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# The Quadratic and Quartic Iterations

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

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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