piecewise functions and integration / arithmetic do not play well together

[was]

On 1/13/08, Hector Villafuerte <> wrote:
> 
> I defined a piecewise function (specifically, a triangular wave) like this:
> 
> sage: f1(x) = -abs(x) + 1
> sage: f2(x) = abs(x - 2) - 1
> sage: tri_wave = piecewise([ [(-1,1), f1], [(1,3), f2]])
> 
> One can plot it and it looks very nice:
> 
> sage: tri_wave.plot()
> 
> But while calculating this integral I get "ValueError: Value not
> defined outside of domain."
> 
> sage: integrate(tri_wave(x)^2, x, -1, 3)
> 
> Is there a way to integrate piecewise-defined functions?
> As always, thanks for your help,

This is clearly broken.  As a band-aide, you can at least
numerically integrate as follows:

sage: integral_numerical(lambda x: tri_wave(x)^2, -1, 3)
(1.3333333333333333, 1.4765966227514582e-14)

The first output (1.3333...) is the answer, and the second is an error bound.

 -- William

[rlm]

This isn't specific to integrate. The problem is that piecewise functions don't play well with *symbolics*

sage: f1(x) = -abs(x) + 1
sage: f2(x) = abs(x - 2) - 1
sage: tri_wave = piecewise([ [(-1,1), f1], [(1,3), f2]])
sage: tri_wave(x)
SAME ERROR

[rlm]

What should maybe happen is in piecewise.py, in the __call__ method, if a SymbolicVariable? is passed in as x0, it should return a CallableSymbolicExpression? which just in turn calls back to the __call__ method again. Does this sound reasonable? If so, how would one implement this?

[kcrisman]

See also #11225, which is about plotting - but sometimes this stuff causes plots to not work, of course.

[kcrisman]

Replying to rlm:

This isn't specific to integrate. The problem is that piecewise functions don't play well with *symbolics*

sage: f1(x) = -abs(x) + 1
sage: f2(x) = abs(x - 2) - 1
sage: tri_wave = piecewise([ [(-1,1), f1], [(1,3), f2]])
sage: tri_wave(x)
SAME ERROR

Or see what happens when we try to multiply piecewise and symbolic.

sage: f = Piecewise([[(0,pi/2),-1],[(pi/2,pi),2]]) 
sage: f*sin(x)
---------------------------------------------------------------------------
AttributeError