piecewise functions and integration / arithmetic do not play well together

[williamstein]

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

CC: @kcrisman @jondo @vbraun @slel @mkoeppe @eviatarbach @rwst

Component: calculus

Author: Marcelo Forets

Branch/Commit: fe8304e

Reviewer: Volker Braun

Issue created by migration from https://trac.sagemath.org/ticket/1773

[rlmill]

comment:2

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

[rlmill]

comment:3

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]

comment:5

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

[kcrisman]

comment:6

Replying to @rlmill:

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              

[ppurka]

Dependencies: #14801

[mkoeppe]

comment:13

While all the examples appearing in the comments are fixed with the new piecewise of #14801, the original bug example still fails (but with a different error message).

Cc'ing #14801's cc list.

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.plot()
Launched png viewer for Graphics object consisting of 1 graphics primitive
sage: integrate(tri_wave(x)^2, x, -1, 3)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-8-9387c34a513c> in <module>()
----> 1 integrate(tri_wave(x)**Integer(2), x, -Integer(1), Integer(3))

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/misc/functional.py in integral(x, *args, **kwds)
    663     """
    664     if hasattr(x, 'integral'):
--> 665         return x.integral(*args, **kwds)
    666     else:
    667         from sage.symbolic.ring import SR

/Users/mkoeppe/cvs/sage/src/sage/symbolic/expression.pyx in sage.symbolic.expression.Expression.integral (/Users/mkoeppe/cvs/sage/src/build/cythonized/sage/symbolic/expression.cpp:60225)()
  11484                     R = ring.SR
  11485             return R(integral(f, v, a, b, **kwds))
> 11486         return integral(self, *args, **kwds)
  11487 
  11488     integrate = integral

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py in integrate(expression, v, a, b, algorithm, hold)
    763         return indefinite_integral(expression, v, hold=hold)
    764     else:
--> 765         return definite_integral(expression, v, a, b, hold=hold)
    766 
    767 integral = integrate

/Users/mkoeppe/cvs/sage/src/sage/symbolic/function.pyx in sage.symbolic.function.BuiltinFunction.__call__ (/Users/mkoeppe/cvs/sage/src/build/cythonized/sage/symbolic/function.cpp:11170)()
    970             res = self._evalf_try_(*args)
    971             if res is None:
--> 972                 res = super(BuiltinFunction, self).__call__(
    973                         *args, coerce=coerce, hold=hold)
    974 

/Users/mkoeppe/cvs/sage/src/sage/symbolic/function.pyx in sage.symbolic.function.Function.__call__ (/Users/mkoeppe/cvs/sage/src/build/cythonized/sage/symbolic/function.cpp:6921)()
    481             for i from 0 <= i < len(args):
    482                 vec.push_back((<Expression>args[i])._gobj)
--> 483             res = g_function_evalv(self._serial, vec, hold)
    484         elif self._nargs == 1:
    485             res = g_function_eval1(self._serial,

/Users/mkoeppe/cvs/sage/src/sage/symbolic/function.pyx in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (/Users/mkoeppe/cvs/sage/src/build/cythonized/sage/symbolic/function.cpp:12417)()
   1059         res = self._evalf_try_(*args)
   1060         if res is None:
-> 1061             return self._eval0_(*args)
   1062         else:
   1063             return res

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py in _eval_(self, f, x, a, b)
    176         for integrator in self.integrators:
    177             try:
--> 178                 return integrator(*args)
    179             except NotImplementedError:
    180                 pass

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/symbolic/integration/external.py in maxima_integrator(expression, v, a, b)
     22         result = maxima.sr_integral(expression,v)
     23     else:
---> 24         result = maxima.sr_integral(expression, v, a, b)
     25     return result._sage_()
     26 

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in sr_integral(self, *args)
    796         """
    797         try:
--> 798             return max_to_sr(maxima_eval(([max_integrate],[sr_to_max(SR(a)) for a in args])))
    799         except RuntimeError as error:
    800             s = str(error)

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in max_to_sr(expr)
   1634         op_max=caar(expr)
   1635         if op_max in special_max_to_sage:
-> 1636             return special_max_to_sage[op_max](expr)
   1637         if not(op_max in max_op_dict):
   1638             op_max_str=maxprint(op_max).python()[1:-1]

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in dummy_integrate(expr)
   1428         integrate(f(x), x, 0, 10)
   1429     """
-> 1430     args=[max_to_sr(a) for a in cdr(expr)]
   1431     if len(args) == 4 :
   1432         return sage.symbolic.integration.integral.definite_integral(*args,

