> On Jul 3, 2014, at 2:30 PM, Gilles <gil...@harfang.homelinux.org> wrote:
>
>> On Thu, 3 Jul 2014 18:14:41 +0200, Axel wrote:
>> Ok,
>> but what about my main question:
>> "Could we have a roots() method in PolynomialFunction class?"
>>
>> Which in the first step delegates to LaguerreSolver#solveAllComplex()?
>
> I guess that people want to be careful before changing the API of
> "PolynomialFunction". [E.g. it would create a dependency towards
> the "o.a.c.m.complex" package. It might be better to add the
> functionality in that package (e.g. in "ComplexUtils").]
>
> Could you explain what you need with more details?
> In particular, what do you expect to get in addition to what the
> following code can already provide?
>
> ---CUT---
> import org.apache.commons.math3.analysis.polynomials.PolynomialFunction;
> import org.apache.commons.math3.analysis.solvers.LaguerreSolver;
> import org.apache.commons.math3.complex.Complex;
>
> public class PolynomialRoots {
> private static final LaguerreSolver solver = new LaguerreSolver();
>
> public static Complex[] solve(PolynomialFunction p) {
> return solver.solveAllComplex(p.getCoefficients(), 0d);
> }
> }
> ---CUT---
I agree with Thomas that it would be good to expose a Complex[] roots() method
directly in the PolynialFunction class. We can then choose whatever numerical
technique we like to back it, starting with Laguerre and refining /
specializing to data as we get better ideas.
Phil
>
>
> Best regards,
> Gilles
>
>
>>
>>> On Sat, Jun 28, 2014 at 8:26 PM, Ted Dunning <ted.dunn...@gmail.com> wrote:
>>> That is one article, but it doesn't actually compare the numerical
>>> stability or efficiency of this method. It just invokes the stability of
>>> "numerical linear algebra". Whether this is a good way to compute roots is
>>> an open question.
>>>
>>> The other major question here is operation count. Computing eigenvalues of
>>> an explicit matrix is likely to be very intensive computationally. The
>>> wikipedia page on root-finding mentions in passing when is says that
>>> matrix-free methods are typically used with the eigenvalue approaches.
>>>
>>> This suggests that the preferable means to implement this idea is not to
>>> build the matrix in question, but to use an abstract linear operator which
>>> implements multiplication making use of the special form of the companion
>>> matrix. I am not sure if this approach is viable in the Commons matrix
>>> framework. I suspect not, but really don't have much of a clue about the
>>> real state of things. If the eigenvalue objects accept something like a
>>> LinearOperator object, then it is likely to work.
>>>
>>> This article[1] seems to suggest that there may be some numerical issues
>>> with large coefficients. That isn't surprising since many algorithms have
>>> similar problems.
>>>
>>> [1]
>>> http://noether.math.uoa.gr/conferences/sla2014/sites/default/files/Dopico.pdf
>>>
>>>
>>>> On Sat, Jun 28, 2014 at 6:33 AM, Axel <axel...@gmail.com> wrote:
>>>>
>>>> On Fri, Jun 27, 2014 at 10:56 PM, Thomas Neidhart
>>>> <thomas.neidh...@gmail.com> wrote:
>>>> ...
>>>> > I did take a look at the stackoverflow question, and there is already a
>>>> > way to do this in Commons Math using the LaguerreSolver via the
>>>> > solveComplex and solveAllComplex methods.
>>>> >
>>>> > But it might be good to support a second way using EigenDecomposition as
>>>> > a stand-alone solver.
>>>> >
>>>> > I like the idea to add a roots() method to PolynomialFunction, but which
>>>> > method to compute the roots is more robust?
>>>>
>>>> The attached link in the stackoverflow question to this paper:
>>>> http://techdigest.jhuapl.edu/TD/td2804/Williams.pdf
>>>>
>>>> has this conclusion:
>>>> "We have discussed some eigenvector methods for finding the roots of multi-
>>>> variate polynomials. Unlike iterative, numerical methods typically
>>>> applied to this
>>>> problem, the methods outlined in this article possess the numerical
>>>> stability of
>>>> numerical linear algebra, do not require a good initial guess of the
>>>> solution, and give all
>>>> solutions simultaneously. Furthermore, if the initial guess is poor
>>>> enough, the methods
>>>> outlined herein may converge more quickly than iterative methods."
>>>>
>>>> So I think the "EigenDecomposition method" is more appropriate if you
>>>> don't have an initial guess to start from getting the roots!?
>>>>
>>>> --
>>>> Axel Kramer
>>>>
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