Re: [Axiom-math] if-expression and variables

[Stefan Karrmann]

Dear all,

we can help ourselves in some cases by:

sign x == (abs x)/x
H x == (1 + sign x)/2
step (a,b,x) == (H(x-a)) * (H(b-x))

The drawback is that we get a 'Division by zero'.

Kind regards,
Stefan

Am Donnerstag, den 05.05.2011, 00:23 -0400 schrieb William Sit:
> 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 <address@hidden> 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.
> > 
> > 
> > _______________________________________________
> > Axiom-math mailing list
> > address@hidden
> > https://lists.nongnu.org/mailman/listinfo/axiom-math
> 
> 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/