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in max_to_sr(expr)
   1651             op=max_op_dict[op_max]
   1652         max_args=cdr(expr)
-> 1653         args=[max_to_sr(a) for a in max_args]
   1654         return op(*args)
   1655     elif expr.symbolp():

/Users/mkoeppe/cvs/sage/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in max_to_sr(expr)
   1652         max_args=cdr(expr)
   1653         args=[max_to_sr(a) for a in max_args]
-> 1654         return op(*args)
   1655     elif expr.symbolp():
   1656         if not(expr in max_sym_dict):

TypeError: __call__() takes exactly 2 arguments (3 given)

[rwst]

comment:14

The problem on first sight here is that operations on piecewise do not get translated to what is expected, i.e., squaring the pieces. Also it seems that Maxima does not convert them, nor is it considering loading pw.mac without which it simply knows nothing about piecewise functions. So we must either load this package when encountering piecewise in the Maxima interface then convert, or the expression must be simplified, i.e., the operations applied piecewise, probably by Pynac. Or both.

Of course, the integral member function is accessible with pure piecewise objects:

sage: tri_wave1 = piecewise([ [(-1,1), f1^2], [(1,3), f2^2]])
sage: tri_wave1.integral(definite=True)
4/3

[mforets]

comment:15

@rwst: does adding a new __pow__ method in piecewise fix the issue or not? (cf. code+test added in this branch)

---

... well it doesn't work for the OP problem, but that's another thing (misuse?) i guess, since:

sage: integrate(piecewise([ [(-1,1), x], [(1,3), x^2]]), x, -1, 3)
...
ValueError: point 3 is not in the domain

although

sage: integrate(piecewise([ [(-1,1), x], [(1,3), x^2]]), x)
piecewise(x|-->1/2*x^2 - 1/2 on (-1, 1), x|-->1/3*x^3 - 1/3 on (1, 3); x)

and with the current patch one has:

sage: integrate(piecewise([ [(-1,1), x], [(1,3), x^2]])**2, x)
piecewise(x|-->1/3*x^3 + 1/3 on (-1, 1), x|-->1/5*x^5 + 7/15 on (1, 3); x)

---

New commits:

115c198

add pow method to piecewise

[mforets]

Branch: u/mforets/1773

[mforets]

Commit: 115c198

[mforets]

comment:16

Hmmm, what i've uploaded doesn't work well with symbolic expressions as exponents!

sage: f = piecewise([ [(-1,1), x], [(1,3), x^2]])
sage: k = var('k')
sage: f**k
piecewise(k|-->x^k on (-1, 1), k|-->(x^2)^k on (1, 3); k)

the independent variable is misunderstood.

[rwst]

comment:17

This seems to depend on if the expression contains a lexicographically earlier symbol:

sage: f^y
piecewise(x|-->x^y on (-1, 1), x|-->(x^2)^y on (1, 3); x)
sage: f^k
piecewise(k|-->x^k on (-1, 1), k|-->(x^2)^k on (1, 3); k)
sage: f^z
piecewise(x|-->x^z on (-1, 1), x|-->(x^2)^z on (1, 3); x)
sage: f^t
piecewise(t|-->x^t on (-1, 1), t|-->(x^2)^t on (1, 3); t)

[sagetrac-git]

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

41c5796

pass independent variable

[sagetrac-git]

Changed commit from 115c198 to 41c5796

[mforets]

comment:19

Replying to @rwst:

This seems to depend on if the expression contains a lexicographically earlier symbol:

sage: f^y
piecewise(x|-->x^y on (-1, 1), x|-->(x^2)^y on (1, 3); x)
sage: f^k
piecewise(k|-->x^k on (-1, 1), k|-->(x^2)^k on (1, 3); k)
sage: f^z
piecewise(x|-->x^z on (-1, 1), x|-->(x^2)^z on (1, 3); x)
sage: f^t
piecewise(t|-->x^t on (-1, 1), t|-->(x^2)^t on (1, 3); t)

Ok, i see. as a workaround i've specified the var argument to the constructor, which gives the same result as self.variables()[0].

[mforets]

Author: Marcelo Forets

[fchapoton]

Changed dependencies from #14801 to none

[fchapoton]

comment:21

does not apply

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

20cb4f4	add pow method to piecewise
ffea990	pass independent variable

[sagetrac-git]

Changed commit from 41c5796 to ffea990

[videlec]

comment:23

update milestone 8.3 -> 8.4

[fchapoton]

Changed branch from u/mforets/1773 to public/ticket/1773

[fchapoton]

New commits:

fe8304e

add pow method to piecewise

[fchapoton]

Changed commit from ffea990 to fe8304e

[vbraun]

Reviewer: Volker Braun

[vbraun]

Changed branch from public/ticket/1773 to fe8304e