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As in section 1.3.2, we use a Taylor series expansion to
see how the error behaves from one step to another. Again we take
where is the exact value of the root, is the value of
the iterate and is the error after
iterations. Hence, fixedpoint iteration can be written as
If is small we may expand the function in a Taylor
series about . Hence,
However, remembering that the root is a fixedpoint and so
satisfies , the leading term in the Taylor series gives

(1.15) 
(1.15) shows us that fixedpoint iteration is a
firstorder scheme provided
. We can
use (1.15) to see if the scheme converges or not.
Assume that the initial error is . Then (1.15)
gives
Repeated use of (1.15) gives
From these first few terms it is clear that in general

(1.16) 
(1.16) shows us that
 the error reduces if
, the
scheme converges,
 the error increases if
, the
scheme diverges.
When the scheme converges we can see that
 the error decreases monotonically if
,
 the error decreases in an oscillatory manner if
.
Example 1. .7Consider the previous example with
. Using the
scheme
we have
Thus,
and so the scheme converges.
However, if you use the scheme
we have
In this case we have
and the scheme does not converge.
Example 1. .8Consider
. The roots are and . We
will express in three different forms and test the
convergence criterion for each form.
 Express as
. Hence, we have
Hence,
and
Thus,
 Express as
. Thus,
In this example
and so
Thus,
 Finally, express as
, so that
Here we have
and so
So the convergence condition gives
Next: Summary
Up: Other Iterative Schemes
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Prof. Alan Hood
20000201