数值分析之 拉格朗日插值、牛顿插值、分段线性插值实现

1、拉格朗日插值法

考虑全局信息的比较经典的插值方法，编程简单，计算量大。

#coding=utf-8
from matplotlib import pyplot as plt

def Lg(data,testdata):
    predict=0
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    if testdata in data_x:
        #print "testdata is already known"
        return data_y[data_x.index(testdata)]
    for i in range(len(data_x)):
        af=1
        for j in range(len(data_x)):
            if j!=i:
                af*=(1.0*(testdata-data_x[j])/(data_x[i]-data_x[j]))
        predict+=data_y[i]*af
    return predict

def plot(data,nums):
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    Area=[min(data_x),max(data_x)]
    X=[Area[0]+1.0*i*(Area[1]-Area[0])/nums for i in range(nums)]
    X[len(X)-1]=Area[1]
    Y=[Lg(data,x) for x in X]
    plt.plot(X,Y,label='result')
    for i in range(len(data_x)):
        plt.plot(data_x[i],data_y[i],'ro',label="point")
    plt.savefig('Lg.jpg')
    plt.show()

data=[[0,0],[1,2],[2,3],[3,8],[4,2]]
print Lg(data,1.5)
plot(data,100)

2、牛顿插值

相比于拉格朗日的优势在于 牛顿插值法在重新引入数据时可以仅仅更新F矩阵进行。（手动维护差商表的话，效率提高很多，但对计算机而言，觉得差不多）。复杂度和拉格朗日一样，O(n2)。

#coding=utf-8
from matplotlib import pyplot as plt

def calF(data):
    #差商计算  n个数据 0-(n-1)阶个差商 n个数据
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    F= [1 for i in range(len(data))]   
    FM=[]
    for i in range(len(data)):
        FME=[]
        if i==0:
            FME=data_y
        else:
            for j in range(len(FM[len(FM)-1])-1):
                delta=data_x[i+j]-data_x[j]
                value=1.0*(FM[len(FM)-1][j+1]-FM[len(FM)-1][j])/delta
                FME.append(value)
        FM.append(FME)
    F=[fme[0] for fme in FM]
    print FM
    return F

def NT(data,testdata,F):
    #差商之类的计算
    predict=0
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    if testdata in data_x:
        return data_y[data_x.index(testdata)]
    else:
        for i in range(len(data_x)):
            Eq=1
            if i!=0:
                for j in range(i):
                    Eq=Eq*(testdata-data_x[j])
            predict+=(F[i]*Eq)
        return predict

def plot(data,nums):
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    Area=[min(data_x),max(data_x)]
    X=[Area[0]+1.0*i*(Area[1]-Area[0])/nums for i in range(nums)]
    X[len(X)-1]=Area[1]
    F=calF(data)
    Y=[NT(data,x,F) for x in X]
    plt.plot(X,Y,label='result')
    for i in range(len(data_x)):
        plt.plot(data_x[i],data_y[i],'ro',label="point")
    plt.savefig('Newton.jpg')
    plt.show()

data=[[0,0],[1,2],[2,3],[3,8],[4,2]]

plot(data,100)

3、分段线性插值

最简单，工业使用最多。

#coding=utf-8
from matplotlib import pyplot as plt
def DivideLine(data,testdata):
    #找到最邻近的
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    if testdata in data_x:
        return data_y[data_x.index(testdata)]
    else:
        index=0
        for j in range(len(data_x)):
            if data_x[j]and  data_x[j+1]>testdata:
                index=j
                break
        predict=1.0*(testdata-data_x[j])*(data_y[j+1]-data_y[j])/(data_x[j+1]-data_x[j])+data_y[j]
        return predict

def plot(data,nums):
    data_x=[data[i][0] for i in range(len(data))]
    data_y=[data[i][1] for i in range(len(data))]
    Area=[min(data_x),max(data_x)]
    X=[Area[0]+1.0*i*(Area[1]-Area[0])/nums for i in range(nums)]
    X[len(X)-1]=Area[1]
    Y=[DivideLine(data,x) for x in X]
    plt.plot(X,Y,label='result')
    for i in range(len(data_x)):
        plt.plot(data_x[i],data_y[i],'ro',label="point")
    plt.savefig('DivLine.jpg')
    plt.show()

data=[[0,0],[1,2],[2,3],[3,8],[4,2]]
print DivideLine(data,1.5)
plot(data,100)


![结果](http://img.blog.csdn.net/20171029205027016?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvc2luYXRfMzM4Mjk4MDY=/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast)