数值计算 利用Python实现牛顿插值法(不调用函数库)

牛顿插值法介绍

牛顿插值公式:

已知对于给定节点$x_0,x_1,...,x_n$，对应函数值$f(x_0),f(x_1),...,f(x_n)$，

则有牛顿插值公式:

$$P_n(x)=f(x_0)+f[x_0,x_1]\cdot (x-x_0)+...+f[x_0,x_1,...,x_n]\cdot (x-x_0)\cdot (x-x_1)\cdot ...\cdot (x-x_n)$$

其中，$f[x_0,x_1,...x_n]$为$n$阶均差。

实现代码

# newton.py
import matplotlib.pyplot as plt
from pylab import mpl
import numpy as np


def newton_dd(x: list, f: list, n: int):  # 此处 n表示多项式阶数，为节点数-1
    dd = [0 for _ in range(0, n)]  # divided difference

    for i in range(0, n):
        for j in range(n-1, i-1, -1):  # n-1,n-2,...,i 必须从后往前
            if i == 0:
                dd[j] = (f[j + 1] - f[j]) / (x[j + 1] - x[j])
            else:
                dd[j] = (dd[j] - dd[j - 1]) / (x[j + 1] - x[j - i])
    return dd


def get_function(data: np.ndarray, x: list, f: list, n: int):
    parm = newton_dd(x, f, n)
    # print(parm)
    res = f[0]
    w = 1
    for i in range(0, n):
        w *= data - x[i]
        res += parm[i] * w

    return res


def draw(data: np.ndarray, x: list, f: list, n: int):
    plt.plot(data, get_function(data, x, f, n), label="牛顿插值拟合曲线", color="red")
    plt.scatter(x, f, label="节点数据", color="red")
    plt.title("%s次牛顿插值法结果"%n)
    mpl.rcParams['font.sans-serif'] = ['SimHei']
    mpl.rcParams['axes.unicode_minus'] = False
    plt.legend(loc="upper right")

使用实例

考虑龙格函数：
$$f(x)=1/(1+25x^2)$$
在$[-1,1]$上取11个等距节点，对这些节点进行牛顿插值。

import numpy as np
import newton as nt # 这是上文的代码文件
import matplotlib.pyplot as plt



def draw(data: np.ndarray, x: list, f:list, n: int):
    nt.draw(data, x, f, n-1)

    x_= list(np.arange(-1, 1.01, 0.01))
    f_= [1 / (1 + 25*pow(num, 2)) for num in x_]
    plt.plot(x_, f_, label="龙格函数", color="black")

    plt.show()


if __name__ == "__main__":
    n = 11
    x = list(np.arange(-1, 1 + 2 / (n - 1), 2 / (n - 1)))
    f = [1 / (1 + 25*pow(num, 2)) for num in x]
    draw(np.arange(-1, 1.01, 0.01), x, f, n)

运行得到结果: