1. Adama Science and Technology
University
Seminar
Title: FIXED POINT ITERATION
METHOD AND ITS APPLICATION
2. outline
β’ INTRODUCTION
β’ FIXED POINT ITERATION THEOREMS
β’ Algorithm of fixed point iteration method
β’ SOME APPLICATIONS OF FIXED POINT
THEOREMS
β’ Some interesting facts about the fixed point
iteration method
4. Introdiction
β’ Fixed point iteration method in numerical analaysis is
used to find an approximate solution to algebric and
transcendential equations
β’ 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
β’ One among those methods is the fixed iteration
method
β’ 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
β’ A fixed point is a point in the domain of a
function is algebrically converted in the form
of g(x)=x
β’ 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)
β’ Choose g(x ) such that Ππβ²
(x)Π < 1 at x=π₯0
where π₯0, is some initial guess called fixed
point iterative scheme.
β’ Then the iterative method is applied by
succesive approximations given by π₯π
=g(π₯πβ1) that is π₯1=g(π₯0), π₯2=g(π₯1) so onβ¦β¦
6. Algorithm of fixed point iteration method
β’ choose the initial value π₯0 for the iterative method
one way to choose π₯0 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 π₯0 as the average
of a and b.
β’ Express the given equation in the form x=g(x) such
that Ππβ²(x)Π < 1 at x=π₯0 if there more than one
possiblity of g(x) which has the minimum value of
πβ²(x) at x=π₯0.
β’ By applying the successive approximations π₯π
=g(π₯πβ1) ,if f is a continous function we get a
sequence of {π₯π} which converges to a point which is
the approximte solution of the given equation.
7. Some interesting facts about the fixed point
iteration method
β’ The form of x=g(x) can be chosen in many ways . But
we choose g(x) for which Ππβ²
(x)Π < 1 at x=π₯0
β’ By the fixed point iteration method we get a
sequence of π₯π which converges to the root of the
given equation.
β’ Lower the value of πβ²(x) .fewer the iteration are
required to get the approximate solution.
β’ The rate of convergence is more if the value of πβ²
(x)
is smaller.
β’ The method is useful for finding the real root of the
equation which is the form of an infinite series.
8. Examples
β’ Find the first approximate root of the equation
2π₯3 β 2x β 5 up to four decimal places
β’ Solution
β’ Given
β’ π π₯ = 2π₯3
β 2x β 5
β’ As per the algorithm we find the value of π₯0 for
which we have to find a and b such that f(a)< 0 and
f(b)>b
9. β’ Now f(0)=-5
β’ F(1)= -5 F(2)= 7
β’ Now we shall find g(x) such that Ππβ²
(x)Π < 1 at
x=π₯0
β’ 2π₯3
β 2x β 5, x=
2π₯+5
2
1
3
β’ g(x)=
2π₯+5
2
1
3
which satisfies Ππβ²
(x)Π < 1 at
x=π₯0
10. β’ At x= 1.5 0n the interval [1,2]
β’ thus a=1 and b=2
β’ therefore x=
π+π
2
=
1+2
2
=1.5
β’ Now applying the iterative method π₯π
=g(π₯πβ1) for n=1,2,3,4,5β¦β¦..
11. Geometric meaning of fixed point iteration
method
β’ the successive approximation of the root are
π₯0, π₯1, π₯2 β¦β¦β¦. Where
β’ π₯1=π(π₯0)
β’ π₯2=π(π₯1)
β’ π₯3=π(π₯2) and so on β¦β¦β¦..
12. β’ Draw the graph of y=x and y=π(x)
β’ Since Ππβ²
(x)Π < 1 near the root the inclination
of the graph of π(x) should be less than
13. APPLICATIONS OF FIXED POINT
THEOREMS
β’ The implicit function theorem
β’ Frβechet diο¬erentiability Let X, Y be (real or
complex) Banach spaces, U β X, U open,
β’ π₯0 β U, and f : U β Y
β’ Deο¬nition f is FrΒ΄echet diο¬erentiable at π₯0 is
there exists T β L(X,Y ) and Ο : X β Y , with
14. β’ The operator T is called the FrΒ΄echet derivative of f at
π₯0, and is denoted by πβ²(π₯0). The function f is said to
be FrΒ΄echet diο¬erentiable in U if it is FrΒ΄echet
diο¬erentiable at every π₯0 β U.
β’ It is straightforward to verify the FrΒ΄echet derivative
at one point, if it exists, is unique.
β’ Ordinary dierential equations in Banach spaces
β’ The Riemann integral
β’ Let X be a Banach space, I = [Ξ±,Ξ²] β R. The notion of
Riemann integral and the related properties can be
extended with no diο¬erences from the case of real-
valued functions to X-valued functions on I. In
particular, if f β C(I,X) then f is Riemann integrable on
I,
15.
16.
17. β’ Also, (d) is always true if X is ο¬nite-
dimensional, for closed balls are compact. In
both cases, setting
β’ we can choose any s < π 0. proof Let r =
min{a,s}, and set
19. β’ We conclude that F maps Z into Z. The last
step is to show that πΉπ
is a contraction on Z
for some n β N. By induction on n we show
that, for every t β πΌπ,
β’ For n = 1 it holds easily. So assume it is true for
nβ1, n β₯ 2. Then, taking t > t0 (the argument
for t < π‘0 is analogous),
β’
21. CONCLUSION
β’ 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 diο¬culties 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.
