*TL;DR :* Sage can produce a symbolic expression :
- that it can't evaluate numerically when correcly substiuted, AND
- that can be evaluated when typed manually.
The problem seems ti be bound to absolute values.
That, IMHO, is a first-class bug...
*Demonstation :* I think that some results produced by fricas' integrator
are wrong, and wanted to show it numerically.
Problem setup :
x,y=var("x,y", domain="real")
assume(x>-1,x<1,y>-1,y<1,x^2+y^2<1)
eps1=var("eps1", latex_name="\\varepsilon_1", domain="positive")
eps2=var("eps2", latex_name="\\varepsilon_2", domain="positive")
assume(eps1<1,eps2<1, eps1^2+eps2^2<1)
f(x,y)=sqrt(1-x^2-y^2)
Define the integral on a given rectangle as a function :
foo(eps1,eps2)=f(x).integrate(x,0,eps1,
algorithm="fricas").integrate(y,0,eps2, algorithm="fricas")
Use it :
sage: bar=foo(1/10,1/10)
sage: bar
299/12000*pi - 1/198*sqrt(11)*pi*abs(-3/2*sqrt(11)) + 7/1500*sqrt(1/2)
+ 1/6*arctan(9799/140*sqrt(1/2)) - 299/6000*arctan(14*sqrt(1/2)) +
299/6000*arctan(1/7*sqrt(1/2))
So far so good. But when I want a numerical approximation :
sage: bar.n()
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-104-425a0225c048> in <module>()
----> 1 bar.n()
/usr/local/sage-8/local/lib/python2.7/site-packages/sage/structure/element.pyx
in sage.structure.element.Element.n
(build/cythonized/sage/structure/element.c:8020)()
859 0.666666666666667
860 """
--> 861 return self.numerical_approx(prec, digits, algorithm)
862
863 def _mpmath_(self, prec=53, rounding=None):
/usr/local/sage-8/local/lib/python2.7/site-packages/sage/symbolic/expression.pyx
in sage.symbolic.expression.Expression.numerical_approx
(build/cythonized/sage/symbolic/expression.cpp:33969)()
5949 res = x.pyobject()
5950 else:
-> 5951 raise TypeError("cannot evaluate symbolic expression
numerically")
5952
5953 # Important -- the we get might not be a valid output for
numerical_approx in
TypeError: cannot evaluate symbolic expression numerically
Let's debug it :
def tryit(x):
try:
return x.n()
except:
return "Failed..."
Let's see :
sage: gee=[[u,tryit(u)] for u in bar.operands()]
sage: gee
[[299/12000*pi, 0.0782780169519457],
[-1/198*sqrt(11)*pi*abs(-3/2*sqrt(11)), 'Failed...'],
[7/1500*sqrt(1/2), 0.00329983164553722],
[1/6*arctan(9799/140*sqrt(1/2)), 0.258432327173397],
[-299/6000*arctan(14*sqrt(1/2)), -0.0732611082333419],
[299/6000*arctan(1/7*sqrt(1/2)), 0.00501690871860373]]
However, when I copy and paste the very same expression (or type it
manually), it can be numerically approximated :
sage: (-1/198*sqrt(11)*pi*abs(-3/2*sqrt(11))).n()
-0.261799387799149
Diving recusively :
sage: [[u,tryit(u)] for u in bar.operands()[1].operands()]
[[sqrt(11), 3.31662479035540],
[pi, 3.14159265358979],
[abs(-3/2*sqrt(11)), 'Failed...'],
[-1/198, -0.00505050505050505]]
Again, the litigious expression can be evaluated by copy 'n paste :
sage: abs(-3/2*sqrt(11)).n()
4.97493718553310
BUT the semi-obvious workaround fails :
sage: SR(repr(bar.operands()[1])).n()
---------------------------------------------------------------------------
TypeError Traceback (most recent call last)
<ipython-input-117-e426b18b4666> in <module>()
----> 1 SR(repr(bar.operands()[Integer(1)])).n()
/usr/local/sage-8/local/lib/python2.7/site-packages/sage/structure/element.pyx
in sage.structure.element.Element.n
(build/cythonized/sage/structure/element.c:8020)()
859 0.666666666666667
860 """
--> 861 return self.numerical_approx(prec, digits, algorithm)
862
863 def _mpmath_(self, prec=53, rounding=None):
/usr/local/sage-8/local/lib/python2.7/site-packages/sage/symbolic/expression.pyx
in sage.symbolic.expression.Expression.numerical_approx
(build/cythonized/sage/symbolic/expression.cpp:33969)()
5949 res = x.pyobject()
5950 else:
-> 5951 raise TypeError("cannot evaluate symbolic expression
numerically")
5952
5953 # Important -- the we get might not be a valid output for
numerical_approx in
TypeError: cannot evaluate symbolic expression numerically
Recurse again :
sage: [[u,tryit(u)] for u in bar.operands()[1].operands()[2].operands()]
[[-3/2*sqrt(11), -4.97493718553310]]
Therefore, the problem seems to be with the numerical approximation of an
absolute value.
I'm stuck.
Two questions :
- Ticket worthy (yes, IMHO)
- Workaround suggestions ?
HTH,
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