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

[Albert Chan]

If we consider polynomial coefficients with geometric progression:

f(x) = 1 + (r*x) + (r*x)^2 + ... + (r*x)^n = ((r*x)^(n+1) - 1) / ((r*x) - 1)

From RHS numerator, all f roots abs = 1/r

P(x) = 2 + 3x + 5x^2 + 7x^3 + 11x^4 + ...

P roots, sorted in abs: (-0.6458±0.4832i), (0.4472±0.7248i), (-0.2853±0.8292i), ...
With corresponding abs: 0.8065, 0.8517, 0.8769, ...

Primes does not grow as fast as geometric progression. (sum of reciprocal primes diverges)

P min abs root is due to ratio, (5/2=2.5) > (11/5=2.2)
If the ratios were about the same, we expected f(x) roots pattern, with similar sized roots.

Example, R(x) = P(x)+0.5, 5/(2+0.5) = 2.

R roots, sorted in abs: (-0.6811±0.5122i), (0.4692±0.7114i), (-0.2905±0.8666i), ...
With corresponding abs: 0.8522, 0.8522, 0.9140, ...

If guess close enough, we can use Newton's method to zeroed in P min abs root.