郭巧 楊兵 吳昌廣

【摘? ?要】? ?利用定積分幾何意義，推導(dǎo)出經(jīng)典牛頓法、算術(shù)平均牛頓法和調(diào)和平均牛頓法，結(jié)合Lagrange插值定義，提出了一類新的六階收斂的平均值牛頓迭代法。該算法每次迭代只需要計算兩個函數(shù)值和兩個一階導(dǎo)數(shù)值，有效避免對函數(shù)進行高階求導(dǎo)。收斂性分析和數(shù)值實例進一步驗證該算法在求解非線性方程迭代時比牛頓迭代法、算術(shù)平均牛頓法和調(diào)和平均牛頓法效率更高、速度更快。

【關(guān)鍵詞】? ?Lagrange插值；非線性方程；定積分；牛頓迭代

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

〔中圖分類號〕? O241.3? ? ? ? ? ? ?〔文獻標(biāo)識碼〕? A ? ? ? ? ? ? ?〔文章編號〕 1674 - 3229（2023）01- 0008 - 05