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
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,
15.
16.
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
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),
•
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.
