PHCpack is probably the best thing to use for numerical solutions of
polynomial equations, or equations that can be converted into
polynomials (e.g.. sin and cos). My interface was recently included
in sage but the PHCpack developers are working on a more sophisticated
one which will hopefully make mine obsolete. But until then, give it
a try and I'll be happy to help if you have problems.
Cheers,
Marshall Hampton
On Oct 29, 4:40 am, Jason Grout <[EMAIL PROTECTED]> wrote:
> William Stein wrote:
> > On 10/28/07, David Joyner <[EMAIL PROTECTED]> wrote:
> >> On 10/26/07, Jason Grout <[EMAIL PROTECTED]> wrote:
> >>> I'm trying to numerically solve a system of equations. Currently I have:
>
> >>> sage: var('x y p q')
> >>> sage: eq1 = p+q==9
> >>> sage: eq2 = q*y+p*x==-6
> >>> sage: eq3 = q*y^2+p*x^2==24
> >>> sage: solve([eq1,eq2,eq3,p==1],p,q,x,y)
> >>> [[p == 3, q == 6, x == (-2*sqrt(10) - 2)/3, y == (sqrt(2)*sqrt(5) -
> >>> 2)/3], [p == 3, q == 6, x == (2*sqrt(10) - 2)/3, y == (-sqrt(2)*sqrt(5)
> >>> - 2)/3]]
>
> >>> I'd like the answer to be a numeric approximation, though (i.e.,
> >>> x==-2.77485177345). I can't find any other solving routines other than
> >>> the symbolic one, though. Is there a way to numerically approximate the
> >>> solution, other than going through each solution and calling n() on the
> >>> left side of each symbolic expression?
>
> >> Examples using numpy and octave are give in the constructions cookbook:
>
> >>http://www.sagemath.org/doc/html/const/node35.html
>
> > Note that his systems are nonlinear, but your examples
> > in the constructions book are linear. So it's sort of
> > a different thing.
>
> William is correct here.
>
>
>
> > Some options for solving systems of nonlinear equations *numerically*
> > include:
>
> > * [100% sage] Use scipy, which wraps minpack:
> > Type
> > sage: import scipy.optimize
> > sage: scipy.optimize.fsolve ?
> > to hopefully get going. Also just try scipy.optimize.[tab]
>
> I saw this late Saturday night and tried to get it working, but haven't
> been successful yet. I think it's a matter of figuring out what fsolve
> is expecting to be passed and how to interpret the results (e.g., it
> appears that fsolve expects the function to take a vector, or list of
> parameters, and not just a bunch of parameters).
>
>
>
> > * [Not sage] Use phcpack, which is open source (GPL'd) but not
> > included in Sage. It is able to do amazing things with solving
> > algebraic systems numerically, using a method called "Polynomial
> > Homotopy Continuation". There is a Sage interface to phcpack, that
> > Marshall Hampton worked on (he frequently posts on sage-devel).
>
> Thanks for this tip as well.
>
> For now, I just ended up making a patch for the functionality that I
> wanted (isn't that what we encourage here? :). Carl merged it into
> 2.8.10 last night. It adds a solution_dict optional parameter to solve
> that returns a dictionary of the solutions. This makes it easy to refer
> to the values in a solution. Using the solution_dict option, my
> original problem may be solved with:
>
> sage: var('x y p q')
> sage: eq1 = p+q==9
> sage: eq2 = q*y+p*x==-6
> sage: eq3 = q*y^2+p*x^2==24
> sage: solutions=solve([eq1,eq2,eq3,p==1],p,q,x,y, solution_dict=True)
> sage: for solution in solutions: print solution[p].n(), solution[q].n(),
> solution[x].n(), solution[y].n()
>
> However, it would still probably be more efficient to use fsolve or the
> other method above since I don't need the symbolic solution. I'll look
> at getting fsolve to work and post an example of working code.
>
> Thanks,
>
> Jason
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