Denis,
Thank you.
If this were really a general solution it would be great, since it
improves the root accuracy by several orders, but I don't fully get the
rationale behind this method.
It seems you are trying to apply a variant of the Raphson-Newton method,
aren't you?
However, in cases like this, in which there are repeated roots, the
derivative approaches zero as you get closer to the root, but the
polynomial goes to zero faster.
In that case the factor 4 may imlpy that the next approximation gets
closer to the actual root. But this is speculation. It could also 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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