Maxima: Piecewise function differentiation and simplification.

Dear Robert,

Thanks for the fast response, detailed analysis and program - much 
appreciated.

One of the program steps does not to seem to be working correctly 
with my maxima 5.10, 5.12:

(%i100) vpen;

(%o100) if u1^2*sin(x3)^2+u1^2*cos(x3)^2 > vmax
            then (u1^2*sin(x3)^2+u1^2*cos(x3)^2-vmax)^2
            elseif u1^2*sin(x3)^2+u1^2*cos(x3)^2 < 0
            then (u1^2*sin(x3)^2+u1^2*cos(x3)^2)^2 else 0
(%i101) odd_args(vpen);

(%o101) [u1^2*sin(x3)^2+u1^2*cos(x3)^2 > vmax,
         u1^2*sin(x3)^2+u1^2*cos(x3)^2 < 0,true]
(%i102) even_args(vpen);

(%o102) [(u1^2*sin(x3)^2+u1^2*cos(x3)^2-vmax)^2,
         (u1^2*sin(x3)^2+u1^2*cos(x3)^2)^2,0]
(%i103) op(vpen);



(%o103) ?mcond
(%i104) 

The step, op(vpen), returns ?mcond, instead of "if", and the relevant part 
of the program is not executed.

Is it possible to modify the program so that it runs correctly (with maxima 
5.10, 5.12) ?

Thanks very much.

Regards,

C. Frangos.





On Thursday 31 July 2008 17:18, you wrote:
> On 7/31/08, Constantine Frangos <cfrangos at telkomsa.net> wrote:
> >  (%i69) vpen : penalty(u1^2*sin(x3)^2+u1^2*cos(x3)^2,0,vsqmax);
> >
> >  (%o69) if u1^2*sin(x3)^2+u1^2*cos(x3)^2 > vmax
> >            then (u1^2*sin(x3)^2+u1^2*cos(x3)^2-vmax)^2
> >            elseif u1^2*sin(x3)^2+u1^2*cos(x3)^2 < 0
> >            then (u1^2*sin(x3)^2+u1^2*cos(x3)^2)^2 else 0
> >
> >  I want to differentiate vpen wrt u1 and then wrt x3 directly, dvpendu1 :
> >  diff(vpen,u1), dvpendx3 : diff(vpen,x3), but could not do this
> > correctly.
>
> Assuming that F(if a then x elseif b then y elseif ... else z) is
> equal to if a then F(x) elseif b then F(y) elseif ... else F(z),
> we can do the following. (There are cases for which distributing
> over the branches of a conditional doesn't yield the correct result.)
>
> ifmap (f, e) := if not atom (e) and op (e) = "if" then apply ("if",
> join (odd_args (e), map (f, even_args (e)))) else e;
>
> :lisp (defun $even_args (e) `((mlist) ,@(odds (rest e) 0)))
> :lisp (defun $odd_args (e) `((mlist) ,@(odds (rest e) 1)))
>
> ifmap (foo, if a then x elseif b then y else z);
>  => if a then foo(x) elseif b then foo(y) else foo(z)
>
> ifmap (lambda ([e], diff (e, u1, 1, x3, 1)), vpen);
> => if u1^2*sin(x3)^2+u1^2*cos(x3)^2 > vmax then 0
>            elseif u1^2*sin(x3)^2+u1^2*cos(x3)^2 < 0 then 0
>            else 0
>
> I think that's correct --- because of the sin and cos terms in vpen,
> the derivative has terms which cancel out.
>
> >  penalty(fx,amin,amax) := block(
> >
> >  [pen],
> >
> >  'if (fx > amax) then
> >  pen : (fx - amax)^2
> >  elseif (fx < amin) then
> >  pen : (fx - amin)^2
> >  else
> >  pen : 0);
>
> For what it's worth, this can be phrased more briefly as
>
> penalty (fx, amin, amax) :=
>   if (fx > amax)
>     then (fx - amax)^2
>   elseif (fx < amin)
>     then (fx - amin)^2
>   else 0;
>
> I'm generally in favor of omitting unneeded stuff; it makes it easier
> to see what' s going on.
>
> best
>
> Robert Dodier