python实现牛顿插值法拟合曲线

牛顿插值法是一种多项式插值方法，它通过已知的数据点来构造一个n次多项式，从而拟合出一条曲线。下面是Python实现牛顿插值法拟合曲线的代码：

import numpy as np

def newton_interpolation(x, y):
    """
    牛顿插值法拟合曲线
    :param x: 数据点的横坐标
    :param y: 数据点的纵坐标
    :return: 返回牛顿插值法拟合的曲线函数
    """
    n = len(x)
    # 初始化差商表
    f = np.zeros((n, n))
    for i in range(n):
        f[i, 0] = y[i]
    # 构造差商表
    for i in range(1, n):
        for j in range(1, i+1):
            f[i, j] = (f[i, j-1] - f[i-1, j-1]) / (x[i] - x[i-j])
    # 构造插值多项式
    def poly(x_val):
        result = f[0, 0]
        for i in range(1, n):
            term = f[i, i]
            for j in range(i-1, -1, -1):
                term = term * (x_val - x[j]) + f[j, j]
            result += term
        return result
    return poly

测试代码：

import matplotlib.pyplot as plt

# 构造测试数据
x = np.linspace(0, 1, 11)
y = np.sin(x * np.pi)

# 拟合曲线
poly = newton_interpolation(x, y)

# 绘制拟合曲线
x_val = np.linspace(0, 1, 101)
y_val = poly(x_val)
plt.plot(x, y, 'ro', label='data')
plt.plot(x_val, y_val, label='interpolation')
plt.legend()
plt.show()

运行结果：