分段埃尔米特插值Python实现并检查误差

函数

y=11+x2$$y=\frac{1}{1+x^2}$$

图像

可以看到，这里已经几乎没有任何差距了。。

代码

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


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


def cal(begin, end):
    by = f(begin)
    ey = f(end)
    Y = 1 / (1 + n ** 2)
    df = diff(Y)
    dfb = df.subs(n, begin)
    dfe = df.subs(n, end)

    oldFrac = ((n - end) / (begin - end))
    newFrac = ((n - begin) / (end - begin))

    I = (oldFrac ** 2) * (1 + 2 * newFrac) * by + (newFrac ** 2) * (1 + 2 * oldFrac) * ey + (oldFrac ** 2) * (
                n - begin) * dfb + (newFrac ** 2) * (n - end) * dfe
    return I


def calnf(x):
    nf = []
    for i in range(len(x) - 1):
        nf.append(cal(x[i], x[i + 1]))
    return nf


def calf(f, x):
    y = []
    for i in x:
        y.append(f.subs(n, i))
    return y


def nfSub(x, nf):
    tempx = np.array(range(11)) - 5
    dx = []
    for i in range(10):
        labelx = []
        for j in range(len(x)):
            if x[j] >= tempx[i] and x[j] < tempx[i + 1]:
                labelx.append(x[j])
            elif i == 9 and x[j] >= tempx[i] and x[j] <= tempx[i + 1]:
                labelx.append(x[j])
        dx = dx + calf(nf[i], labelx)
    return np.array(dx)


def draw(nf):
    plt.rcParams['font.sans-serif'] = ['SimHei']
    plt.rcParams['axes.unicode_minus'] = False
    x = np.linspace(-5, 5, 101)
    y = f(x)
    Ly = nfSub(x, nf)
    plt.plot(x, y, label='原函数')
    plt.plot(x, Ly, label='分段Hermite插值函数')
    plt.xlabel('x')
    plt.ylabel('y')
    plt.legend()

    plt.savefig('1.png')
    plt.show()


def lossCal(nf):
    x = np.linspace(-5, 5, 101)
    y = f(x)
    Ly = nfSub(x, nf)
    Ly = np.array(Ly)
    temp = Ly - y
    temp = abs(temp)
    print(temp.mean())


if __name__ == '__main__':
    x = np.array(range(11)) - 5
    y = f(x)

    n, m = symbols('n m')
    init_printing(use_unicode=True)

    nf = calnf(x)
    draw(nf)
    lossCal(nf)