Python插值法(3)————hermite插值多项式(拉格朗日型)

只讨论含一阶导数的情况.

代码

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


def hermite(x, y, dy):
    p, la = symbols('p  la')
    f = 0
    n = len(x)
    for i in range(n):
        la = 1
        lp = 0
        for j in range(n):
            if j != i:
                la = la*(p - x[j])/(x[i] - x[j])
                lp = lp + 1/(x[i] - x[j])
        temp1 = 1 - 2 * (p - x[i])*lp
        temp2 = y[i] * temp1 * la * la
        temp3 = dy[i] * (p - x[i]) * la * la
        f = f + temp2 + temp3
        f = simplify(f)
    return f


if __name__ == "__main__":
    x = [-1, 1, 2, 5, 10]
    y = [1, 1, 4, 25, 100]
    dy = [-2, 2, 4, 10, 20]
    p, la = symbols('p  la')
    s = hermite(x, y, dy)
    x1 = np.linspace(-1, 10, 50)
    x1 = list(x1)
    y1 = []
    for i in x1:
        y1.append(s.subs(p, i))
    plt.figure(figsize=(15, 8), dpi=80)
    plt.plot(x, y, 'r1', markersize=40, label='original')
    plt.plot(x1, y1, 'kD:', markersize=8, label='newton')
    plt.xlabel('x')
    plt.ylabel('y')
    plt.legend()
    plt.savefig('hermite.jpg')
    plt.show()

结果如下图所示