On 7/13/07, sen1 at math.msu.edu <sen1 at math.msu.edu> wrote:
> Hello,
> Using the function 'find_root' one can find an intersection of
> two explicit curves
> x -> [x, g(x)]
> x -> [x, h(x)]
>
> in an interval [a,b] such that g - h have different signs at the
> endpoints.
>
> Does maxima have an analogous routine to find the intersection of two
> parametrized curves
>
> g: t -> [x(t), y(t)], 0 < t < 1
> h: s -> [u(s), v(s)], 0 < s < 1
>
> where it is known that such an intersection exists?
In many cases, I'm guessing the answer is "yes", for the following reasons.
If x,y,u,v are polynomials in t then Grobner bases will
put these curves in algebraic form. Maxima has some Grobner bases routines.
An example is in
http://cadigweb.ew.usna.edu/~wdj/book/node86.html
Once they are in algebraic form, again Grobner bases can be used
to find 0-dimensional solutions.
Does this make sense?
>
> Note: the parameters are different, so any intersection requires the
> solution of a pair of equations If not, does anyone know a good
> reference for such a routine.
>
> I am interested in the case of transcendental functions or high degree
> polynomials. In the cases of interest to me, newton's method has degeneracies or
> numerical overflows and does not work. So, I would prefer some sort
> of 'bisection' method.
>
> One special case of interest to me is the one in which the curves have
> a non-degenerate crossing; i.e., have linearly independent tangent
> vectors at the crossing.
>
> TIA,
>
> -sen
>
>
> --
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> | Sheldon E. Newhouse | e-mail: sen1 at math.msu.edu |
> | Mathematics Department | |
> | Michigan State University | telephone: 517-355-9684 |
> | E. Lansing, MI 48824-1027 USA | FAX: 517-432-1562 |
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