# sympy: Problem with integration/differentiation of symbolic functions.

Reported by: Owned by: charpent critical sage-9.1 symbolics integrate, integral, sympy Markus Wageringel Vincent Delecroix N/A 57a8e83 57a8e83d6e1f1b6a667b444817f1314effb0f95e

This ticket fixes several problems related to the conversion of derivatives to and from sympy:

• the derivative arguments were permuted when converting to sympy (comment:7)
• the conversion from Sage to sympy did not account for the case of variables with multiple occurences (comment:1)
• the first argument of the derivative was skipped when converting from sympy (comment:6)
• sympy returns arguments of type `sympy.core.containers.Tuple` which was not handled
• Sage's derivative does not accept tuples, so the arguments must be flattened
• sympy's integer type is converted to Sage's rational type, but should be converted to Sage's integer type

(Original description below.)

Minimal case of a problem reported on ask.sagemath.org.

```sage: reset()
sage: var("x,t")
(x, t)
sage: f=function("f") ## Note : no formal arguments
sage: diff(f(x,t),x)
diff(f(x, t), x)
sage: diff(f(x,t),x).integrate(x)
f(x, t)
```

As expected. But :

```sage: diff(f(x,t),x).integrate(t)
f(x, t)
```

No, no, no. And no.

Slightly better:

```sage: reset()
sage: var("x,t")
(x, t)
sage: f=function("f")(x,t)  ## Note : specified formal arguments.
sage: diff(f(x,t),x)
diff(f(t, x), x)
sage: diff(f(x,t),x).integrate(x)
f(t, x)
sage: diff(f(x,t),x).integrate(t)
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/all_cmdline.py in <module>()
----> 1 diff(f(x,t),x).integrate(t)

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.integral (build/cythonized/sage/symbolic/expression.cpp:64575)()
12389                     R = ring.SR
12390             return R(integral(f, v, a, b, **kwds))
> 12391         return integral(self, *args, **kwds)
12392
12393     integrate = integral

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/integration/integral.py in integrate(expression, v, a, b, algorithm, hold)
927         return integrator(expression, v, a, b)
928     if a is None:
--> 929         return indefinite_integral(expression, v, hold=hold)
930     else:
931         return definite_integral(expression, v, a, b, hold=hold)

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/function.pyx in sage.symbolic.function.BuiltinFunction.__call__ (build/cythonized/sage/symbolic/function.cpp:12262)()
1136             res = self._evalf_try_(*args)
1137             if res is None:
-> 1138                 res = super(BuiltinFunction, self).__call__(
1139                         *args, coerce=coerce, hold=hold)
1140

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/function.pyx in sage.symbolic.function.Function.__call__ (build/cythonized/sage/symbolic/function.cpp:7039)()
600                     (<Expression>args)._gobj, hold)
601         elif self._nargs == 2:
--> 602             res = g_function_eval2(self._serial, (<Expression>args)._gobj,
603                     (<Expression>args)._gobj, hold)
604         elif self._nargs == 3:

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/integration/integral.py in _eval_(self, f, x)
100         for integrator in self.integrators:
101             try:
--> 102                 A = integrator(f, x)
103             except (NotImplementedError, TypeError, AttributeError):
104                 pass

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/integration/external.py in sympy_integrator(expression, v, a, b)
66     else:
67         result = sympy.integrate(ex, (v, a._sympy_(), b._sympy_()))
---> 68     return result._sage_()
69
70 def mma_free_integrator(expression, v, a=None, b=None):

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/interfaces/sympy.py in _sympysage_integral(self)
307     """
308     from sage.misc.functional import integral
--> 309     f, limits = self.function._sage_(), list(self.limits)
310     for limit in limits:
311         if len(limit) == 1:

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/interfaces/sympy.py in _sympysage_derivative(self)
333     f = self.args._sage_()
334     args = [[a._sage_() for a in arg] if isinstance(arg,tuple) else arg._sage_() for arg in self.args[2:]]
--> 335     return derivative(f, *args)
336
337 def _sympysage_order(self):

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/calculus/functional.py in derivative(f, *args, **kwds)
129     """
130     try:
--> 131         return f.derivative(*args, **kwds)
132     except AttributeError:
133         pass

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.derivative (build/cythonized/sage/symbolic/expression.cpp:25806)()
4186             ValueError: No differentiation variable specified.
4187         """
-> 4188         return multi_derivative(self, args)
4189
4190     diff = differentiate = derivative

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/misc/derivative.pyx in sage.misc.derivative.multi_derivative (build/cythonized/sage/misc/derivative.c:3014)()
217     if not args:
218         # fast version where no arguments supplied
--> 219         return F._derivative()
220
221     for arg in derivative_parse(args):

/usr/local/sage-P3-2/local/lib/python3.7/site-packages/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression._derivative (build/cythonized/sage/symbolic/expression.cpp:26148)()
4249             else:
-> 4250                 raise ValueError("No differentiation variable specified.")
4251         if not isinstance(deg, (int, long, sage.rings.integer.Integer)) \
4252                 or deg < 1:

ValueError: No differentiation variable specified.
```

No again. But at least, this is an explicit error, not a silently accepted wrong result.

