Since algorithms for anti-differentiation need
to know where the break-points are, and integrate
separately, all that matters is that the interval
(and for multivariable, intervals) can be broken up
into disjoint intervals that add up to the whole
interval.
This requires some work describing geometry. Whether
it is done with if/then/else or delta function or ...
is probably not critical. The work is not just a bunch
of programming of easy ideas. There have been people
worrying about this or related geometry problems
for decades. Some of it may be related to the
problem of cylindrical algebraic decomposition, which
has a long history, and in some ways is too simple and
also too expensive.
This is not to say that some hacked up heuristic cannot
work. It will just fall apart sooner or later.
RJF
Robert Dodier wrote:
> Fellow Maximists,
>
> About functions defined piecewise, these are sometimes
> defined using a function denoted U(x) := 1 if x > 0 and
> 0 otherwise (i.e., a unit step function). Then
> U(x-a) - U(x-b) is a "boxcar" from a to b,
> (U(x-a) - U(x-b))*f(x) is a restriction of f to [a,b],
> and so on. Sometimes these constructions are used in
> engineering.
>
> If some such functions appear in an integrand, it
> seems plausible that one could reformulate the integral
> as a collection of integrals in which U doesn't appear,
> but the limits of integration are modified.
> Maybe if a derivative and antiderivative for U are
> added to what Maxima knows, expressions involving U
> could be handled.
>
> A problem which I've attempted to solve before,
> without success -- let V(x) := U(x) - U(x-1).
> The convolution V*V*V*...*V is a piecewise polynomial
> approximation to something proportional to e^{-x^2}.
> What is the polynomial in each piece?
>
> I don't know if defining piecewise functions via U
> is more fruitful than via IF or COND, but it could
> be worth investigating.
>
> For what it's worth,
> Robert Dodier
>
>
>
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