求和的导数

Question

I am using sympy from time to time, but am not very good at it. At the moment I am stuck with defining a list of indexed variables, ie n1 to nmax and performing a summation on it. Then I want to be able to take the derivative:

So far I tried the following:

numSpecies = 10
n = IndexedBase('n')
i = symbols("i",cls=Idx)
nges = summation(n[i],[i,1,numSpecies])

However, if i try to take the derivative with respect to one variable, this fails:

diff(nges,n[5])

I also tried to avoid working with IndexedBase .

numSpecies = 10
n = symbols('n0:%d'%numSpecies)
k = symbols('k',integer=True)
ntot = summation(n[k],[k,0,numSpecies])

However, here already the summation fails because mixing python tuples and the sympy summation.

How I can perform indexedbase derivatives or some kind of workaround?

Answer 1

With SymPy's development version, your example works.

To install SymPy's development version, just pull it down with git :

git clone git://github.com/sympy/sympy.git
cd sympy

Then run python from that path or set the PYTHONPATH to include that directory before Python's default installation.

Your example on the development version:

In [3]: numSpecies = 10

In [4]: n = IndexedBase('n')

In [5]: i = symbols("i",cls=Idx)

In [6]: nges = summation(n[i],[i,1,numSpecies])

In [7]: nges
Out[7]: n[10] + n[1] + n[2] + n[3] + n[4] + n[5] + n[6] + n[7] + n[8] + n[9]

In [8]: diff(nges,n[5])
Out[8]: 1

You could also use the contracted form of summation:

In [9]: nges_uneval = Sum(n[i], [i,1,numSpecies])

In [10]: nges_uneval
Out[10]: 
  10      
 ___      
 ╲        
  ╲   n[i]
  ╱       
 ╱        
 ‾‾‾      
i = 1     

In [11]: diff(nges_uneval, n[5])
Out[11]: 
  10      
 ___      
 ╲        
  ╲   δ   
  ╱    5,i
 ╱        
 ‾‾‾      
i = 1     

In [12]: diff(nges_uneval, n[5]).doit()
Out[12]: 1

Also notice that in the next SymPy version you will be able to derive symbols with symbolic indices:

In [13]: j = symbols("j")

In [13]: diff(n[i], n[j])
Out[13]: 
δ   
 j,i

Where you get the Kronecker delta .

If you don't feel like installing the SymPy development version, just wait for the next full version (probably coming out this autumn), it will support derivatives of IndexedBase .

Answer 2

I don't know why the IndexedBase approach does not work (I would be interested to know also). You can, however, do the following:

import sympy as sp

numSpecies = 10
n = sp.symbols('n0:%d'%numSpecies)   # note that n equals the tuple (n0, n1, ..., n9)

ntot = sum(n)       # sum elements of n using the standard
                    # Python function for summing tuple elements
#ntot = sp.Add(*n)  # same result using Sympy function

sp.diff(ntot, n[5])

Answer 3

I'm not clear about what you want to do. However, perhaps this will help. Edited in response to two comments received.

from sympy import *

nspecies = 10
[var('n%s'%_) for _ in range(nspecies)]

expr = sympify('+'.join(['n%s'%_ for _ in range(nspecies)]))
expr
print ( diff(expr,n1) )

expr = sympify('n0**n1+n1**n2')
expr
print ( diff(expr,n1) )

Only the first expression responds to the original question. This is the output.

1
n0**n1*log(n0) + n1**n2*n2/n1