Contents
** Next:** A Sixteenth Order
**Up:** Approximations to via the
** Previous:** The Septic Iteration

In this section we explore nonic (ninth order) iterations.
Of course, we may construct a ninth order iteration by iterating
the cubic. We have found a nonic iteration which does not seem to
come from the cubic.
Our nonic iteration
is based on a
symbolic discovery
that is a nice eta-product:

This identity can be proved easily from results in Ramanujan's notebook.
From (3.15), (3.16) we have
by [1, Entry 1(iv), Chapter 20, p. 345,].
In the last step we have used the classical cubic modular equation for the
eta-function:
See [7, Corollary 2.5,]. Using (6.5), (6.6)
may be written terms of either or
(defined in (6.2), (6.3)), which were involved in the
cubic iteration. We have

This time we let
where , , are defined in (6.1)--(6.3).
We define the following functions:
Using the results of Section 6 it can be shown that
where . We can also utilize the initial values
for the cubic and obtain a nonic iteration.
### Theorem 8.1

Define sequences , , and by
where
and where
Then converges nonically to .

Contents
** Next:** A Sixteenth Order
**Up:** Approximations to via the
** Previous:** The Septic Iteration