Hello,
After a quick search on the internet I have found and translated in
Scilab "multroot", a Matlab Package computing polynomial roots and
multiplicities. If you are interested proceed to this url:
http://bugzilla.scilab.org/show_bug.cgi?id=15349#c9
Enjoy !
S.
Le 14/01/2019 à 21:07, Federico Miyara a écrit :
Denis,
What I meant is that convergence is a limiting process. On average, as
the number of iterations rises you´ll be closer to the limit, bu there
is no guarantee that any single iteration will bring you any closer;
it may be a question of luck. Maybe (though it would require a proof,
it is not self-evident for me) in the exact case of a single multiple
root as (x - a)^n the convergence process is monotonous, but what if
you have (x - a1)* ... * (x - an) where ak are all very similar but
not identical, say, with relative differences of the order of those
reported by the application of the regular version of roots.
Regards,
Federico Miyara
On 14/01/2019 13:47, CRETE Denis wrote:
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-boun...@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-boun...@lists.scilab.org] De la part de Federico Miyara
Envoyé : jeudi 10 janvier 2019 00:32
À :users@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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