It's been twelve days since I got the Wildberger-Rubine PDF. After some adventures in C#, I can now report on my first W-R construction of my first quintic. This is the easy one: -4+x+x5. Eisenstein picked that one in 1844, which holds a powerseries to solve x for 0.
Newton, Raphson & Co. told me the answer: 1.22634... ish, for precision 0.000001. Given start 1, this took five (5) iterations. How does Eisenstein hold against that?
Eisenstein actually forced negatives on the first two quintic-coefficients: -4 > 4, +x > -1. So I get 4 - 4^5 + (10/2!)4^9 - (15*14/3!)4^13 + (20*19*18/4!)4^17 [at m5=4].
At precision 4 this thing is running away and I'm not seeing relief as I increase that. "Extremely tedious", indeed.
Also, to get the series-coefficients, we have the choice of calculating every time we do it, or else compiling a "geode" based on the polynomial. Quintics get a 4-dimensional hypercube. As precision "number go up", hypercube go up by, what, 4^n? Tell you what, there's no storing all that in RAM. Precision 7 is about my limit before this machine starts wheezing. This needs to be dumped into the file-system, whence the function must stream it.
By that point to find the correct factor is getting slow. I'm sure I could index it... at cost of still more space. Eisenstein can drop a few coeffs.
But astronomers want quintics for Lagrange. The physics are the physics.
To sum up (heh) I don't see the utility in Wildberger-Rubine's expanded-Eisenstein powerseries for quintics ... let alone beyond. Newton remains archangel of the three halo-orbits.
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