[Scilab-users] improve accuracy of roots

[CRETE Denis]

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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