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 sets 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)=1in=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)=1in =1 This method
can be derived directly by the Taylor expansion f(x) in the
neighborhood of the root of
. The starting
approximation
to
is to be properly chosen so that
the first order Taylor series approximation of
in the
neighbour of
leads to
, an improved approximation
to
. i.e
Remark(3) =1in =1 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.
Newton Raphson Method | |||
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Iteration no. | ![]() |
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0 | 2.0000000000 | 1.7209302187 | 0.8910911679 |
1 | 1.7209302187 | 1.6625729799 | 0.0347661413 |
2 | 1.6625729799 | 1.6601046324 | 0.0000604780 |
3 | 1.6601046324 | 1.6601003408 | 0.0000002435 |
Newton Raphson Method | |||
---|---|---|---|
Iteration no. | ![]() |
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0 | 0.5000000000 | 0.6934901476 | -0.1086351126 |
1 | 0.6934901476 | 0.7013291121 | -0.0005313741 |
2 | 0.7013291121 | 0.7013678551 | -0.0000003363 |