[Scilab-users] improve accuracy of roots

[CRETE Denis]

Thank you Frederico!
According to the page you refer to, the method seems to converge more rapidly with this factor equal to the multiplicity of the root. 
About overshoot, it is well known to occur for |x|^a where a <1. But for a>1, the risk of overshoot with the Newton-Raphson method seems to be very small...
Best regards
Denis 

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-----Message d'origine-----
De : users [mailto:users-bounces at lists.scilab.org] De la part de Federico Miyara
Envoyé : samedi 12 janvier 2019 07:52
À : Users mailing list for Scilab
Objet : Re: [Scilab-users] improve accuracy of roots


Denis,

I've found the correction here,

https://en.wikipedia.org/wiki/Newton%27s_method

It is useful to accelerate convergence in case of multiple roots, but I 
guess it is not valid to apply it once to improve accuracy because of 
the risk of overshoot.

Regards,

Federico Miyara


On 10/01/2019 10:32, CRETE Denis wrote:
> Hello,
> I tried this correction to the initial roots z:
>
> z-4*(1+z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))')
>   ans  =
>
>    -1. - 1.923D-13i
>    -1. + 1.189D-12i
>    -1. - 1.189D-12i
>    -1. - 1.919D-13i
>
> // Evaluation of new error, (and defining Z as the intended root, i.e. here Z=-1):
> z2=z-4*(z-Z).^4 ./([ones(z),z,z.^2,z.^3]*(C(2:5).*(1:4))')
> z2 - Z
>   ans  =
>
>     2.233D-08 - 1.923D-13i
>    -2.968D-08 + 1.189D-12i
>    -2.968D-08 - 1.189D-12i
>     2.131D-08 - 1.919D-13i
>
> The factor 4 in the correction is a bit obscure to me, but it seems to work also for R=(3+p)^4, again with an accuracy on the roots of a ~2E-8.
>
> HTH
> Denis
>
> -----Message d'origine-----
> De : users [mailto:users-bounces at lists.scilab.org] De la part de Federico Miyara
> Envoyé : jeudi 10 janvier 2019 00:32
> À : users at lists.scilab.org
> Objet : [Scilab-users] improve accuracy of roots
>
>
> Dear all,
>
> Consider this code:
>
> // Define polynomial variable
> p = poly(0, 'p', 'roots');
>
> // Define fourth degree polynomial
> R = (1 + p)^4;
>
> // Find its roots
> z = roots(R)
>
> The result (Scilab 6.0.1) is
>
>    z  =
>
>     -1.0001886
>     -1. + 0.0001886i
>     -1. - 0.0001886i
>     -0.9998114
>
> It should be something closer to
>
>     -1.
>     -1.
>     -1.
>     -1.
>
> Using these roots
>
> C = coeff((p-z(1))*(p-z(2))*(p-z(3))*(p-z(4)))
>
> yield seemingly accurate coefficients
>    C  =
>
>      1.   4.   6.   4.   1.
>
>
> but
>
> C - [1  4  6 4 1]
>
> shows the actual error:
>
> ans  =
>
>      3.775D-15   1.243D-14   1.155D-14   4.441D-15   0.
>
> This is acceptable for the coefficients, but the error in the roots is
> too large. Somehow the errors cancel out when  assembling back the
> polynomial but each individual zero should be closer to the theoretical
> value
>
> Is there some way to improve the accuracy?
>
> Regards,
>
> Federico Miyara
>
>
>
>
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