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FIXED POINT ITERATIONBy Raj Nandkeolyar
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BACKGROUND
Definition: A number p is said to be a fixed point
of a function g( x) defined on [a, b] if g( p) = pfor
some .
For example:
1. g(x) = x2 has two fixed points x = 0 and x = 1.
2. g(x) = 1/x has two fixed points x = 1 and x = -1.
3. g(x) = x2 – 2 has two fixed points x = -1 and 2.
. Definition: The problem of finding the fixed
points of a function g( x) is called a fixed point
problem.
],[ ba p∈
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FINDING FIXED POINTS AND
GEOMETRICAL INTERPRETATION
Find the fixed points of the function g(x) = x2 – 2.
Solution: Let p be the fixed point of g(x), then by
the definition of fixed points:
2 ,1
02
2
)(
2
2
−=⇒
=−−⇒
=−⇒
=
p
p p
p p
p p g
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-3 -2 -1 0 1 2 3-3
-2
-1
0
1
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7
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FIXED POINT ITERATION
In the previous example we have seen that the fixed
points of g(x) are the roots of the equation x –
g(x) = 0,
or, f(x) = 0.
Conversely, the problem of finding the roots of the
equation is equivalent to the fixed
point problem
Thus, for a given root finding problem f(x) = 0 we
can construct a corresponding fixed point
problem x = g(x), such that the solution of both
the problems are same.
022 =−− x x
.2)( 2 −== x x g x
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-3 -2 -1 0 1 2 3-4
-2
0
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10
-3 -2 -1 0 1 2 3-3
-2
-1
0
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Graph of f(x)=x2 -x-2
Graph of y = x, and y = g(x)
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AN EXAMPLE OF FIXED POINT
ITERATION
Example:
Solution:Write x = cos x
such that g( x) = cos x
0cos
root positivesmallestfor theequationfollowingtheSolve
=− x x
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x0x1
,...2,1,0),(1 ==+ n x g x nn
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FIXED POINT ITERATION SCHEME
For the root finding problem f(x) = 0,
We write the corresponding fixed point problem
x = g(x)
Then the fixed point iteration scheme is given by
,...2,1,0),(1 ==+ n x g x nn
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CHOICE OF ITERATION FUNCTION
Finding a suitable iteration function g(x) is
critical:
Example: Find the smallest positive root of the
equation x2 - 2x +1 = 0
Solution:
Writing
Such that
Taking x0 = 0.8, we obtainx1 = 0.7746,
x2 = 0.7411,
x3 = 0.6944…The sequence of iterates is diverging
))((12 x g x x =−=
,...2,1,0 ,121 =−=+ n x x nn
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0.5 1 1.5 2 2.50
0.5
1
1.5
2
2.5
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Writing
We obtain the scheme:
Taking x0 = 0.8, we obtain
x1 = 0.82,
x2 = 0.8362,
x3 = 0.8496,
x4 = 0.8609,
x5 = 0.8706,…
The sequence of iterates is approaching towards the
exact root x = 1.
2
12+
= x
x
,...2,1,0,2
12
1 =+
=+
n x
x nn
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-3 -2 -1 0 1 2 3-3
-2
-1
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Assumption 1: g(x) is defined on [a,b]
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],[allfor],[)(:2ssumption ba xba x g ∈∈
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Conclusion: g(x) has a fixed point in [a, b]
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