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In the previous section we showed how to construct a quadratic iteration
from the case p=2.
In this section we show how the cases p=2 and p=4
relate to the Borwein's quadratic and quartic iterations.
For an overview of the method see [8]. For more complete
details see [4], [6]. In this and the
next section we summarise the results needed and omit the proofs.
Before stating these results, we first give a brief overview.
In the previous section we saw how a quadratic iteration can be
constructed using the modular equation (4.6) which relates
and
. It is useful it introduce auxiliary functions.
These functions are modular forms and are usually denoted by
,
and
. In each case, these auxiliary functions have some
remarkable properties:
- (1)
-
is some k-th root of
- (2)
-
There exist linear relationships between the functions
,
,
,
,
and
.
- (3)
-
There is a simple polynomial equation relating
,
and
.
In fact, for the quadratic and quartic iterations of this section
and the cubic iteration of the next section this relation is precisely

Instead of defining a sequence
which comes from the function
we define two auxiliary sequences
and
which will come from the functions

In this way, (5.1) becomes

It will turn out that the multiplier
can be written
simply in terms of
.
We now consider the first case p=2. By [4, p.698,] we have

Following [4] we define

Then we have the following results

Now we let

Then from (5.7), (5.9)--(5.11) we have

and

Now, by (5.8),(5.12) and (5.13) we have

and, by (3.18), and (5.15) we have

where

Now, from (3.19), (3.20) we know that

and so

By letting,

we have the following
Theorem 5.1
(Borwein and Borwein [6, Iteration 3.6, p. 700,])
Define sequences
,
, and
by

Then
converges quadratically to
.
We now consider the case p=4 and show how it is
related to Borwein and Borwein quartic iteration [8, Algorithm1,]
and how it coincides with another Borwein and Borwein quartic
iteration [6, Iteration 3.4, p. 700,].
From (3.15), [4, Chapter 9,], [6, Theorem 2.2,] we have

Following [4] we define

Then we have the following results

As usual we let

Then, by (5.34), (5.36)--(5.37) we have

and

Now, from (5.35), (5.39)--(5.41) we have

and, by (3.18), (5.42) we have

where

From (3.19), (3.20) we know that

and so

By letting,

we have the following
Theorem 5.2
(Borwein and Borwein [6, Iteration 3.4, p. 700,])
Define sequences
,
, and
by

Then
converges quartically to
.

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