The problem seems to point to handling formal arguments of undefined functions. I do not (yet) understand this code...

IMHO, at least the first case is critical (possibly blocker), because it silently returns a wrong result.

Note that the handling of defined functions seems to be correct. For example:

```sage: sin(t*x).diff(x)
t*cos(t*x)
sage: sin(t*x).diff(x).integrate(x)
sin(t*x)
sage: sin(t*x).diff(x).integrate(t)
(t*x*sin(t*x) + cos(t*x))/x^2
```

### comment:1 follow-up: ↓ 2 Changed 21 months ago by nbruin

Weird. Round-tripping to maxima seems to do the right thing, so that's not where the problem seems to lie:

```sage: diff(f(x,t),x)._maxima_().integrate(t)._sage_()
integrate(diff(f(x, t), x), t)
```

If you make the variable combinations a little more complicated:

```sage: diff(f(x,x,t),x).integrate(t)
ValueError: No differentiation variable specified.
sage: diff(f(x,x,t),t).integrate(x)
ValueError:
Since there is more than one variable in the expression, the
variable(s) of differentiation must be supplied to differentiate f(x,
x, t)
```

These are Pynac errors. So I suspect that there is a differentiation that doesn't have its variable specified, and that with only two variables around (one for the integration!) it ends up choosing that variable. So my guess is either Pynac itself or the way we interface with it.

The same problem seems to arise the other way around as well:

```sage: integrate(f(x,t),x).diff(t)
f(x, t)
```

we can make that cause an error by fiddling with some more variables and then we see that the error traces back in something being called from `_sympysage_integral()`, so I think this error was introduced relatively recently, when sympy was added as an integrator by default.

Last edited 21 months ago by nbruin (previous) (diff)

### comment:2 in reply to: ↑ 1 ; follow-up: ↓ 3 Changed 21 months ago by charpent

[ Snip... ]

we can make that cause an error by fiddling with some more variables and then we see that the error traces back in something being called from `_sympysage_integral()`, so I think this error was introduced relatively recently, when sympy was added as an integrator by default.

According to a comment posted around Jan 7, 2020 07:00 by dsejas in the original ask.sagemath question, the problem might have been introduced between 8.8 and 8.9.

It should be possible to `git bisect` this range to pinpoint the commit. I don't have the necessary resources on hand...

### comment:3 in reply to: ↑ 2 Changed 21 months ago by egourgoulhon

we can make that cause an error by fiddling with some more variables and then we see that the error traces back in something being called from `_sympysage_integral()`, so I think this error was introduced relatively recently, when sympy was added as an integrator by default.

According to a comment posted around Jan 7, 2020 07:00 by dsejas in the original ask.sagemath question, the problem might have been introduced between 8.8 and 8.9.

The culprit might be #27958.

### comment:5 Changed 21 months ago by gh-bryanso

I am the original poster in ask.sagemath.org for this issue. Hope I am not out of line to add a test case because I hope the eventual fix will solve my original problem:

```var('x t')
f = function('f')(x, t)
g = function('g')(x, t)
integrate(g * diff(f, x), x)
...
ValueError: No differentiation variable specified.
```

Used to work in Sage 8.8. Not working in 8.9 or 9.0.

Thanks a lot.

### comment:6 follow-up: ↓ 7 Changed 21 months ago by nbruin

Translation of derivatives to sympy is definitely wonky:

```sage: f=function('f')
sage: F=f(x,y,t).diff(t)
sage: F
diff(f(x, y, t), t)
sage: F._sympy_()
Derivative(f(x, y, t), y)
sage: F=f(x,x,t).diff(t)
sage: F._sympy_()
ValueError:
Since there is more than one variable in the expression, the
variable(s) of differentiation must be supplied to differentiate f(x,
x, t)
```

There are typos in `sage/interfaces/sympy.py`:

```def _sympysage_derivative(self):
"""
EXAMPLES::

sage: from sympy import Derivative
sage: f = function('f')
sage: sympy_diff = Derivative(f(x)._sympy_(), x._sympy_())
sage: assert diff(f(x),x)._sympy_() == sympy_diff
sage: assert diff(f(x),x) == sympy_diff._sage_()
"""
from sage.calculus.functional import derivative
f = self.args._sage_()
args = [[a._sage_() for a in arg] if isinstance(arg,tuple) else arg._sage_() for arg in self.args[2:]]
return derivative(f, *args)
```

I'm pretty sure the second-last line should be iterating over `self.args[1:]`, but changing that doesn't actually seem to fix the problem.

