Solving quintics

Fat i

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Sep 6, 2022, 2:43:42 AM9/6/22

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Hello all,

if you are interested in solving polynomial equations, quintics in particular, this might be of use.

With a class library I developed, it is now possible to solve irreducible solvable Bring-Jerrard quintics, i.e. f(x) = x^5 + ax + b, based on the Spearman-Williams approach. Coefficients are calculated up to a certain limit which is based on the Cantor counting scheme of rational numbers with default maxValue = 20. Higher number coefficients may be generated if needed; however, due to O(n^4) complexity in generating the Spearman Williams coefficients, this is limited to maxValue = 100 currently. Contact the author for an extension if higher number coefficients are required.

You may have a look at

CoCalc -- Development

including class files and a test script called TestWorksheetBJ.sagews as well as a README file which contains how to get started. Let me know if you have questions.

Any feedback is appreciated. As an additional feature, one might imagine to look at Tschirnhaus transformations of general quintics to find conditions which yield solvable Bring-Jerrard quintics. Currently I don´t know how complicated that would be - just an idea.

Best regards

Achim

Achim Plum, independent Computer Scientist (with a focus on Mathematics)

Germany

David Roe

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Sep 6, 2022, 10:34:10 AM9/6/22

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Can you explain what benefit your solver has over Sage's built-in root finding (which is not restricted to Bring-Jerrard quintics)?

David

sage: R.<x> = CC[]
sage: f = x^5 + 5*x + 1
sage: %time L = f.roots()
CPU times: user 3.55 ms, sys: 1.73 ms, total: 5.28 ms
Wall time: 11.1 ms
sage: L
[(-0.199936102171220, 1),
(-1.00449745579684 - 1.06094650640604*I, 1),
(-1.00449745579684 + 1.06094650640604*I, 1),
(1.10446550688245 - 1.05982966915252*I, 1),
(1.10446550688245 + 1.05982966915252*I, 1)]

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Fat i

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Sep 7, 2022, 12:12:50 PM9/7/22

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Hi David,

I am wondering if the root finding in sage is a numeric algorithm. May be it is just an academic exercise, but the roots found here are "exact" roots as symbolic expressions. The rounding is only done in the test script at the end to check that the found expressions are indeed roots because the symbolic expressions become rather complicated.

Just to mention that I used this project to learn to develop some software in sage and python. For sure there is still much to learn for me about the sage environment and all its classes and capabilities.

Kind regards

Achim

Fat i

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Oct 9, 2022, 2:50:21 AM10/9/22

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In the last weeks I took a deeper look into publications of S.B. Buya (2017, 2021). As a result and as I had expected, I think the methods cannot be used for general solutions of quintic equations. Thus, I discontinued efforts in developing some software in Sage for it.