equations; definite integral; Newton iteration

〔中圖分类号〕  O241.3             〔文献标识码〕  A              〔文章编号〕 1674 - 3229（2023）01- 0008 - 05