[VA] SRC #012b - Then and Now: Root

[PeterP]

(11-09-2022 04:37 PM)C.Ret Wrote:  So there is a real root between -.996 and -.997 and another one between -.999 and -1, in accordance to my guess that real roots (if existing) would be close to -1.

That is very perplexing. At first I immediately gave up given then large number of coefficients and powers thinking of the solve algorithm we have for the hp41. But then I decided to just play around a little bit, using simpler polynomials.

I made my way in a somewhat (ok, very) manual sleuthing fashion across the first 30 or so polynomials. And then looked at the real roots. In all cases, I was not able to find a real root for a polynomial which ended in a even power. Which peter-intuitively (ie inferior, wrong, intuition) made sense - the last term has the highest exponent and highest multiplier and as such dominates the prior term. And this is true for all pairs of "even, odd" powers. So each pair creates a positive overhang, making it impossible to converge. Or so my logic went. Clearly, contra factum non est discudandum so I need to think more about this (or get some help from the team).

Using the real roots for the odd-ending polynomials, I saw a really nice curve that looked like a logarithmic curve. And low and behold, a logarithmic fit gives some 92% R^2. And would point to a solution slightly bigger than -1. So I need to think about why that pretty smooth and steady fit for the odd-ending polynomials would have a minimum and turn back up. Only thing I can think of is the space between primes is getting larger, creating bigger gaps between the even-odd pairs.

However, all my hopes sank when I realized that we are looking for the smallest absolute solution. And there is no way I could think of (so far) to find the smallest absolute value directly, out of 5000 pairs of conjugated roots for the full polynomial.

I will think how to do that in the smaller polynomial case.

it is also possible that I can find a way to show that the 10 digit precision of the hp41 only goes to a much smaller polynomial and as such I can just solve a much smaller polynomial as an approximation inside the accuracy of the HP chosen.