and 
and define the cubic AGM by
which converge cubically to a common limit
where the hypergeometric function 
. In
particular, the hypergeometric function possesses the simple cubic
functional equation 
This can be validated
symbolically once known! As an example 
and 4 iterations of 
will compute the hypergeometric function at
to 25 significant digits. Any direct computation so near the
radius of convergence is doomed.
and 
Theorem 2. The functions 
and 
``parametrize'' the
cubic AGM in the sense that if 
and 
then
and
while 
.
Thus a step of the iteration has the effect of sending q to 
. From
this, one is led to an easy to state but hard to derive
Cubic iteration for 
. Let 
, 
and set
then 
converges cubically to 
.
This iteration gives 1, 5,
21, 70, 
digits correct and more than triples accuracy at each
step.
The Computational Component. This is the most challenging and most satisfying of our three examples for computer assisted analysis. We began with one of Ramanujan's typically enigmatic entries in Chapter 20 of his notebook, now decoded in [1]. It told us that a ``quadratic modular equation'' relating to F was
>From this we gleaned that some function R should exist so that 
and 
would solve 
. We formally solved for the
coefficents of R and learned nothing. Motivated by the analogy with the
classical theory of the AGM iteration [2] we looked at 
which
produced
This was ``pay-dirt'' since the coefficients were sparse and very regular.
Some analysis suggested that they related to the number of representations
of the form 
. From this we looked at theta function
representations and were rewarded immediately by the apparent identity
. Given the truth of this it was relatively easy to
determine that 
with M and L as in 
.
It was now clear that the behaviour as q goes to 
should be at least
as interesting as 
. Indeed, motivated by the modular properties of
L we observed symbolically that
At this stage in [4] we resorted rather unsatisfactorily to a classical
modular function proof of 
and so to a proof of Theorem 2. Later we
returned with Frank Garvan [8] to a search for an elementary proof. This
proved successful. By searching for product expansions for M we were lead
to an entirely natural computer-guided proof --- albeit with human insight
along the way.
It is actually possible, as described in [8], to search for, discover and
prove 
modular identities of the type of 
and 
in an entirely automated fashion. Again, this is possible
because we have ultimately reduced most of the analytic questions to algebra
through the machinery of modular forms.
As a final symbolic challenge we observe that 
may be recast as
saying that
where
This invariance should be susceptible to a direct --- hopefully
experimentally guided --- proof.
![[Annotate]](/organics/icons/sannotate.gif){kind=icon}
![[Shownotes]](../gif/annotate/shide-51.gif)