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Fixed point Iteration:

Let be a root of and be an associated iteration function. Say, is the given starting point. Then one can generate a sequence of successive approximations of as:

...............

...............

.................

This sequence is said to converge to iff as .
Now the natural question that would arise is what are the conditions on s.t. the sequence    as
Here, we state few important comments on such a convergence:
(i)Suppose on an interval is defined and . i.e. g(x) maps I into itself.
(ii) The iteration function is continuous on I=[a,b].
(iii)The iteration function g(x) is differentiable on and s.t.

Theorem :
Let g(x) be an iteration function satisfying (i), (ii) and (iii) then g(x) has exactly one fixed point in I and starting with any , the sequence generated by fixed point iteration function converges to .

(iv) If then . For rapid convergence it is desirable that . Under this condition for the Newton Raphson method one can show that (i.e. quadratic convergence).

Remark 1:     One can generalize all the iterative methods for a system of nonlinear equations. For instance, if we have two non-linear equations then given a suitable starting point , the Newton-Raphson algorithm may be written as follows:
For i=1,2... until satisfied , do

Exercises: Solve the following systems of equations by Newton Raphson Method.

(1)

Use the initial approximation

(2)

Use the initial approximation

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