Editing Borwein's algorithm (section)

===Quartic convergence (1985)===

Start by setting<ref>{{cite book | title=The Java Programmers Guide to Numerical Computation |last=Mak |first=Ronald |publisher=Pearson Educational |year=2003 |isbn=0-13-046041-9 |page=353}}</ref>

:<math> \begin{align} a_0 & = 2\left(\sqrt{2}-1\right)^2 \\
                      y_0 & = \sqrt{2}-1
        \end{align}
</math>

Then iterate

: <math> \begin{align} y_{k+1} & = \frac{1-\left(1-y_k^4\right)^\frac14}{1+\left(1-y_k^4\right)^\frac14} \\
                       a_{k+1} & = a_k\left(1+y_{k+1}\right)^4 - 2^{2k+3} y_{k+1} \left(1 + y_{k+1} + y_{k+1}^2\right)
          \end{align}
</math>

Then ''a''<sub>''k''</sub> converges quartically against {{sfrac|1|{{pi}}}}; that is, each iteration approximately quadruples the number of correct digits. The algorithm is ''not'' self-correcting; each iteration must be performed with the desired number of correct digits for {{pi}}'s final result.

One iteration of this algorithm is equivalent to two iterations of the [[Gauss–Legendre algorithm]].
A proof of these algorithms can be found here:<ref>{{citation|title=Easy Proof of Three Recursive {{pi}}-Algorithms|last1=Milla|first1=Lorenz|arxiv=1907.04110|year=2019}}</ref>