牛顿插值多项式（python实现）

理论知识

牛顿插值多项式（理论知识）

目标函数

f(x)=11+x2$$f(x)=\frac{1}{1+x^2}$$

插值点为[-10, 10]上的整数点。

图片

代码实现

import sympy
import numpy as np
from matplotlib import pyplot as plt


def f(X):
    return 1 / (X ** 2 + 1)


def ff(X=list()):
    if len(X) < 2:
        raise ValueError('X\'s length must be bigger than 2')
    ans = 0
    for i in range(len(X)):
        temp = 1.0
        for j in range(len(X)):
            if j == i:
                continue
            temp *= (X[i] - X[j])
        ans += (f(X[i]) / temp)
    return ans


def draw():
    plt.rcParams['font.sans-serif'] = ['SimHei']
    plt.rcParams['axes.unicode_minus'] = False
    X = np.linspace(-10, 10, 100)

    TargetY = f(X)
    GetY = [Px.subs(x, i) for i in X]

    plt.plot(X, TargetY, label=r'$\frac{1}{x^2+1}$')
    plt.plot(X, GetY, label='$L(x)$')
    plt.legend()
    plt.show()


def generatePx(DataX):
    ans = f(DataX[0])
    if len(DataX) == 1:
        return ans
    else:
        temp = 1
        for i in range(len(DataX) - 1):
            temp *= (x - DataX[i])
            ans += ff(DataX[:i + 2]) * temp
        return ans


if __name__ == '__main__':
    x = sympy.symbols('x')

    DataX = np.linspace(-10, 10, 11)  # 插值点

    Px = sympy.expand(generatePx(DataX))
    draw()