.is_zero() method raises error for symbolic expression involving derivative

[golam-m-hossain]

If a symbolic expression contains  symbolic derivative then
checking whether it is zero, raises error:

sage: x.diff(x,2).is_zero()
True

sage: f(x) = function('f',x)
sage: f(x).diff(x).is_zero()
....
NotImplementedError: derivative

This fails because new symbolics tries to convert it to maxima
expression for checking the relation.

Update:

***** A patch to fix the issue is attached. The patch
adds a new method ".has_fderivative()" for symbolic expressions
and in __nonzero__ method adds a check whether it has fderivative.

Comments (for future works):
A simple timing comparison that illustrates why we should
avoid calling maxima to assert nonzero even for symbolic
functions

sage: f(x) = function('f',x)
sage: timeit('sin(f(x)).is_zero()')
5 loops, best of 3: 85.8 ms per loop
sage: timeit('sin(f(x).diff(x)).is_zero()')
625 loops, best of 3: 132 µs per loop

It seems pynac is 400 times faster than maxima in this case.

CC: @mwhansen

Component: symbolics

Author: Burcin Erocal

Reviewer: Mike Hansen

Merged: sage-4.3.rc0

Issue created by migration from https://trac.sagemath.org/ticket/6523

[golam-m-hossain]

Attachment: trac_6523-fixes-error-in-is_zero_method_for_symbolic_derivative.patch.gz

[golam-m-hossain]

Description changed:

--- 
+++ 
@@ -14,8 +14,24 @@
 This fails because new symbolics tries to convert it to maxima
 expression for checking the relation.
 
-It works fine for any other expression not involving symbolic
-derivative and without invoking maxima.
+**Update:**
 
-It seems to me, pynac relational test needs to be fixed.
+***** A patch to fix the issue is attached. The patch 
+adds a new method ".has_fderivative()" for symbolic expressions 
+and in `__nonzero__` method adds a check whether it has fderivative.
 
+**Comments**  (for future works):
+A simple timing comparison that illustrates why we should
+avoid calling maxima to assert nonzero even for symbolic 
+functions
+
+```
+sage: f(x) = function('f',x)
+sage: timeit('sin(f(x)).is_zero()')
+5 loops, best of 3: 85.8 ms per loop
+sage: timeit('sin(f(x).diff(x)).is_zero()')
+625 loops, best of 3: 132 µs per loop
+```  
+
+It seems pynac is 400 times faster than maxima in this case.
+

[burcin]

comment:2

Unfortunately, the fact that an expression contains a symbolic derivative doesn't guarantee that it is nonzero:

sage: t = f(x).derivative(x)
sage: (x*t +(1-x)*t - t)
-(x - 1)*D[1](f)(x) + x*D[1](f)(x) - D[1](f)(x)
sage: (x*t +(1-x)*t - t).collect(x)
0

The right fix for this is to either implement the .derivative() method in sage/symbolic/expression_conversions.py or to change pynac to allow different parents in evalf(), so that conversion to CIF can be done without the code in expression_conversions.pyx.

I was planning to do this for #6243, but ended up using a different/better fix there.

[burcin]

Attachment: trac_6523-derivative_is_zero.patch.gz

add doctests

[burcin]

Author: Burcin Erocal

[burcin]

comment:3

This is fixed by #7490, since we don't use the expression_conversions module to convert to RIF any more.

attachment: trac_6523-derivative_is_zero.patch adds a doctest with the example in the description.

[mwhansen]

comment:4

Looks good to me.

Merged trac_6523-derivative_is_zero.patch.

[mwhansen]

Reviewer: Mike Hansen

[mwhansen]

Merged: sage-4.3.rc0