Here’s a lovely iteration which converges quadratically to . Start with:

.

Then:

for

for

for

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:

and 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:

`(%1) fpprec:200;`

(%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...

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Possible typo?

> the number of correct digits multiples by five at each step.

> the number of correct digits multiplies by five at each step.

Wow, I just added Brent-Salamin to my blog … then took a look at yours. Your algorithm looks very similar to mine, though seems simpler?