Dear Stefan:
You posed a legitimate problem: how should symbolic
computation handle piecewise defined functions, and in
particular, how to integrate such a function.
Maple and Mathematica both can handle piecewise functions.
Look up "piecewise" from Maple Help. You can easily define
a piecewise function, and differentiate or integrate it.
Indeed, Maple says:
The piecewise function can be differentiated, integrated,
simplified, plotted, and used in the following types of
differential equations: constant coefficients and
discontinuous perturbation function, general first-order
linear, Riccati, and some other classes which are handled
by integration or variation of parameter. See
dsolve[piecewise] for more details. series, limit, abs,
and signum can handle the piecewise function.
As example, the desired solution the problem of
integrating f(x) from 0 to t, where f(x) is 2x if x < 10
and 5x^2 otherwise, should be the function g(t), defined
as t^2 if t < 10 and -4000/3 +(5 t^3)/3 otherwise. Maple
does exactly that. In fact, I even tried to integrate
f(f(x)) and f(f(x+1)) and Maple does it with no problems
with all the cases covered.
Mathematica has a similar function called Piecewise to
construct piecewise functions, and like Maple, Piecewise
can be used in such functions as Integrate, Minimize,
Reduce, DSolve and Simplify, as well as their numeric
analogs.
This may be an uncovered domain in Axiom. A search for
"piecewise" shows no hits. I think piecewise functions
have to be separately handled, particularly in case
analysis (possibly involving semi-algebraic sets and CAD)
if there is any indefiniteness in the answer (like an
indefinite integral). There is some evidence that if the
user does not use the built-in "piecewise" or "Piecewise"
function, but uses an if-then-else construction, neither
Maple nor Mathematica can handle subsequent mathematical
calculations. For example, the system would not do the
case analysis, much less the "simplification"
automatically, but would present the result as the
integral of If[x < 10, 2 x, 5 x^2] (in Mathematica; I did
not try Maple). Even when the case analysis is done, it
would still not simplify or evaluate the integrals:
h[x_] := If[x < 10, Integrate[2 y, {y, 0, x}],
Integrate[2 y, {y, 0, 10}] + Integrate[5 y^2, {y, 10,
x}]]
when h[x] is called. It will evaluate on numerical inputs.
In our earlier discussions, we were "lured" into using
"if-the-else" constructions and thus got the feeling that
this is difficult to handle. The confusion is that we
interpret "x < 10" as an binary relation, whereas it
should be handled as a semi-algebraic set (in one
dimension, this is just an interval)!
However, the algorithms seem to be there, and someone
should implement them in Axiom if it is not already done
but hidden in some obscure packages.
William
On Wed, 04 May 2011 22:37:48 +0200
Stefan Karrmann <[email protected]> wrote:
Dear all,
thanks for your answers. They clears a lot.
I actually want to integrate test1 and solve an
differential equation
with it.
E.g.
test2 x == rho * test1 x
y = operator 'y
odeq := D(y x) = test2 x
solve(odeq, y, x)
Obviously, the solution is "formally"
y_sol x == integrate(test2 x,x)
Kind regards,
Stefan
Am Dienstag, den 03.05.2011, 11:21 +0200 schrieb Ralf
Hemmecke:
Dear Stefan,
as others already have pointed out, for Axiom, your
question is not
really well posed.
In Axiom
if x<10 then 2*x else 5*x^2
is *not* an expression (as you might know it from other
untyped CAS like
Mathematica or Maple), but rather a programming language
construct. In
other words, if Axiom sees this, it is evaluated. So the
result is
either 2*x or 5*x^2 depending on the (boolean) outcome
of the evaluation
of x<10.
I think, Bill suggested to use something like InputForm.
There it would
be possible to represent an if-expression unevaluated.
But you should rather say what you actually want (it's
not the same what
you expect).
In order for us to suggest you a proper way to handle
your use case, you
should tell us why you want a piecewise function and
(more important)
what you later want to do with that function.
Until we have that information, everything would be just
digging in the
dark.
Ralf
On 04/30/2011 08:40 PM, Stefan Karrmann wrote:
> Dear all,
>
> I'm new to axiom and have a problem with piecewise
functions.
>
> test1 (x | x< 10) == 2*x
> test1 (x | x >= 10) == 5*x^2
> [was typo: test1 (x | x< 10) == 5*x^2]
> test1
> ->
> test1 (x | x< 10) == 2x
> test1 (x | ^ x< 10) == 5x
>
Type:
FunctionCalled
> test1 y
> ->
> 2
> 5y
>
> I expected something like (if y< 10 then 2*y else
5*y**2).
>
> How is it possible to pass a Variable to a piecewise
function respecting
> the pieces?
>
> PS: Using a block and => or explicit if-then-else
does not help.
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William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
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