+26404 To: heureka and others..........
If you calculate the value of yk+1 in (1) and feed it into (2) and repeat the iterations at least 2-3 times, what does the result converge to and what is the rate of convegence, i.e., is it: quadratic, cubic, quartic.....etc. Also, what is the source of this algorithm? Thanks and have fun!.
Set a0 = 6−4√2 and y0 = √2−1 and k=0, 1, 2, 3................... Iterate:
yk+1 = 1−(1−y^4 k)^1/4 / 1 + (1−y^4 k)^1/4 ..............................(1)
and ak+1 = ak(1 + yk+1)^4 − 2^(2k+3)yk+1(1 + yk+1 + y^2 k+1)..(2)
\(\begin{array}{rcll} \text{Borwein's algorithm:}\\\\ \end{array} \\ \begin{array}{rcll} \text{Quartic algorithm (1985)}\\\\ \end{array} \\ \text{Start out by setting[1]}\\ \begin{array}{rcll} a_0 & =& 2\big(\sqrt{2}-1\big)^2 = 6-4\sqrt{2}\\ y_0 & =& \sqrt{2}-1 \end{array} \\ \begin{array}{rcll} \\ \text{Then iterate}\\ \end{array} \\ \\ \begin{array}{rcll} y_{k+1} & =& \frac{1-(1-y_k^4)^{1/4}}{1+(1-y_k^4)^{1/4}} \\ a_{k+1} & =& a_k(1+y_{k+1})^4 - 2^{2k+3} y_{k+1} (1 + y_{k+1} + y_{k+1}^2) \end{array} \\ \\ \text{Then }~a_k \text{ converges quartically against }~ 1/\pi; \\ \text{ that is, each iteration approximately quadruples the number of correct digits.} \)
see: https://en.wikipedia.org/wiki/Borwein's_algorithm
