问题：
给定若干个三维离散点，任意求出一个方程使得尽可能的将这些点包含在其中，并用python画出函数图像
思路：
最小二乘法，根据最优性表示出一个表达式E，为了让E最小，那么对任意一个系数求偏导后的值应该为0，列出若干条式子后，再用内置的高斯消元，求解系数。百度文库中有若干篇论文描述最小二乘法的实现方法。
代码：
%matplotlib inlinefrom matplotlib import pyplot as pltimport numpy as npfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib import cmdef fun(x): #处理符号问题round(x,2)if x>=0: return '+'+str(x)else: return str(x)#主函数if __name__ == '__main__':n=int(input("请输入一个n，代表点的个数"))#print(n)X=[]Y=[]Z=[]#读入n个点的坐标for i in range(n):a=[int(i) for i in input('please input: ').split()]X.append(a[0])Y.append(a[1])Z.append(a[2])#print(X) #print(Y)#print(Z)#fig = plt.figure() #建立一个新的图像#ax = Axes3D(fig) #建立一个三维坐标系#ax.scatter(X,Y,Z,c='b')#ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='rainbow')# plt.show()#求方程系数sigma_x = 0for i in X : sigma_x+=isigma_y = 0for i in Y : sigma_y+=isigma_z = 0for i in Z : sigma_z+=isigma_x2 = 0for i in X : sigma_x2+=i*isigma_y2 = 0for i in Y : sigma_y2+=i*isigma_x3 = 0for i in X : sigma_x3+=i*i*isigma_y3 = 0for i in Y : sigma_y3+=i*i*isigma_x4 = 0for i in X : sigma_x4+=i*i*i*isigma_y4 = 0for i in Y : sigma_y4+=i*i*i*isigma_x_y = 0for i in range(n) : sigma_x_y+=X[i]*Y[i]#print(sigma_xy)sigma_x_y2 = 0for i in range(n) : sigma_x_y2+=X[i]*Y[i]*Y[i]sigma_x_y3 = 0for i in range(n) : sigma_x_y3+=X[i]*Y[i]*Y[i]*Y[i]sigma_x2_y = 0for i in range(n) : sigma_x2_y+=X[i]*X[i]*Y[i]sigma_x2_y2 = 0for i in range(n) : sigma_x2_y2+=X[i]*X[i]*Y[i]*Y[i]sigma_x3_y = 0for i in range(n) : sigma_x3_y+=X[i]*X[i]*X[i]*Y[i]sigma_z_x2 = 0for i in range(n) : sigma_z_x2+=Z[i]*X[i]*X[i]sigma_z_y2 = 0for i in range(n) : sigma_z_y2+=Z[i]*Y[i]*Y[i]sigma_z_x_y = 0for i in range(n) : sigma_z_x_y+=Z[i]*X[i]*Y[i]sigma_z_x = 0for i in range(n) : sigma_z_x+=Z[i]*X[i]sigma_z_y = 0for i in range(n) : sigma_z_y+=Z[i]*Y[i]#print("-----------------------")#给出对应方程的矩阵形式a=np.array([[sigma_x4,sigma_x3_y,sigma_x2_y2,sigma_x3,sigma_x2_y,sigma_x2],[sigma_x3_y,sigma_x2_y2,sigma_x_y3,sigma_x2_y,sigma_x_y2,sigma_x_y],[sigma_x2_y2,sigma_x_y3,sigma_y4,sigma_x_y2,sigma_y3,sigma_y2],[sigma_x3,sigma_x2_y,sigma_x_y2,sigma_x2,sigma_x_y,sigma_x],[sigma_x2_y,sigma_x_y2,sigma_y3,sigma_x_y,sigma_y2,sigma_y],[sigma_x2,sigma_x_y,sigma_y2,sigma_x,sigma_y,n]])b=np.array([sigma_z_x2,sigma_z_x_y,sigma_z_y2,sigma_z_x,sigma_z_y,sigma_z])#高斯消元解线性方程res= np.linalg.solve(a,b)#print(a)#print(b)#print(x)# print("-----------------------")#输出方程形式print("z=%.6s*x^2%.6s*xy%.6s*y^2%.6s*x%.6s*y%.6s"%(fun(res[0]),fun(res[1]),fun(res[2]),fun(res[3]),fun(res[4]),fun(res[5])))#画曲面图和离散点fig = plt.figure()#建立一个空间ax =fig.add_subplot(111,projection='3d')# 3D坐标n = 256u = np.linspace(-20,20,n) # 创建一个等差数列x,y = np.meshgrid(u,u) #转化成矩阵#给出方程z=res[0]*x*x+res[1]*x*y+res[2]*y*y+res[3]*x+res[4]*y+res[5]#画出曲面ax.plot_surface(x,y,z,rstride=3,cstride=3,cmap=cm.jet)#画出点ax.scatter(X, Y, Z,c='r')
参考网站：
/qq_39735236/article/details/79066032
/NanShan/p/5493429.html
/jukan/p/7826851.html
/qq5q13638/article/details/78398997
/eddy_zheng/article/details/48713449
