MA 132
Julia Bielaski0163140
Sequences: Newton’s Method
11/21/08
1.
The Newton Method or the NewtonRaphson Method is a numerical root finder.
The
method is used when to find the root, r, using a graph or a polynomial expression.
The
first approximation made is labeled x
1
and it is found by either an educated guess or from
a sketch of the graph of a function, f.
Basically, the point of the method is that the
tangent line is close the f equation and therefore the xintercept of the tangent line is close
to the xintercept of f.
A simple equation is used to calculate this approximation; x
n+1
=x
n
– [f(x
n
)/f
1
(x
n
)].
If the x
n
numbers continue to grow closer to the value of r and the value
of n increases then the sequence of x
n
converges to r.
However, the Newton method may
not always be the best method to use.
For instance, if the approximation does not fit
within the domain of f then another method needs to be used.
When using the Newton
Method, the idea is to get the approximation as accurate as possible.
The general rule is
that the value of x
n
and x
n+1
should agree to eight decimal places.
Lastly, this is known as
an iterative process, which basically means that it is predominantly agreeable for use with
a graphing calculator or a computer.
2.
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 Spring '09
 FOWLER
 Calculus, Numerical Analysis, Approximation, Mile, Rootfinding algorithm, newton method

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