Viète à grande Vitesse or Accelerated Viète (kind of)

[Gerson W. Barbosa]

(06-21-2024 10:27 AM)Albert Chan Wrote:  Using x/asin(x) coefficients, we may divide by correction instead of multiply

\(\displaystyle \pi \approx \frac{\pi{'}}{
\left[1-\frac{1}{6}\left(\frac{\pi{'}}{2^{n+1} }\right)^2\right]
\left[1-\frac{17}{360}\left(\frac{\pi{'}}{2^{n+1} }\right)^4\right]
\left[1-\frac{9}{280}\left(\frac{\pi{'}}{2^{n+1} }\right)^6\right]
\left[1-\frac{37579}{1814400}\left(\frac{\pi{'}}{2^{n+1} }\right)^8\right]}\)

That's interesting, only the 3ⁿ⁻¹/(2ⁿ×COMB(2n+1, n)) pattern for the first three factors is lost. Viète produces 0.6 digits per iteration and each additional correction factor adds another 0.6 digits per iteration to that count. Viète computed \(\pi\) to 9 digits, so he needed at least fifteen square root extractions, not a time consuming task with pencil and paper. I take he kept a record of all intermediate results. A few more simple operations and the first factor could be found and he would double his resusts. But that would be too boring a task compared to the neat result he had achieved.

A continued-fraction correction would soon form a pattern, but these seem to be possible only when the original series or product has sublinear convergence rate, not the case. Anyway, for me that has been a good exercise, even if the result is somewhat irrelevant for this millenium. Thank you for your interest and contributions!