Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]

11012021, 10:42 PM
Post: #16




RE: Evaluation of ζ(2) by the definition (sort of) [HP42S & HP71B]
(10262021 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 sidebyside) Code: [1.0, 12.0, 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 

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