[英]SymPy not simplifying "enough" while differentiating an implicit function
I am trying to find the 1st and 2nd derivatives of the implicit function y = f(x)
我试图找到隐式 function
y = f(x)
的一阶和二阶导数
which is defined by the equation: exp(sin(x)) - x * exp(sin(y)) = 0
由以下等式定义:
exp(sin(x)) - x * exp(sin(y)) = 0
SymPy calculates the 1st derivative and gives this answer: SymPy 计算一阶导数并给出以下答案:
But this expression can be written much simpler as:但是这个表达式可以更简单地写成:
(x * cos(x) - 1) / (x * cos(y))
using the fact that x = exp(sin(x)-sin(y))
使用
x = exp(sin(x)-sin(y))
的事实
The answer given for the 2nd derivative is also quite complicated.对二阶导数给出的答案也相当复杂。
Of course the second derivative can also be simplified当然二阶导数也可以化简
quite a lot using the same fact x = exp(sin(x)-sin(y))
.很多使用相同的事实
x = exp(sin(x)-sin(y))
。
How can I make/force SymPy apply these additional simplifications?我怎样才能使/强制 SymPy 应用这些额外的简化?
Is that possible even?这甚至可能吗?
Here is my script.这是我的脚本。
#!/usr/bin/env python
# coding: utf-8
# ### Differentiating an implicit function using SymPy
# In[1]:
import sympy as sp
# In[2]:
sp.__version__
# In[3]:
sp.init_printing(use_latex='mathjax') # use pretty mathjax output
# In[4]:
sp.var('x y z')
F = sp.exp(sp.sin(x)) - x * sp.exp(sp.sin(y))
# In[5]:
f1 = sp.idiff( F, y, x ) # First derivative of y w.r.t. x
f1
# In[6]:
sp.simplify(f1)
# In[7]:
f2 = sp.idiff( F, y, x, 2) # Second derivative of y w.r.t. x
f2
sp.simplify(f2)
# In[ ]:
And also, here is an even simpler example which shows this undesired behavior.而且,这里有一个更简单的例子,它显示了这种不受欢迎的行为。
#!/usr/bin/env python
# coding: utf-8
# ### Differentiating an implicit function using SymPy
# In[1]:
import sympy as sp
# In[2]:
sp.__version__
# In[3]:
sp.init_printing(use_latex='mathjax') # use pretty mathjax output
# In[4]:
sp.var('x y')
F = sp.ln(sp.sqrt(x**2 + y**2)) - sp.atan(y / x)
# In[5]:
f1 = sp.idiff( F, y, x ) # First derivative of y w.r.t. x
f1
# In[6]:
sp.simplify(f1)
# In[7]:
f2 = sp.idiff( F, y, x, 2) # Second derivative of y w.r.t. x
f2
sp.simplify(f2)
# In[ ]:
The second derivative here is given as:这里的二阶导数为:
This expression obviously can be simplified further even without using any special facts.即使不使用任何特殊事实,这个表达式显然也可以进一步简化。
You can simplify the expressions yourself.您可以自己简化表达式。 In the first example you can just choose a term to eliminate and solve
F
for that:在第一个示例中,您可以选择一个项来消除并解决
F
:
In [42]: F
Out[42]:
sin(y) sin(x)
- x⋅ℯ + ℯ
In [43]: solve(F, exp(sin(y)))
Out[43]:
⎡ sin(x)⎤
⎢ℯ ⎥
⎢───────⎥
⎣ x ⎦
In [44]: [esy] = solve(F, exp(sin(y)))
In [45]: f2.subs(exp(sin(y)), esy)
Out[45]:
2
x⋅sin(y)⋅cos (x) 2⋅sin(y)⋅cos(x) sin(y) 1
-x⋅sin(x) + ──────────────── - ─────────────── + ───────── + ─
2 2 2 x
cos (y) cos (y) x⋅cos (y)
──────────────────────────────────────────────────────────────
x⋅cos(y)
You can apply further simplification operations from there.您可以从那里应用进一步的简化操作。
In the second example you can just call factor
:在第二个示例中,您可以调用
factor
:
In [47]: f2
Out[47]:
⎛ 2 2⎞
2⋅⎝x + y ⎠
─────────────────────────
3 2 2 3
x - 3⋅x ⋅y + 3⋅x⋅y - y
In [48]: factor(f2)
Out[48]:
⎛ 2 2⎞
2⋅⎝x + y ⎠
───────────
3
(x - y)
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