导数+偏导数+方向梯度+梯度下降法(python)

主要内容：导数+偏导数+方向梯度+梯度下降法+极限(python)

1.导数

import sympy
from sympy.abc import x,y
y=sympy.asin(sympy.sqrt(sympy.sin(x)))
result=sympy.diff(y)
print(result)
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2.偏导数

方式一：
import sympy
from sympy.abc import x,y
f=x**2+3*x*y+y**2
fx=sympy.diff(f,x)
fy=sympy.diff(f,y)
print(fx.evalf(subs={x:1,y:2}))
print(fy.evalf(subs={x:1,y:2}))
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方式二：
import sympy
from sympy.abc import x,y
f=x**2+3*x*y+y**2
fx=sympy.diff(f,x)
fy=sympy.diff(f,y)
print(fx.subs({x:1,y:2}))
print(fy.subs({x:1,y:2}))
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3.方向导数

方式一：
import sympy
from sympy.abc import x,y,z
z=x*sympy.exp(2*y)
zx=sympy.diff(z,x)
zy=sympy.diff(z,y)
result=zx.evalf(subs={x:1,y:0})*sympy.cos(-sympy.pi/4)+zy.evalf(subs={x:1,y:0})*sympy.sin(-sympy.pi/4)
print(result)
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方式二：
import sympy
from sympy.abc import x,y,z
z=x*sympy.exp(2*y)
zx=sympy.diff(z,x)
zy=sympy.diff(z,y)
result=zx.subs({x:1,y:0})*sympy.cos(-sympy.pi/4)+zy.subs({x:1,y:0})*sympy.sin(-sympy.pi/4)
print(result)
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4.梯度

方式一：（梯度类型一般选择方式一）
import sympy
from sympy.abc import x,y,z,u
u=x*y*z+z**2+5
ux=sympy.diff(u,x)
uy=sympy.diff(u,y)
uz=sympy.diff(u,z)
gradu=[ux.subs({x:0,y:1,z:-1}),uy.subs({x:0,y:1,z:-1}),uz.subs({x:0,y:1,z:-1})] #这里subs后有个括号
print(gradu)
maxgradu=sympy.sqrt(gradu[0]**2+gradu[1]**2+gradu[2]**2)
print(maxgradu)
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方式二：
import sympy
from sympy.abc import x,y,z,u
u=x*y*z+z**2+5
ux=sympy.diff(u,x)
uy=sympy.diff(u,y)
uz=sympy.diff(u,z)
gradu = [ux.evalf(subs={x:0,y:1,z:-1}),uy.evalf(subs={x:0,y:1,z:-1}),uz.evalf(subs={x:0,y:1,z:-1})]
maxgradu = sympy.sqrt(gradu[0]**2+gradu[1]**2+gradu[2]**2)
print(gradu)
print(maxgradu)
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5.梯度下降法求最小值

方式一：直接计算，无图像
#先定义这个函数
def Fun(x,y):
    return x-y+2*x**2+2*x*y+y**2
#需要朝着哪个方向滚，计算梯度
def PxFun(x,y):
    return 1+4*x+2*y
def PyFun(x,y):
    return -1+2*x+2*y
#滚多远，需要知道步长（学习率）
step=0.0008#如果大于1，越来越大；等于1，来回摆渡不下降；在0到1之前，若太小，迭代次数越多
x=0
y=0
new_x=x
new_y=y
Over=False
#准备好了哪个方向+步长，开始滚
while Over==False:
    new_x-=step*PxFun(x,y)
    new_y-=step*PyFun(x,y)
#滚到什么程度结束
    if Fun(x,y)-Fun(new_x,new_y)<7e-9:
        Over=True
    x=new_x
    y=new_y
print(x,y)
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方式二：有图像
import matplotlib.pyplot as plt  #  导入matplotlib 库
from mpl_toolkits.mplot3d import Axes3D # 导入Axes3D
import numpy as np  #  导入numpy 库
def  Fun(x,y):
    return x-y+2*x*x+2*x*y+y**2      # 定义函数表达式
def  PxFun(x,y):
    return 1+4*x+2*y     # 定义沿x方向偏导数
def  PyFun(x,y):
    return -1+2*x+2*y   # 定义沿y方向偏导数

