introduction to fixed point iteration method and its ... · introduction fixed point iteration...
TRANSCRIPT
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Introduction to Fixed Point Iteration Method andits application
Damodar Rajbhandari
St. Xavier’s College
Nepal, 2016
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Table of contents
1 Introduction
2 Fixed Point Iteration Method
3 Condition for Convergence
4 Application
5 Appendix
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Introduction
What is fixed point?Does anybody knows, What is it?
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Introduction
What is fixed point?→If a function φ(x) intersected by y = x then, the x-co-ordinate of an intersection is known as fixed point.
Formal definition
”A fixed point of a function is an element of function’s domainthat is mapped to itself by the function.”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Introduction
What is fixed point?→If a function φ(x) intersected by y = x then, the x-co-ordinate of an intersection is known as fixed point.
Formal definition
”A fixed point of a function is an element of function’s domainthat is mapped to itself by the function.”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Lets see an example 1 [See its matlab code in Appendix Section];
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
In the previous figure, what are the fixed points?
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Fixed Point’s response on your answer:
”Finally, you knew me!”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Fixed Point’s response on your answer:
”Finally, you knew me!”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is ”Fixed Point Iteration Method”?
In Numerical analysis, It is a method of computing fixedpoints by doing no. of iteration to the functions.
So, What?
You’ll get roots for your desire equation.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is ”Fixed Point Iteration Method”?
In Numerical analysis, It is a method of computing fixedpoints by doing no. of iteration to the functions.
So, What?
You’ll get roots for your desire equation.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is ”Fixed Point Iteration Method”?
In Numerical analysis, It is a method of computing fixedpoints by doing no. of iteration to the functions.
So, What?
You’ll get roots for your desire equation.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is ”Fixed Point Iteration Method”?
In Numerical analysis, It is a method of computing fixedpoints by doing no. of iteration to the functions.
So, What?
You’ll get roots for your desire equation.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the reason behind changing my equation into x = φ(x)form?
→ You’re in the right place! This presentation is for you!Hehe, Thank Me!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the reason behind changing my equation into x = φ(x)form?
→ You’re in the right place! This presentation is for you!Hehe, Thank Me!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Fixed Point Iteration Method
Let me answer your question first!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the reason behind changing my equation into x = φ(x)form?
Suppose, We have a function like f (x) = x − φ(x).(Not mandatory!)
If we want to find a root of this equation then, we haveto do like this;
f (x) = 0
So, the function becomes x = φ(x)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the reason behind changing my equation into x = φ(x)form?
Suppose, We have a function like f (x) = x − φ(x).(Not mandatory!)
If we want to find a root of this equation then, we haveto do like this;
f (x) = 0
So, the function becomes x = φ(x)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the reason behind changing my equation into x = φ(x)form?
Suppose, We have a function like f (x) = x − φ(x).(Not mandatory!)
If we want to find a root of this equation then, we haveto do like this;
f (x) = 0
So, the function becomes x = φ(x)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the reason behind changing my equation into x = φ(x)form?
Suppose, We have a function like f (x) = x − φ(x).(Not mandatory!)
If we want to find a root of this equation then, we haveto do like this;
f (x) = 0
So, the function becomes x = φ(x)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Let me ask you a question!
In the previous example 1, y = x is a line. Lets say, y = φ(x) then,the function φ(x) = x becomes y = x . Is this an equation of a linethat will intersect the equation φ(x)?
ANSWER: No! It is a fixed point that will satisfy f (x) = 0.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Let me ask you a question!
In the previous example 1, y = x is a line. Lets say, y = φ(x) then,the function φ(x) = x becomes y = x . Is this an equation of a linethat will intersect the equation φ(x)?
ANSWER: No! It is a fixed point that will satisfy f (x) = 0.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Let me ask you a question!
In the previous example 1, y = x is a line. Lets say, y = φ(x) then,the function φ(x) = x becomes y = x . Is this an equation of a linethat will intersect the equation φ(x)?
ANSWER: No! It is a fixed point that will satisfy f (x) = 0.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the primary algorithm for this method?
1. First, Convert your function f (x) into x = φ(x) form.
2. Iterate the value of x to the function φ(x) until you get thedesire precision such that the precision value should berepeated after you again iterate it.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the primary algorithm for this method?
1. First, Convert your function f (x) into x = φ(x) form.
2. Iterate the value of x to the function φ(x) until you get thedesire precision such that the precision value should berepeated after you again iterate it.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
What is the primary algorithm for this method?
1. First, Convert your function f (x) into x = φ(x) form.
2. Iterate the value of x to the function φ(x) until you get thedesire precision such that the precision value should berepeated after you again iterate it.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Can you show me in geometrical perspective?
CLICK THE ”ICON” TO GLORIFY YOURSELF!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Can you show me in geometrical perspective?
CLICK THE ”ICON” TO GLORIFY YOURSELF!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Condition for Convergence
We can make lots of functions in the form of x = φ(x) by using thef (x). Then, Can every functions will give me the root of the f (x)?
