Evaluation of ζ(2) by the definition (sort of) [HP-42S & HP-71B]

[Albert Chan]

(10-26-2021 02:12 AM)Gerson W. Barbosa Wrote:  That gives linear convergence ( 25/12 digits per iteration ), as you can see by the Decimal Basic code and output ...

This may have explained the mysterious 25/12 digits per iteration.

According to Loch's theorem, for "typical" real number between 0 and 1
Asymptote when accuracy is extremely good:

Gain in decimal accuracy (per iteration) ≈ pi^2/(6*ln(2)*ln(10)) ≈ 1.0306 digit

With Decimal Basic code, I checked for for digits accuracy, difference to ζ(2) = pi^2/6
Note: digits = 1 - log10(abs(pi^2/6 - x)). So, if x=1, it is 1.1905 digits accurate.
Anyway, we are only interested in differences.

n=100: 210.6882
n=101: 212.7781      → gained 2.0899 digit
n=102: 214.8680      → gained 2.0899 digit

As experiment, I added an extra CF term to b, and compare against original b=0

n=100: 211.9306      → gained 1.2424 digit
n=101: 214.0206      → gained 1.2425 digit
n=102: 216.1107      → gained 1.2427 digit

This is better than 1.0306 digits. Perhaps CF's not "typical".
Simple CF coefs is decreasing (both odd and even, showing side-by-side)

Code:

[1.0,               12.0,
 0.3125,            5.53086419753,
 0.177978515625,    3.55998024691,
 0.123977661133,    2.62034818825,
 0.0950344838202,   2.07210260024,
 0.0770232130308,   1.71322636883,
 0.0647422966949,   1.46016607257 ] .* N

The difference (~ 0.85 digit) may be due to changes to CF constants.
Going from n to n+1, all the simple CF coefs scaled up, by factor (N+1)/N

We can estimate the gain in accuracy I learn from giac source
When CF coefs increases, estimate is better, roughly by its square. (and, we have n+1 of them)

log10(((N+1)/N)^(2*(n+1))) = 2*(n+1) * log10(1+1/N) = (2N+1)/ln(10) * log1p(1/N)

limit((2N+1)/ln(10) * log1p(1/N), N=inf) = 2/ln(10) ≈ 0.8686 digit