[Scilab-users] improve accuracy of roots

[Federico Miyara]

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-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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