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In this section we show how a computer algebra package like MAPLE
can be used to generate p-th order iterations that converge
to from the functions .
For a fixed initial value and a fixed p
we define the sequence by
Of course, as it stands, although it is clear that
converges to to p-th order, it is not very practical.
The idea is to write the sequence recursively. Therefore there are
two problems to solve:
-
Find initial values .
-
Get in terms of .
Table 1 contains some functions we defined in MAPLE and that we
used to find and prove initial values and iterations.
Table 1: MAPLE functions
Our function etaq utilizes the expansion due to Euler:
We are then able to effectively compute q-series expansions of , ,
. We have written MAPLE procedures to compute all the necessary
functions.
Contents
Next: Initial values
Up: Approximations to via the
Previous: The functions