Is there any way to build a differentiable sum of indexed types?

[Jason Moore]

I'd like to do this:

>>> r = IndexedBase['m']

>>> i = idx('i')

>>> h = Symbol('h')

>>> N = Symbol('N', integer=True)

>>> J = Sum(h * r[i]**2, (i, 1, N))

>>> J.diff(r[i])

2 * h * r[i]

This currently doesn't work (Sum doesn't like idx). But are there other ways to do this in SymPy? Or should this work?

The big picture is that I want to be able write a scalar expression that involves integrals of continuous functions, but then numerically approximate it with various numerical integration patterns. And finally I need to find the gradient of the expression with respect to discrete variables, like r[i] above.

Jason
moorepants.info
+01 530-601-9791

[Jason Moore]

Here is as an attempt but the derivative fails:

In [5]: i = sm.symbols('i', integer=True)

In [6]: s=sm.Sum(sm.Indexed('r',i),(i,1,3))

In [7]: s
Out[7]: Sum(r[i], (i, 1, 3))

In [8]: s.diff(sm.Indexed('r',i))
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-8-f8ca8cff3c71> in <module>()
----> 1 s.diff(sm.Indexed('r',i))

/home/moorepants/anaconda/lib/python2.7/site-packages/sympy/core/expr.pyc in diff(self, *symbols, **assumptions)
   2773         new_symbols = list(map(sympify, symbols))  # e.g. x, 2, y, z
   2774         assumptions.setdefault("evaluate", True)
-> 2775         return Derivative(self, *new_symbols, **assumptions)
   2776
   2777     ###########################################################################

/home/moorepants/anaconda/lib/python2.7/site-packages/sympy/core/function.pyc in __new__(cls, expr, *variables, **assumptions)
   1016                 from sympy.utilities.misc import filldedent
   1017                 raise ValueError(filldedent('''
-> 1018                 Can\'t calculate %s-th derivative wrt %s.''' % (count, v)))
   1019
   1020             if all_zero and not count == 0:

ValueError:
Can't calculate 1-th derivative wrt r[i].

Jason
moorepants.info
+01 530-601-9791

[Francesco Bonazzi]

If you include this PR:
https://github.com/sympy/sympy/pull/9314

Your second example gives 0, which doesn't look quite correct.

I would rather expect a Sum(delta[i, j], (i, 1, 3)) as a result.

[Jason Moore]

I've checked out that PR briefly, but don't really understand it yet. I will look closer. I'm not sure what delta represents.

Is there even a way to represent a series like this in SymPy:

\sum^{N}_{i=1} r_i^2 = r_1^2 + r_2^2 + ... + r_N^2

where r is a sequence of values {r_1, r_2, ..., r_N}

I was assuming that the Indexed types were appropriate for that.

This seems like it does the correct thing:

In [5]: import sympy as sm

In [6]: i = sm.symbols('i', integer=True)

In [7]: sm.Sum(sm.Indexed('r', i), (i, 1, 3)).doit()
Out[7]: r[1] + r[2] + r[3]

Jason
moorepants.info
+01 530-601-9791

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[Jason Moore]

Here is a related issue: https://github.com/sympy/sympy/issues/6967

Jason
moorepants.info
+01 530-601-9791