11-05:Python 插值

1.函数式

import numpy as np

'''
(X,Y):需要插值的一系列点，array数据类型
point:需要计算插值的点
'''

def _Lagrange(X,Y,point):
    '''挑战一行代码实现Lagrange插值'''
    return np.sum([(((np.prod(point-X)/(point-X[i]))/np.prod(np.delete((X[i]-X),i)))*Y[i]) for i in range(len(X)) if point != X[i]])

def Lagrange(X,Y,point):
    '''正常写法'''
    result = 0.0
    for i in range(len(X)):
        if point != X[i]:
            Li = (np.prod(point-X)/(point-X[i])) / np.prod(np.delete((X[i]-X),i))
            result = result + Li*Y[i]
        else:
            result = Y[i]
    return result

def Difference(X,Y):
    '''差分函数'''
    return np.sum([Y[i]/np.prod(np.delete((X[i]-X),i)) for i in range(len(X))])

def Newton(X,Y,point):
    '''牛顿插值'''
    result = Y[0]
    for i in range(1,len(X)):
        result = result + Difference(X[:i+1],Y[:i+1])*np.prod(point-X[:i])
    return result

X = np.arange(10)
Y = X**2
point = 3.5
print('真实值 ',point**2)
print('拉格朗日插值1：',_Lagrange(X,Y,point))
print('拉格朗日插值2：',Lagrange(X,Y,point))
print('牛顿插值',Newton(X,Y,point))

2.面向对象式

import numpy as np
class Interpolation(object):
    def __init__(self,X,Y,point):
        '''
        @author:zengwei
        (X,Y):给定的用于插值的点
        point:需要用插值函数计算值的点
        '''
        self.X = np.array(X)
        self.Y = Y
        self.point = point
        
    def _Lagrange(self):
        '''挑战一行代码实现Lagrange插值'''
        return np.sum([(((np.prod(self.point-self.X)/(self.point-self.X[i]))/np.prod(np.delete((self.X[i]-self.X),i)))*self.Y[i]) for i in range(len(self.X)) if self.point!=self.X[i]])
        
    def Lagrange(self):
        '''Lagrange插值的正常写法'''
        result = 0.0
        for i in range(len(self.X)):
            if self.point != self.X[i]:
                Li = (np.prod(self.point-self.X)/(self.point-self.X[i]))/np.prod(np.delete((self.X[i]-self.X),i))
                result = result + Li*self.Y[i]
            else:
                result = self.Y[i]
        return result
    
    def Difference(X,Y):
        return np.sum([Y[i]/np.prod(np.delete((X[i]-X),i)) for i in range(len(X))])
    
    def Newton(self):
        '''牛顿插值'''
        result = self.Y[0]
        for i in range(1,len(self.X)):
            result = result + Interpolation.Difference(self.X[:i+1],self.Y[:i+1])*np.prod(self.point-self.X[:i])
        return result
        
    def Hermite(self):
        '''Hermite插值'''
        pass
    
    def CubicSpline(self):
        '''三次样条插值'''
        pass

X = np.arange(10)
Y = X**2
point = 3.5
myop = Interpolation(X,Y,point)
print('真实值 ',point**2)
print('拉格朗日插值1：',myop._Lagrange())
print('拉格朗日插值2：',myop.Lagrange())
print('牛顿插值',myop.Newton())
print('计算差分',Interpolation.Difference(X,Y))