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1. Adama Science and Technology
University
Seminar
Title: FIXED POINT ITERATION METHOD AND ITS
APPLICATION

2. • Contents
• INTRODICTION
• 1.1 Roots of Nonlinear Equations
• 1.2 Mathematical analysis
• FIXED POINT ITERATION METHOD
• 2.1 FIXED POINT ITERATION METHOD THEOREM:
• 2.2 Algorithm of fixed point iteration method
• 2.3 properties of fixed point iteration method
• 2.4 Geometric meaning of fixed point iteration method
• APPLICATION OF FIXED POINT ITERATION METHOD
• 3.1 Approximation by Fixed Point Iteration
• Illustrative example
• CONCLUSION

3. Introdiction
• Fixed point iteration method in numerical analaysis is used to find an
approximate solution to algebric and transcendential equations
• Sometimes it becomes veery tedious to find solutions to qubic quadratic
and trancedential equations then we can apply specific numerical
methods to find the solution
• One among those methods is the fixed iteration method
• The fixed point iteration method uses the concept of a fixed point in a
repeated manner to compute the solution of the given equation.

4. • 1.1 Roots of Nonlinear Equations
• Let say we want to find the solution of f (x) = 0. For example:
These equations can not be solved directly.
• We need numerical methods to compute the approximate solutions.
• First we write f (x) = 0 in the form x = F (x).
• Note that F (x) is not unique. For instance, see the following.

5. • 1.2 Mathematical analysis
• If f (p) = p, then we say that p is a fixed point of the function f (x). We
note a strong relation between root finding and finding fixed points:
• If f (p) = p, then we say that p is a fixed point of the function f (x). We
note a strong relation between root finding and finding fixed points:
• To convert a fixed-point problem
• g(x) = x,
• to a root finding problem, define f (x) = g(x)−x, and look for roots of f
(x) = 0.
• If f (p) = p, then we say that p is a fixed point of the function f (x). We
note a strong relation between root finding and finding fixed points:
• To convert a fixed-point problem
• g(x) = x,

6. • FIXED POINT ITERATION METHOD
• 2.1 FIXED POINT ITERATION METHOD THEOREM:
• Let 𝛼 be a root of f(x) =0.in a neighbourhood I of 𝛼 ,let the equation be
written as x=𝜑(x) lf 𝜑(x) is continuous in I and 𝜑 1
(𝑥) < 1∀𝑥 ∈ 𝐼, then
the sequence of approximation 𝑥0, 𝑥1. 𝑥3 … . . ,given by 𝑥𝑛 = 𝜑(𝑥𝑛)
converges to 𝛼 ,where 𝑥0 ∈ 𝐼.
Algorithm of fixed point iteration method
• choose the initial value 𝑥0 for the iterative method one way to choose 𝑥0
is is to find the values of x=a and x=b for which f(a)< 0 and f(b)>b by
narrowing down the selection of a and b take 𝑥0 as the average of a and b.
• Express the given equation in the form x=g(x) such that І𝑔′
(x)І < 1 at x=𝑥0
if there more than one possiblity of g(x) which has the minimum value of
𝑔′
(x) at x=𝑥0.
• By applying the successive approximations 𝑥𝑛 =g(𝑥𝑛−1) ,if f is a continous
function we get a sequence of {𝑥𝑛} which converges to a point which is the
approximte solution of the given equation.

7. Properties of fixed point iteration method
• The form of x=g(x) can be chosen in many ways . But we choose g(x) for
which І𝑔′(x)І < 1 at x=𝑥0
• By the fixed point iteration method we get a sequence of 𝑥𝑛 which
converges to the root of the given equation.
• Lower the value of 𝑔′(x) .fewer the iteration are required to get the
approximate solution.
• The rate of convergence is more if the value of 𝑔′
(x) is smaller.
• The method is useful for finding the real root of the equation which is the
form of an infinite series.

8. Examples
• Find the first approximate root of the equation 2𝑥3
− 2x − 5 up to
four decimal places
• Solution
• 𝑓 𝑥 = 2𝑥3 − 2x − 5
• As per the algorithm we find the value of 𝑥0 for which we have to find
a and b such that f(a)< 0 and f(b)>b
• Now f(0)=-5
• F(1)= -5 F(2)= 7
• Now we shall find g(x) such that І𝑔′
(x)І < 1 at x=𝑥0
• 2𝑥3 − 2x − 5, x=
2𝑥+5
2
1
3
• g(x)=
2𝑥+5
2
1
3
which satisfies І𝑔′
(x)І < 1 at x=𝑥0

9. • At x= 1.5 0n the interval [1,2]
• thus a=1 and b=2
• therefore x=
𝑎+𝑏
2
=
1+2
2
=1.5
• Now applying the iterative method 𝑥𝑛 =g(𝑥𝑛−1) for n=1,2,3,4,5……..
Geometric meaning of fixed point iteration method
• the successive approximation of the root are 𝑥0, 𝑥1, 𝑥2 ………. Where
• 𝑥1=𝜑(𝑥0)
• 𝑥2=𝜑(𝑥1)
• 𝑥3=𝜑(𝑥2) and so on ………..

10. APPLICATION OF FIXED POINT ITERATION METHOD
Approximation by Fixed Point Iteration
(Based on Example 2)
In Example 2, we have obtained various forms of x = F(x). Now referring to
xi+1 = F(xi), we put subscripts as follows.
10

1.185138
1.367163
1.458333
:
compute
can
we
(1)
From
value.
initial
the
be
1.5
Let
(1)
3
1
3
1
)
3
2
1
0
3
1
3

=

=

=

=

=


= +
x
x
x
x
x
x
x
x
i i
i

11. 11
The distance between xi+1 and xi increases,
i.e. |xi+1 – xi| > |xi – xi1|.
This iteration fails (since it diverges).

0.347296
0.347296
0.347294
0.347271
0.346992
0.343643
:
compute
can
we
(2)
From
0.3.
Put
(2)
3
1
3
1
)
6
5
4
3
2
1
0
2
1
2

=

=

=

=

=

=

=

=


= +
x
x
x
x
x
x
x
x
x
x
x
ii
i
i
The distance between xi+1 and xi decreases,
i.e. |xi+1 – xi | < | xi – xi1 |.
This iteration converges (succeeds).

12. CONCLUSION
Fixed Point Iteration Method is used for finding ROOTS of Non-Linear Equations.
fzero command can be used to find solution of Non-linear equations directly (Initial
Guess value should be chosen carefully).
Graphs can be used to visualize the solution of the equation.
Loops (both For & While) are tools which can be implemented in program code to
find solution of equations, with following steps:
Initial Guess value
Algorithm for finding solution
Criteria for stopping computations
Generally,Fixed point theory is a fascinating subject,with an enormous number of
applications in various fields of mathematics. Maybe due to this transversal character,
I have always experienced some difficulties to find a book (unless expressly devoted
to fixed points) treating the argument in a unitary fashion. In most cases, I noticed
that fixed points pop up when they are needed.

13. Thank you