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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 . For more complete details see , . 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  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  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 .  Contents Next: The Cubic Iteration Up: Approximations to via the Previous: Iteration Construction