The document discusses the fixed point iteration method, which is a numerical method used to find approximate solutions to algebraic and transcendental equations. It presents the theorems and algorithm of the fixed point iteration method, and provides examples of its applications. Some key points covered include expressing equations in the form x=g(x) such that the derivative is less than 1, using successive approximations xn=g(xn-1) to generate a converging sequence, and illustrating the geometric interpretation of the method graphically. The document concludes that fixed point theory has many applications in mathematics.

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

2. outline
• INTRODUCTION
• FIXED POINT ITERATION THEOREMS
• Algorithm of fixed point iteration method
• SOME APPLICATIONS OF FIXED POINT
THEOREMS
• Some interesting facts about the fixed point
iteration method

3. Con..
• Geometric meaning of fixed point iteration
method
• CONCLUSION

4. 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.

5. FIXED POINT ITERATION THEOREMS
• A fixed point is a point in the domain of a
function is algebrically converted in the form
of g(x)=x
• Suppose we have an equation f(x)=0 for which
we have to find the solution . The equation
can be expressed as x=g(x)
• Choose g(x ) such that І𝑔′
(x)І < 1 at x=𝑥0
where 𝑥0, is some initial guess called fixed
point iterative scheme.
• Then the iterative method is applied by
succesive approximations given by 𝑥𝑛
=g(𝑥𝑛−1) that is 𝑥1=g(𝑥0), 𝑥2=g(𝑥1) so on……

6. 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. Some interesting facts about the 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
• Given
• 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

9. • 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

10. • 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……..

11. 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 ………..

12. • Draw the graph of y=x and y=𝜑(x)
• Since І𝜑′
(x)І < 1 near the root the inclination
of the graph of 𝜑(x) should be less than

13. APPLICATIONS OF FIXED POINT
THEOREMS
• The implicit function theorem
• Fr’echet differentiability Let X, Y be (real or
complex) Banach spaces, U ⊂ X, U open,
• 𝑥0 ∈ U, and f : U → Y
• Definition f is Fr´echet differentiable at 𝑥0 is
there exists T ∈ L(X,Y ) and σ : X → Y , with

14. • The operator T is called the Fr´echet derivative of f at
𝑥0, and is denoted by 𝑓′(𝑥0). The function f is said to
be Fr´echet differentiable in U if it is Fr´echet
differentiable at every 𝑥0 ∈ U.
• It is straightforward to verify the Fr´echet derivative
at one point, if it exists, is unique.
• Ordinary dierential equations in Banach spaces
• The Riemann integral
• Let X be a Banach space, I = [α,β] ⊂ R. The notion of
Riemann integral and the related properties can be
extended with no differences from the case of real-
valued functions to X-valued functions on I. In
particular, if f ∈ C(I,X) then f is Riemann integrable on
I,

17. • Also, (d) is always true if X is finite-
dimensional, for closed balls are compact. In
both cases, setting
• we can choose any s < 𝑠0. proof Let r =
min{a,s}, and set

18. Observe that

19. • We conclude that F maps Z into Z. The last
step is to show that 𝐹𝑛
is a contraction on Z
for some n ∈ N. By induction on n we show
that, for every t ∈ 𝐼𝑇,
• For n = 1 it holds easily. So assume it is true for
n−1, n ≥ 2. Then, taking t > t0 (the argument
for t < 𝑡0 is analogous),
•

20. • Therefore from we get

21. CONCLUSION
• 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.

22. •Thank you