# The Arithmetic-Geometric Mean Iteration  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  and  where These are modular forms. So for example  # A Cubic Analogue of the AGM

Let

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  and  Then  and  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  where Let    Then  and    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.