- 用数值微分对简单函数进行求导
# coding: utf-8
import numpy as np
import matplotlib.pylab as plt
#微分方程
def numerical_diff(f, x):
h = 1e-4 # 0.0001
return (f(x+h) - f(x-h)) / (2*h)
#函数的表达形式
def function_1(x):
return 0.01*x**2 + 0.1*x
#切线 x 为传入点5的斜率
def tangent_line(f, x):
d = numerical_diff(f, x)
print(d)#0.1999999999990898 接近于2
y = f(x) - d*x#计算x=5 y的值
return lambda t: d*t + y#函数表达式 d*t + y
x = np.arange(0.0, 20.0, 0.1)
y = function_1(x)
plt.xlabel("x")
plt.ylabel("f(x)")
tf = tangent_line(function_1, 5)
y2 = tf(x)#与曲线用同样的点
plt.plot(x, y)
plt.plot(x, y2)
plt.show()
曲面函数求解
2) 偏导数求导
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np
fig = plt.figure()
ax1 = plt.axes(projection="3d")
x1 = np.arange(-10,10,0.1)
x2 = np.arange(-10,10,0.1)
#网格数据采集函数,否则是一条曲线不是曲面
X,Y=np.meshgrid(x1,x2)
Z = X**2 +Y**2
ax1.plot_surface(X,Y,Z,cmap="binary")#,alpha=0.5
#ax1.plot3D(X,Y,Z) 去掉meshgrid使用就是现实曲线
plt.show()
def numerical_diff(f, x):
h = 1e-4 # 0.0001
return (f(x + h) - f(x - h)) / (2 * h)
def function_tmp1(x0):
return x0*x0 + 4.0** 2.0
#x1处的偏导
def function_tmp2(x1):
return 3.0*2.0 + x1*x1
var = numerical_diff(function_tmp1,3.0)
print(var)#6.00000000000378
var = numerical_diff(function_tmp2,4.0)
print(var)#7.999999999999119
#和解析解的值基本一致
3）梯度求导
# coding: utf-8
# cf.http://d.hatena.ne.jp/white_wheels/20100327/p3
import numpy as np
import matplotlib.pylab as plt
from mpl_toolkits.mplot3d import Axes3D
#梯度算法核心
def _numerical_gradient_no_batch(f, x):
h = 1e-4 # 0.0001
grad = np.zeros_like(x)
for idx in range(x.size):
tmp_val = x[idx]
x[idx] = float(tmp_val) + h
fxh1 = f(x) # f(x+h)
x[idx] = tmp_val - h
fxh2 = f(x) # f(x-h)
grad[idx] = (fxh1 - fxh2) / (2*h)
x[idx] = tmp_val # 还原值
return grad
def numerical_gradient(f, X):
if X.ndim == 1:
return _numerical_gradient_no_batch(f, X)
else:
grad = np.zeros_like(X)
for idx, x in enumerate(X):
grad[idx] = _numerical_gradient_no_batch(f, x)
return grad
def function_2(x):
if x.ndim == 1:
return np.sum(x**2)
else:
return np.sum(x**2, axis=1)
def tangent_line(f, x):
d = numerical_gradient(f, x)
print(d)
y = f(x) - d*x
return lambda t: d*t + y
if __name__ == '__main__':
x0 = np.arange(-2, 2.5, 0.25)
x1 = np.arange(-2, 2.5, 0.25)
X, Y = np.meshgrid(x0, x1)
X = X.flatten()
Y = Y.flatten()
#va = numerical_gradient(function_2, np.array([3.0, 4.0]))
#print(va)#[6. 8.] 越远梯度越大
#va = numerical_gradient(function_2, np.array([0.0, 2.0]))
#print(va)# [0. 4.]越近梯度越小
grad = numerical_gradient(function_2, np.array([X, Y]) )
plt.figure()
plt.quiver(X, Y, -grad[0], -grad[1], angles="xy",color="#666666")#,headwidth=10,scale=40,color="#444444")
plt.xlim([-2, 2])
plt.ylim([-2, 2])
plt.xlabel('x0')
plt.ylabel('x1')
plt.grid()
plt.legend()
plt.draw()
plt.show()
程序运行结果
def function_2(x):
return x[0]**2 + x[1]**2
init_x = np.array([-3.0, 4.0])#初始值选择
#寻找最小值
var = gradient_descent(function_2, init_x=init_x, lr=0.1, step_num=100)
print(var)#逐渐趋向于[0,0]
#输出结果，循环100 次逐渐趋向于[0,0]
[-3.00000000e+00, 4.00000000e+00],
[-2.40000000e+00, 3.20000000e+00],
[-1.92000000e+00, 2.56000000e+00],
...
[-1.19357577e-09, 1.59143436e-09],
[-9.54860614e-10, 1.27314749e-09],
[-7.63888491e-10, 1.01851799e-09]
# coding: utf-8
import numpy as np
import matplotlib.pylab as plt
from gradient_2d import numerical_gradient
def gradient_descent(f, init_x, lr=0.01, step_num=100):
x = init_x
x_history = []
for i in range(step_num):
x_history.append( x.copy() )
grad = numerical_gradient(f, x)
x -= lr * grad# x值逐渐变化 趋向于[0,0]
return x, np.array(x_history)
def function_2(x):
return x[0]**2 + x[1]**2
init_x = np.array([-3.0, 4.0])
lr = 0.1
step_num = 200
x, x_history = gradient_descent(function_2, init_x, lr=lr, step_num=step_num)
print(x)#-1.25592487e-19 1.65436123e-19
plt.plot( [-5, 5], [0,0], '--b')
plt.plot( [0,0], [-5, 5], '--b')
plt.plot(x_history[:,0], x_history[:,1], 'o')
plt.xlim(-3.5, 3.5)
plt.ylim(-4.5, 4.5)
plt.xlabel("X0")
plt.ylabel("X1")
plt.show()
值逐渐向[0,0]靠近