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`
The Newton method is useful for solving nonlinear equations of the
form
`

`
More exactly, given , an initial, rough guess of the solution,
the method builds a sequence of points
`

`
which converge to the true solution of the problem
`

`
where
`

`We start with an intial guess . can be expanded in Taylor
series around the guessed solution
`

`
Our purpose is to derive a better approximation for . From the above formula
we have
`

`
In this exact formula the higher order terms of the right hand side depend on the unknown .
We simply ignore them to arrive at the approximate relation
`

`
The obtained is (hopefully) a better approximation to than the initial guess
was. To obtain an even better approximation we repeat the procedure
with the ``guess'' and arrive at
`

`
It is clear now that we can repeat the steps as many times as we need, untill is
sufficiently close to .
`
`The sequence of succesive approximations is built recursively
using Newton's formula
`

`From the formula we infer that
`

- each iteration requires one evaluation of the function
and one evaluation of its derivative ;
- the iteration step cannot be performed if .

`Usually, if and sufficiently close to then
Newton iterations converge very fast.
`

`The order of convergence of an iterative procedure is
the largest number for which
`

`
(here is some constant which does not depend on ).
The larger is the faster the method converges.
`
`For Newton's method ; we say that `*Newton's method
converges quadratically*. To see this
we develop
in Taylor series around :

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Adrian Sandu
2001-08-26