→ May be! But, the rate of convergence would be slow.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Condition for Convergence
We can make lots of functions in the form of x = φ(x) by using thef (x). Then, Can every functions will give me the root of the f (x)?→ May be! But, the rate of convergence would be slow.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
So, How can I decide whether the function φ(x) would convergesfast or not?
→ You have to need a test called ”Condition for Convergence”Lets talk about it!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
So, How can I decide whether the function φ(x) would convergesfast or not?→ You have to need a test called ”Condition for Convergence”
Lets talk about it!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
So, How can I decide whether the function φ(x) would convergesfast or not?→ You have to need a test called ”Condition for Convergence”Lets talk about it!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Condition for Convergence
”If α be the root of the f (x) which is equivalent to x = φ(x) andα is contained in the interval I of φ(x) and |φ′(x)| < 1 ∀ x ∈ I .Then, if x0 is any point in I so that, the sequence defined by
xn = φ(xn−1) | n ≥ 1will converges to the unique fixed point x in I . Thus, that uniquefixed point x of φ(x) will be the root of f (x).”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Basic Preliminaries
1. Visualization of xn = φ(xn−1)
2. Mean Value Theorem
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Visualization of xn = φ(xn−1)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Mean Value Theorem
Lagrange’s MVT
If φ(x) is a real valued function continuous on the closed interval[x , xn−1], and differentiable in the open interval (x , xn−1) then,there exists a point ε ∈ (x , xn−1) such that
φ(xn−1)− φ(x) = φ′(ε)(xn−1 − x)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
By Uniqueness Theorem, ”A unique fixed point exists in I”.
Since, φ(x) maps I into itself. The sequence {x}∞n=0 is defined forall n ≥ 0 and xn ∈ I for all n.If x is fixed point then, it should follow x = φ(x). Also, fromabove;
xn = φ(xn−1)Now, lets write as ;|xn − x | = |φ(xn−1)− φ(x)|or , |xn − x | = |φ′(ε)(xn−1 − x)| Applying Lagrange MVT in rightpart.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
By Uniqueness Theorem, ”A unique fixed point exists in I”.Since, φ(x) maps I into itself. The sequence {x}∞n=0 is defined forall n ≥ 0 and xn ∈ I for all n.
If x is fixed point then, it should follow x = φ(x). Also, fromabove;
xn = φ(xn−1)Now, lets write as ;|xn − x | = |φ(xn−1)− φ(x)|or , |xn − x | = |φ′(ε)(xn−1 − x)| Applying Lagrange MVT in rightpart.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
By Uniqueness Theorem, ”A unique fixed point exists in I”.Since, φ(x) maps I into itself. The sequence {x}∞n=0 is defined forall n ≥ 0 and xn ∈ I for all n.If x is fixed point then, it should follow x = φ(x). Also, fromabove;
xn = φ(xn−1)
Now, lets write as ;|xn − x | = |φ(xn−1)− φ(x)|or , |xn − x | = |φ′(ε)(xn−1 − x)| Applying Lagrange MVT in rightpart.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
By Uniqueness Theorem, ”A unique fixed point exists in I”.Since, φ(x) maps I into itself. The sequence {x}∞n=0 is defined forall n ≥ 0 and xn ∈ I for all n.If x is fixed point then, it should follow x = φ(x). Also, fromabove;
xn = φ(xn−1)Now, lets write as ;|xn − x | = |φ(xn−1)− φ(x)|
or , |xn − x | = |φ′(ε)(xn−1 − x)| Applying Lagrange MVT in rightpart.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
By Uniqueness Theorem, ”A unique fixed point exists in I”.Since, φ(x) maps I into itself. The sequence {x}∞n=0 is defined forall n ≥ 0 and xn ∈ I for all n.If x is fixed point then, it should follow x = φ(x). Also, fromabove;
xn = φ(xn−1)Now, lets write as ;|xn − x | = |φ(xn−1)− φ(x)|or , |xn − x | = |φ′(ε)(xn−1 − x)| Applying Lagrange MVT in rightpart.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.
Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |. . .∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |
Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |. . .∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |
⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |. . .∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |
⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |. . .∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |
. . .∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |. . .
∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
or , |xn − x | ≤ |φ′(ε)||xn−1 − x | Applying Cauchy-Schwarz’sinequality in right part.Since, ε ∈ (x , xn−1) so, assume |φ′(ε)| = k .∴ |xn − x | ≤ k |xn−1 − x |Applying the inequality of the hypothesis inductively gives,|xn − x | ≤ k |xn−1 − x |⇒ |xn − x | ≤ k .k |xn−2 − x | = k2|xn−2 − x |⇒ |xn − x | ≤ k2.k|xn−3 − x | = k3|xn−3 − x |. . .∴ |xn − x | ≤ kn|xn−n − x | = kn|x0 − x |
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
Suppose, k < 1 and taking limn→∞
in both part.
limn→∞
|xn − x | ≤ limn→∞
kn|x0 − x |⇒ lim
n→∞|xn − x | < 0 Since, lim
n→∞kn → 0
Proof.