Last edited 21 months ago by nbruin (previous) (diff)

### comment:7 in reply to: ↑ 6 Changed 21 months ago by gh-mwageringel

Translation of derivatives to sympy is definitely wonky:

```sage: f=function('f')
sage: F=f(x,y,t).diff(t)
sage: F
diff(f(x, y, t), t)
sage: F._sympy_()
Derivative(f(x, y, t), y)
sage: F=f(x,x,t).diff(t)
sage: F._sympy_()
ValueError:
Since there is more than one variable in the expression, the
variable(s) of differentiation must be supplied to differentiate f(x,
x, t)
```

The following fixes this particular problem:

• ## src/sage/symbolic/expression_conversions.py

 a class SympyConverter(Converter): # retrieve order order = operator._parameter_set # arguments _args = ex.arguments() _args = ex.operands() sympy_arg = [] for i, a in enumerate(_args):

`arguments` contains the operands, but in a different order – alphabetically I assume.

### comment:8 Changed 19 months ago by gh-mwageringel

• Description modified (diff)
• Summary changed from Problem with integration/differentiation of symbolic functions. to sympy: Problem with integration/differentiation of symbolic functions.

### comment:9 Changed 19 months ago by gh-mwageringel

• Authors set to Markus Wageringel
• Branch set to u/gh-mwageringel/28964
• Commit set to 0bfd840d432c82fa84d7015736d8caff8cef8832
• Status changed from new to needs_review

The patches in comment:6 and comment:7 almost fix the problem, but there were a few more issues in the conversion of derivatives between sympy and Sage. These are now fixed by this branch.

Most notably, sympy's arguments are encoded as `sympy.core.containers.Tuple` which was missed in a typecheck `isinstance(arg, tuple)`, and Sage's derivative does not accept tuples as arguments, so they need to be flattened.

Moreover, the conversion from Sage to sympy was very broken and has been rewritten completely, especially to account for the case of variables with multiple occurences as in comment:1.

As far as I can see, all the test cases mentioned on this ticket work correctly now.

New commits:

 ​42d400b `28964: fix sympy conversions of derivative` ​0bfd840 `28964: sympy: derivatives by variables with multiple occurences`

### comment:10 follow-up: ↓ 13 Changed 19 months ago by vdelecroix

• Reviewers set to Vincent Delecroix
• Status changed from needs_review to needs_work

Handling the Rational this way leads to problem

```sage: cos(x).diff(x, 3/2)
-sin(x)
```

If this is really needed (I don't quite understand why yet), then you should check that there is no denominator.

### comment:11 Changed 19 months ago by git

• Commit changed from 0bfd840d432c82fa84d7015736d8caff8cef8832 to 57a8e83d6e1f1b6a667b444817f1314effb0f95e

Branch pushed to git repo; I updated commit sha1. New commits:

 ​57a8e83 `28964: fix conversion of integers from sympy to sage`

### comment:12 Changed 19 months ago by gh-mwageringel

• Description modified (diff)

### comment:13 in reply to: ↑ 10 Changed 19 months ago by gh-mwageringel

• Status changed from needs_work to needs_review

You are right. That was not a good solution. It turns out the problem is that elements of type `sympy.core.numbers.Integer` (a subtype of `sympy.core.numbers.Rational`) are converted to Sage's `Rational`. For example, this causes problems here:

```sage: import sympy
sage: sympy.sympify('Derivative(f(x, t), x, x)')
Derivative(f(x, t), (x, 2))
sage: _._sage_()            # the 2 is converted to Rational here
...
```

I have just fixed this by implementing the missing conversion routine for integers. Thanks for pointing me in the right direction. I will squash the commits if you prefer.

### comment:14 Changed 19 months ago by vdelecroix

• Status changed from needs_review to positive_review

Good enough as it is.

### comment:15 Changed 19 months ago by vbraun

• Branch changed from u/gh-mwageringel/28964 to 57a8e83d6e1f1b6a667b444817f1314effb0f95e
• Resolution set to fixed
• Status changed from positive_review to closed
Note: See TracTickets for help on using tickets.