Re: [Help-gsl] solution of 3d quadratic algebraic system

[FARKAS, Illes]

Thanks Juan,

The equations are smooth (const, linear and quadratic terms on the r.h.s.),
so the best I can come up with right now is to iterate with the following
method.

(A) Initialize cube size to (1,1,1) and cube center to (x1=0.5, x2=0.5,
x3=0.5)
(B) Compute the function (f1,f2,f3) in all equidistant 11 * 11 * 11 grid
points of the cube
(C) Find shortest (f1,f2,f3) vector and based on that decide if we're done
(D) If we're not yet done, then:
   - center cube on the (x1,x2,x3) location of the shortest (f1,f2,f3)
vector
   - reduce cube size to 60%
   - goto (B)

Do you think this makes sense?

Thanks
Illes
-- 
http://hal.elte.hu/fij


2012/10/30 Juan Pablo Amorocho <address@hidden>

> Hi Illes,
> As far as I know, you can't tell Newton to stay within a "region of
> interest" ( @help-gnu If I'm wrong please correct me). All you can do is to
> provide the solver with a reasonable starting point. You could also take a
> look at your function around that region of interest, and see if it behaves
> nasty.  Maybe this could help:
>
> http://en.wikipedia.org/wiki/Newton_iteration#Practical_considerations
>
> - Juan Pablo
>
> On Oct 30, 2012, at 6:40 AM, "FARKAS, Illes" <address@hidden> wrote:
>
> > 2012/10/29 FARKAS, Illes <address@hidden>
> >
> >> 2012/10/28 Rhys Ulerich <address@hidden>
> >>
> >>>> Can you please suggest a fast GSL method / algorithm to find the
> >>> solutions
> >>>> of a quadratic 3d system of algebraic equations?
> >>>>
> >>>> In the reduced form all 3 equations have zero on the l.h.s., and on
> the
> >>>> r.h.s. there are constant, linear and quadratic terms composed of x1,
> >>> x2,
> >>>> x3 (the three variables).
> >>>
> >>> Newton iteration, especially if you provide analytic Jacobian, should
> do
> >>> well here. There may be faster things that can return multiple
> solutions,
> >>> however.
> >>>
> >>> - Rhys
> >>>
> >>
> >> Thanks, I've chosen the hybrid algorithm<
> http://www.gnu.org/software/gsl/manual/html_node/Example-programs-for-Multidimensional-Root-finding.html
> >
> >> .
> >
> >
> > Hello,
> >
> > The 3 variables represent biochemical concentrations normalized to the
> > interval [0,1]. Can I tell the solver (or another solver) that it should
> > stay inside this interval? I keep receiving solutions outside this
> > interval, which are mathematically OK, but biochemically really nonsense.
> > Also, with some help I'm ready to read/write the source code.
> >
> > Many thanks!
> >
> > Illes
>
>