Here’s a lovely iteration which converges quadratically to . Start with:
This remarkable iteration comes from Borwein and Borwein’s “Pi and the AGM“; this is the first formula for of the many in the book. It is based on the arithmetic-geometric mean, which is defined as follows:
Given define the sequences by and for all
It can be shown that these sequences both converge to the same value, called the arithmetic-geometric mean of and , and denoted
The derivation of the iteration above starts with the complete elliptic integral of the first kind:
The importance of these integrals for the AGM includes the useful result
It can also be shown that:
(In these results, the upper dot indicates the derivative. It is conventional to write for but to keep the symbols down I won’t use that here.) Finally, and this is the basis for the iteration:
Full details are given in the book above.
Now, let’s see this in operation. Using Maxima, with floating point precision set at 200 digits; we start with:
(%i2) [x,y,pi]:bfloat([sqrt(2),2^(1/4),2+sqrt(2)])latex pilatex pilatex pi$, including one which exhibits quintic convergence; that is, the number of correct digits multiplies by five at each step. To find that you'll have to read their book...