牛顿法、梯度下降法、最小二乘法的原理以及利用它们解决实际问题的python编程
牛顿法、梯度下降法、最小二乘法的原理以及利用它们解决实际问题的python编程
一、牛顿法原理
1、产生背景
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import math
from mpl_toolkits.mplot3d import Axes3D
import warnings
3、画出函数图像
def f2(x1,x2):
return x1 ** 2 + 2 * x2 ** 2 -4*x1- 2 * x1*x2
X1 = np.arange(-4,4,0.2)
X2 = np.arange(-4,4,0.2)
X1, X2 = np.meshgrid(X1, X2) # 生成xv、yv，将X1、X2变成n*m的矩阵，方便后面绘图
Y = np.array(list(map(lambda t : f2(t[0],t[1]),zip(X1.flatten(),X2.flatten()))))
Y.shape = X1.shape # 1600的Y图还原成原来的（40,40）
%matplotlib inline
#作图
fig = plt.figure(facecolor='w')
ax = Axes3D(fig)
ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
ax.set_title(u'$ x1 ** 2 + 2 * x2 ** 2 -4*x1- 2 * x1*x2 $')
plt.show()
# 解决中文显示问题
mpl.rcParams['font.sans-serif'] = [u'SimHei']
mpl.rcParams['axes.unicode_minus'] = False
%matplotlib inline
# 二维原始图像
def f2(x, y):
return x ** 2 + 2 * y ** 2 -4*x- 2 * x*y
## 偏函数
def hx1(x, y):
return 2*x-4-2*y
def hx2(x, y):
return 4*y-2*x
x1 = 4
x2 = 4
alpha = 0.001
#保存梯度下降经过的点
GD_X1 = [x1]
GD_X2 = [x2]
GD_Y = [f2(x1,x2)]
# 定义y的变化量和迭代次数
y_change = f2(x1,x2)
iter_num = 0
while(iter_num < 10000) :
tmp_x1 = x1 - alpha * hx1(x1,x2)
tmp_x2 = x2 - alpha * hx2(x1,x2)
tmp_y = f2(tmp_x1,tmp_x2)
f_change = np.absolute(tmp_y - f2(x1,x2))
x1 = tmp_x1
x2 = tmp_x2
GD_X1.append(x1)
GD_X2.append(x2)
GD_Y.append(tmp_y)
iter_num += 1
print(u"最终结果为:(%.5f, %.5f, %.5f)" % (x1, x2, f2(x1,x2)))
print(u"迭代过程中X的取值，迭代次数:%d" % iter_num)
print(GD_X1)
# 作图
fig = plt.figure(facecolor='w',figsize=(20,18))
ax = Axes3D(fig)
ax.plot_surface(X1,X2,Y,rstride=1,cstride=1,cmap=plt.cm.jet)
ax.plot(GD_X1,GD_X2,GD_Y,'ko-')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')
ax.set_title(u'函数;\n学习率:%.3f; 最终解:(%.3f, %.3f, %.3f);迭代次数:%d' % (alpha, x1, x2, f2(x1,x2), iter_num))
plt.show()
最终结果为:(4.00043, 2.00027, -8.00000)
迭代过程中X的取值，迭代次数:10000
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