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:
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(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(Borwein and Borwein [6, Iteration 3.4, p. 700,]) Define sequences , , and by Then converges quartically to .