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The method can be derived in a variety of different ways. Here we use
a graphical approcah. We start with an initial guess for the root .
Call this initial estimate . Next we construct the tangent to the
curve at
. Then, we calculate where this tangent
crosses the axis. This provides the next estimate, , for the
root . This is illustrated in Figure 1.3.
Figure 1.3:
The tangent at
cuts the axis at .
The tangent at
cuts the axis at .

The equation of the tangent at is
where
Hence,

(1.9) 
The tangent intersects the axis at and this value of
gives us our next estimate of the root. Thus,

(1.10) 
Using (1.10) we obtain . By replacing by
in (1.10) generates , which could be a better
estimate for the root than . Repeating this process generates
a sequence of iterates
which possibly converges to the root . Thus,
Knowing we can generate using (1.10) or
specifically

(1.11) 
This is the NewtonRaphson method for obtaining a root of
the function . This is the most widely used practical technique
for solving algebraic equations. It has fast convergence and this will
be discussed later.
Example 1. .5Consider
Then using the product rule for differentiating a product
with and , the derivative of is
This is positive if is positive. Now since
there is only one root in the interval . Then the NewtonRaphson
formula is
Take as our intial guess . Probably a better guess would
be to take the midpoint of the interval, i.e. . However,
if the method converges, the initial guess is not too crucial. The
results of the iteration are presented in the table.
Table 1.4:
The NewtonRaphson method applied to
.




1 
1.71828 
5.43656 
0.68394 
0.68394 
0.35534 
3.33701 
0.57745 
0.57445 
0.02872 
2.810211 
0.56723 
0.56723 

2.763615 
0.567143296 
0.567143296 

2.763223 
0.567143289 
When do we stop the iteration procedure?
Subsections
Next: Stopping Criterion
Up: Solutions of Equations
Previous: Approximate Numerical Methods
Prof. Alan Hood
20000201