fig=plt.figure()
ax=Axes3D(fig)
X,Y=np.mgrid[-2:2:40j,-2:2:40j]
Z=Fun(X,Y)
ax.plot_surface(X,Y,Z,rstride=1,cstride=1,cmap="rainbow")
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')

step = 0.0008;  
x = 0;  
y = 0 ;  
tag_x = [x];     
tag_y = [y];     
tag_z = [Fun(x,y)]
new_x = x ;   
new_y = y;   
Over = False
while Over == False:
    new_x -= step*PxFun(x,y);  new_y -= step*PyFun(x,y)
    if   Fun(x,y)-Fun(new_x,new_y) < 7e-9:
         Over = True
    x = new_x;    y = new_y;  
    tag_x.append(x); 
    tag_y.append(y); 
    tag_z.append(Fun(x,y))
ax.plot(tag_x,tag_y,tag_z,'r')
plt.title('(x,y)~('+str(x)+","+str(y)+')')
plt.show()
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import matplotlib.pyplot as plt  #  导入matplotlib 库
from mpl_toolkits.mplot3d import Axes3D # 导入Axes3D
import numpy as np  #  导入numpy 库
def  Fun(x,y):
    return x-y+2*x*x+2*x*y+y**2      # 定义函数表达式
def  PxFun(x,y):
    return 1+4*x+2*y     # 定义沿x方向偏导数
def  PyFun(x,y):
    return -1+2*x+2*y   # 定义沿y方向偏导数

fig=plt.figure()
ax=Axes3D(fig)
X,Y=np.mgrid[-2:2:40j,-2:2:40j]
Z=Fun(X,Y)
ax.plot_surface(X,Y,Z,rstride=1,cstride=1,cmap="rainbow")
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('z')

step = 0.0008;  
x = 0;  
y = 0 ;  
tag_x = [x];     
tag_y = [y];     
tag_z = [Fun(x,y)]
new_x = x ;   
new_y = y;   
Over = False
while Over == False:
    new_x -= step*PxFun(x,y);  new_y -= step*PyFun(x,y)
    if   Fun(x,y)-Fun(new_x,new_y) < 7e-9:
         Over = True
    x = new_x;    y = new_y;  
    tag_x.append(x); 
    tag_y.append(y); 
    tag_z.append(Fun(x,y))
ax.plot(tag_x,tag_y,tag_z,'r')
plt.title('(x,y)~('+str(x)+","+str(y)+')')
plt.show()
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6.求极限（实验部分的补充）

1.

import sympy
from sympy import oo     #这里不是sympy.abc
#from sympy.abc import x
x=sympy.Symbol('x')
f=sympy.sin(x)/x
result=sympy.limit(f,x,oo)
print(result)
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2.

import sympy
from sympy.abc import x
f=(x**2-1)/(x-1)
result=sympy.limit(f,x,1)
print(result)
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实验部分的练习题：

1.

import sympy
from sympy.abc import x
f=sympy.sin(sympy.ln(x))
result=sympy.limit(f,x,1)
print(result)
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2.

import sympy
from sympy.abc import x
f=(x**(1/3)-2)/(x-8)
result=sympy.limit(f,x,8)
print(result)
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3.求导数

import sympy
from sympy.abc import x,y
y=x**4-2*x**3+5*sympy.sin(x)+sympy.ln(3)
result=sympy.diff(y)
print(result)
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4.

import sympy
from sympy.abc import x,y,z
z=x**2+y**2
zx=sympy.diff(z,x)
zy=sympy.diff(z,y)
print(zx.evalf(subs={x:1,y:2}))
print(zy.evalf(subs={x:1,y:2}))
gradu=[zx.subs({x:1,y:2}),zy.subs({x:1,y:2})]
print(gradu)
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（理论课无习题）