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I have a polynomial of the 6th degree. For this Polynom, I want to get all real roots. My problem is that all the Methods I read about are only to get one root. Is there a way how I could get all the roots?

For example: With a polynom of 5th degree I can get the derivation. Solve it explicit and then use the solutions to get the intervals for the Newton Method.

Is there maybe also a way to solve this problem with a polynomial of 6th degree?

Thanks for your Help

IlPad
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  • Why not just do it as for the degree 5 polynomials? Compute derivatives, find zeros as you did before, then you have the intervals. – M. Winter Jul 11 '18 at 10:12
  • Would this help? – a concerned citizen Jul 11 '18 at 10:12
  • @M.Winter Because if i take the derivative i get an polynom of 5th degree. Then i have to solve this polynom again numerical :/ I think it's possible but i thought that maybe there would be a faster solution. @a concernced citizien i cannot reach the website from your link. – IlPad Jul 11 '18 at 10:15
  • Have a look at https://en.wikipedia.org/wiki/Jenkins%E2%80%93Traub_algorithm – Claude Leibovici Jul 11 '18 at 10:20
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    @aconcernedcitizen Can you please not post an onion link? – Parcly Taxel Jul 11 '18 at 10:21
  • @ClaudeLeibovici Thanks for the answer. I read about this Method but the problem here is that i have to create an Matlab implementation. I searched for an Matlab Implementation but there was only Python and c++. And i doubt it that i am able to translate this code to an Matlab Code because it seems really complicated. There is also the possibility to use the mex function of Matlab but i want to use only Matlab skripts without other implementations. That's why i want to try to solve this problem with "easy" Methods – IlPad Jul 11 '18 at 10:27
  • @aconcernedcitizen. it seems that you diud not choose the most appropriate user name. Please, do not play this game on this site. – Claude Leibovici Jul 11 '18 at 10:45
  • I am afraid that there is nothing really simple. What you could do is to find one root (try to bracket a root) and then deflate the polynomial. – Claude Leibovici Jul 11 '18 at 10:51
  • @ClaudeLeibovici Sorry for the link, I thought I grabbed the wikipedia link (the former link is to duckduckgo.onion, and the search phrase is what is seen). Why do you say that about the username? What games are you talking about? – a concerned citizen Jul 11 '18 at 14:28

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The Durand-Kerner and Aberth-Ehrlich methods find all roots simultaneously and are as easy to implement as Newton’s method. Both methods are mentioned on this overview page.

WimC
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  • Thanks for the answer. I read also about this methods but after some research i found that the Durand Kerner Method is unstable and doesn't converge everytime – IlPad Jul 11 '18 at 11:55
  • Have you tried it? I haven’t come across a situation where it did not give all roots. You should take care to start with “random” initial complex roots. If all roots are known to be real then you can start with random real initial roots. The Aberth method converges quite a bit faster though in my experience. – WimC Jul 11 '18 at 12:24