龙格现象及分段线性插值基于python实现

拉格朗日插值的龙格现象

1. 依据x的值以及所对应的y的值，分别求解出不同的插值点个数n所对应的格朗日插值多项式，并将函数图像可视化。
2. 可视化原函数图像。

3.与原函数的图像相比较，当插值节点数n增加时，由于插值多项式的次数也随之增加，拉格朗日插值多项式逼近函数的效果随之下降，插值函数相较于原函数更加失真，振荡现象严重。

 

 

代码 

import numpy as np
import matplotlib.pyplot as plt


def Lagrange(arr_x, arr_y, _x):
    l = [0 for j in range(len(arr_x))]
    result = 0
    for i in range(0, len(arr_x)):
        denominator = 1
        molecular = 1
        for j in range(0, len(arr_x)):
            if i != j:
                denominator = denominator * (arr_x[i] - arr_x[j])
                molecular = molecular * (_x - arr_x[j])
        l[i] = molecular / denominator
        result = result + l[i] * arr_y[i]
    return result


original_x = np.arange(-5.0, 5.01, 0.01)
original_y = [0.0 for j in range(len(original_x))]
for i in range(len(original_y)):
    original_y[i] = 1 / (1 + original_x[i] * original_x[i])

x_arr = np.arange(-5.0, 5.5, 1)
y_arr = [0.0 for i in range(len(x_arr))]

for i in range(len(x_arr)):
    y_arr[i] = 1 / (1 + x_arr[i] ** 2)
x = np.arange(-5.0, 5.01, 0.01)
y = [0.0 for j in range(len(x))]
for i in range(len(y)):
    y[i] = Lagrange(x_arr, y_arr, x[i])

plt.plot(original_x, original_y, label='f(x) = 1 / (1 + x2)')
plt.scatter(x_arr, y_arr, label='The interpolation points')
plt.plot(x, y, label='Lagrange interpolation')
plt.plot([-5.5, 5.5], [0, 0], linestyle='--')
plt.plot([0, 0], [-0.5, 2], linestyle='--')
plt.plot(x_arr, y_arr, linestyle='--', label='piecewise linear interpolation')

plt.title("Runge phenomenon, piecewise linear interpolation")
plt.legend(loc="lower left")
plt.xlabel("x")
plt.ylabel("y")
plt.show()