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Newton-Raphson Method:
Unlike the earlier methods, this method requires only one
appropriate starting point as an initial assumption of
the root of the function
. At
a tangent
to
is drawn. Equation of this tangent is given by
The various steps involved in calculating the root of by
Newton Raphson Method are described compactly in the algorithm
below.
Algorithm:
Given a continuously differentiable function
and an initial approximation
to the root of
, the steps involved in calculating an approximation
to the
root of
s.t.
are:
(1) Calculate
and set
(2) For n = 0,1,2... until convergence criteria is satisfied
,do:
Calculate
Remark (1): This method converges faster than the earlier methods. In fact the method converges at a quadratic rate. We will prove this later.
Remark (2): This method
can be derived directly by the Taylor expansion f(x) in the
neighbourhood of the root of
. The starting
approximation
to
is to be properly chosen so that
the first order Taylor series approximation of
in the
neighbourhood of
leads to
, an improved approximation
to
. i.e
Remark(3) : One may also derive the above iteration formulation starting with the iteration formula for the secant method. In a way this may help one to visualize Newton-Raphson method as an improvement over the secant method. So, let us consider the iteration formula for the secant method i.e.
or ,
Since,
Therefore repeat the process.
Results are tabulated below:
Newton Rahpson Method
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Iteration no. |
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Say,
The results are tabulated below:
Newton Raphson Method
Iteration no. | ![]() |
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0 | 0.5000000000 |
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1 | 0.6934901476 |
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0.0005313741 |
2 | 0.7013291121 |
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0.0000003363 |
Exercise: Find the solutions accurate to within for the following problems using Newton-Raphson Method.
(1) for
and
(2) for
and