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Error Estimate for the NewtonRaphson
Method
In this section we estimate how the error varies from one iteration
to the next. This gives us an idea on the speed of convergence of
the method. Using the definition of absolute error in (1.6)
we have the following relation between the exact value of the root
, the iterate and the error after iterations
,
Similarly after iterations we have
These expressions are substituted into the NewtonRaphson Method
(1.11)
This may be rearranged to give

(1.12) 
To proceed we assume that the error at each step is small. Thus, if
is small we may expand and
in a Taylor series about . The trick is to
keep the first two nonzero terms. Thus,
The first term on the right hand side is zero since is a root.
Similarly we have
Here we only need the first two terms as they are assumed to be
nonzero. Hence, substituting into (1.12) we obtain
To obtain the last line we expand the denominator using the binomial
expansion and then neglect all terms that have a higher power of
than the leading term. Thus, we neglect
and all higher powers. Thus,

(1.13) 
The error after iterations is proportional to square of the
error after iterations. Once the relationship between
and is known then the order
of the iterative scheme (which is bascially the speed of convergence)
is the power of . Thus, NewtonRaphson is a
second order scheme and we have fast convergence.
To illustrate the speed of convergence of the NewtonRaphson method,
assume that the error at a particular iteration is . After
the next step the error is proportional to and after two
iterations it is now proportional to . Thus, the error
reduces extremely quickly.
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Prof. Alan Hood
20000201