分段埃尔米特插值Python实现并检查误差

港控/mmm° 2022-05-24 12:44 210阅读 0赞

## 函数 ##

y=11\+x2  y = 1 1 + x 2

## 图像 ##

![这里写图片描述][70]  
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)

[70]: /images/20220524/e22f50ba3af14e089f6f2b001ae0b78e.png