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The Quadratic and Quartic Iterations

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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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Contents Next: The Cubic Iteration Up: Approximations to via the Previous: Iteration Construction