fixed point algorithm

8/20/2019 Fixed Point algorithm

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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

2

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5

6

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

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-3 -2 -1 0 1 2 3-3

-2

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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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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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fixed point algorithm

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