The Arithmetic-Geometric Mean Iteration


If and

In 1799, Gauss observed this purely numerically and wrote that this result

''will surely open a whole new field of analysis.''

Equivalent Modular Parameterization

This is equivalent to the identities



These are modular forms. So for example

A Cubic Analogue of the AGM


and then the common limit is

The convergence is cubic.
Proof: The proof is opaque. It works because satisfies

In the above notation.

where is, as above, with, Taking k+1 terms of the sum and limit gives a cubically convergent algorithm. Twenty one terms gives three billion digits.The underlying cubic transformation is beautiful:

Equivalent Modular Parameterization If




Note that if then

This is a cubic modular equation for R.

See Some cubic modular identities of Ramanujan by J. Borwein, P. Borwein & F. Garvan in Trans. A.M.S. 343 (1994) 35-47.

Some Explanations





With we get

If we can compute iteratively we can compute and by a second linked iteration.

There are four particularly interesting cases.

Inverting the ratios gives elliptic modular functions. Which reduces much of this to an algebraic theory.

Finding and proving these iterations can (at least in principal) be effected entirely computationally.

The Quadratic s=1/4 Iteration

Let and Then the common limit is

Caveat Emptor: Let and Then the common limit is not differentially algebraic.