I don’t know anything about the internals of the float package.
For that you will have to contact it’s maintainer Laurent Bartholdi. See the
package documentation for details.
I do observe though, that the imaginary parts of the roots are very small —
less that 10^-7, so I imagine that the “correct” roots are very close together
and rounding error is then shifting them off the real line.
You could try setting a higher floating point precision.
Steve
> On 19 Apr 2015, at 04:36, Dr. Kashyap Rajeevsarathy <kash...@iiserb.ac.in>
> wrote:
>
> Dear Forum,
>
>
> Kindly disregard my earlier email:
>
>
> I have been having some issues computing the roots of polynomials using
> "RootsFloat" function of the "Float" package. Please consider the following
> example:
>
> gap> a := last;
> [ [ 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
> [ 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
>
> [ 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
> [ 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
> [ 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
> [ 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0 ],
> [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1 ],
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1 ],
> [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0 ],
> [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0 ],
> [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1 ],
> [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0 ] ]
>
> gap> a = TransposedMat(a);
> true
> gap> LoadPackage("float");
> Loading FLOAT 0.6.2 ...
> true
> gap> SetFloats(MPFR);
> gap> pol := CharacteristicPolynomial(a);
> x_1^20-40*x_1^18+610*x_1^16-8*x_1^15-4640*x_1^14+40*x_1^13+
> 19475*x_1^12-560*x_1^11-46380*x_1^10+5560*x_1^9+61610*x_1
> ^8-19600*x_1^7-42220*x_1^6+25784*x_1^5+8265*x_1^4-11560*
> x_1^3+3700*x_1^2-400*x_1
> gap> coeff := CoefficientsOfUnivariatePolynomial(pol);
> [ 0, -400, 3700, -11560, 8265, 25784, -42220, -19600, 61610, 5560, -46380,
> -560, 19475, 40, -4640, -8, 610, 0, -40, 0, 1 ]
> gap> RootsFloat(coeff*1.0);
> [ .0e0, .381966e0, .381966e0, .618034e0+.218005e-7ⅈ, .618034e0-.218005e-7ⅈ,
> .618034e0+.807401e-7ⅈ, .618034e0-.807401e-7ⅈ, -.138197e1, .2e1, -.138197e1,
> -.161803e1, -.161803e1-.141802e-8ⅈ, .2e1, -.161803e1+.136277e-8ⅈ,
> -.161803e1, .261803e1, .261803e1, -.361803e1, -.361803e1, .4e1 ]
> Note that that the RootsFloat function is showing complex roots
> (eigenvalues) for the characteristic polynomial of a real symmetric matrix.
>
> It would greatly help of you could me what's going wrong here.
>
> Best regards,
> Kashyap
>
>
> --
> Kashyap Rajeevsarathy
> Assistant Professor,
> Indian Institute of Science Education and Research (IISER) Bhopal,
> Indore By-pass Road,
> Bhauri, Bhopal - 462066,
> Madhya Pradesh, India.
> Phone: +91-755-669-2364
> Website: https://home.iiserb.ac.in/~kashyap
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