Fixed point iteration

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Fixed Point iteration
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Fixed point iteration
Simple fixed point iteration
Other names-
Method of iteration
Picards iteration
Linear iteration
To find the root of the equation
f(x)0..(1) By fixed point iteration , we
write the equation (1) as
Then the successive approximations can be found by
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Note-
1) The equation (2) can be written in many forms.
2) It is not sure that the sequence of
approximations x1,x2,x3xn obtained by equation
(3) for some representation of (2) will converges
to desired root or some other number.
3) We have to choose initial approximation (x0)
and representation (2) in such a way that the
successive approximations x1, x2, x3,xn.
Converges to desired root.
The following theorem will helps in making the
right choice of x0 and one of the form from (2).
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Theorem-
If(1) a be the root of f(x)0 which is equivalent
to a?(a)..
2) I be any interval containing the point xa.
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Then the sequence of approximations x1,x2, x3xn
will converges to the root a provides the initial
condition x0 chosen in I
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Algorithm for fixed point iteration.
To find the root of the function f(x)0. we need
to follow the following steps.
Step-1 Find the interval a,b such that
f(a).f(b)lt0.
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Example-
Find the real root of x3-x-10 near x1 by fixed
point iteration method
OR
Find the real root of x3-x-10 with x01 by
fixed point iteration method.
OR
Find the real root of x3-x-10 by fixed point
iteration method.
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Solution
Clearly the root of given equation x3-x-1 is lies
between 1,2. i.e. a,b1,2
So we can take x01
Given equation may be written as
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Continue
Now we need to choose ?(x) such that it will give
convergent sequence of approximations. So
Case-1
So this form will give divergence sequence.
Case-2-
So this form will give divergence sequence.
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Continue
Case-3
This form may give convergence sequence.
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Continue..
In similar way we can get
So clearly the root of the given equation is 1.32
up to 3 significant figures.
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Assignment
Use simple fixed point iteration to locate the
root of f(x)e-x-x. With initial guess 0.
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Thank You ALL

• Presented by
• Boina Anil Kumar
• Asst.Prof in Mathematics
• MITS, Rayagada.
• Odisha, India
• E-mail anil.anisrav_at_gmail.com
• Visit at http//www.mits.edu.in/academics.phpfac
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Thank You ALL