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

[J-F Garnier]

(11-09-2022 08:13 PM)PeterP Wrote:  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).

Let's take the last two terms:
104729 * X^9999 + 104743 *X^10000
for X=-1 you get +14
but for X=-0.999 you get -0.0041
do the sum of 5000 such pairs , and you can get something quite negative, large enough to overcome the constant 2 factor.
Matter of fact, I found real roots for polynomials of 5000 and 10000 terms, but not with only 2000 terms or less.

J-F