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 producedThis 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.