So, {x}∞n=0 converges to x.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
Suppose, k < 1 and taking limn→∞
in both part.
limn→∞
|xn − x | ≤ limn→∞
kn|x0 − x |
⇒ limn→∞
|xn − x | < 0 Since, limn→∞
kn → 0
Proof.
So, {x}∞n=0 converges to x.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
Suppose, k < 1 and taking limn→∞
in both part.
limn→∞
|xn − x | ≤ limn→∞
kn|x0 − x |⇒ lim
n→∞|xn − x | < 0 Since, lim
n→∞kn → 0
Proof.
So, {x}∞n=0 converges to x.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Proof
Suppose, k < 1 and taking limn→∞
in both part.
limn→∞
|xn − x | ≤ limn→∞
kn|x0 − x |⇒ lim
n→∞|xn − x | < 0 Since, lim
n→∞kn → 0
Proof.
So, {x}∞n=0 converges to x.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
1. Find the roots of the equation f (x) = x3 + x − 1? Correctupto two decimal places.
→ Given, equation f (x) = x3 + x − 1. To find the root of thisequation, we say f (x) = 0 i.e.
x3 + x − 1 = 0By testing on f (a).f (b) < 0. We estimated, the root shouldlies on [0, 1].
There are many ways to change the equation to the fixed pointform x = φ(x) using simple algebraic manipulation.
My possible choices for φ(x) are;x = φ1(x) = 1− x3
x = φ2(x) = (1− x)1/3
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
1. Find the roots of the equation f (x) = x3 + x − 1? Correctupto two decimal places.→ Given, equation f (x) = x3 + x − 1. To find the root of thisequation, we say f (x) = 0 i.e.
x3 + x − 1 = 0By testing on f (a).f (b) < 0. We estimated, the root shouldlies on [0, 1].
There are many ways to change the equation to the fixed pointform x = φ(x) using simple algebraic manipulation.
My possible choices for φ(x) are;x = φ1(x) = 1− x3
x = φ2(x) = (1− x)1/3
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
1. Find the roots of the equation f (x) = x3 + x − 1? Correctupto two decimal places.→ Given, equation f (x) = x3 + x − 1. To find the root of thisequation, we say f (x) = 0 i.e.
x3 + x − 1 = 0By testing on f (a).f (b) < 0. We estimated, the root shouldlies on [0, 1].
There are many ways to change the equation to the fixed pointform x = φ(x) using simple algebraic manipulation.
My possible choices for φ(x) are;x = φ1(x) = 1− x3
x = φ2(x) = (1− x)1/3
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
1. Find the roots of the equation f (x) = x3 + x − 1? Correctupto two decimal places.→ Given, equation f (x) = x3 + x − 1. To find the root of thisequation, we say f (x) = 0 i.e.
x3 + x − 1 = 0By testing on f (a).f (b) < 0. We estimated, the root shouldlies on [0, 1].
There are many ways to change the equation to the fixed pointform x = φ(x) using simple algebraic manipulation.
My possible choices for φ(x) are;x = φ1(x) = 1− x3
x = φ2(x) = (1− x)1/3
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
Previous theorem will suggests us, which form of φ(x) willconverges reliably and rapidly! So that, we can be able to rejectthe one which converges slowly.
Taking the derivative of above forms with respect to x . We getφ′1(x) = −3x2
φ′2(x) = −13(1−x)
23
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
Previous theorem will suggests us, which form of φ(x) willconverges reliably and rapidly! So that, we can be able to rejectthe one which converges slowly.
Taking the derivative of above forms with respect to x . We getφ′1(x) = −3x2
φ′2(x) = −13(1−x)
23
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
Test on the value of x = 0.8. On putting in above equation andtaking absolute value of it. We get
|φ′1(0.8)| = | − 1.92| > 1|φ′2(0.8)| = | − 0.97| < 1
Since,|φ′1(0.8)| > 1 So,we reject φ1(x) and we choose φ2(x) foriteration.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
Test on the value of x = 0.8. On putting in above equation andtaking absolute value of it. We get|φ′1(0.8)| = | − 1.92| > 1|φ′2(0.8)| = | − 0.97| < 1
Since,|φ′1(0.8)| > 1 So,we reject φ1(x) and we choose φ2(x) foriteration.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
Test on the value of x = 0.8. On putting in above equation andtaking absolute value of it. We get|φ′1(0.8)| = | − 1.92| > 1|φ′2(0.8)| = | − 0.97| < 1
Since,|φ′1(0.8)| > 1 So,we reject φ1(x) and we choose φ2(x) foriteration.
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
Taking initial approximation of x as 0.5. We can see in theattached video, how it converges!
CLICK THE ”ICON” TO GLORIFY YOURSELF!
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Application
From the ”Fixed Point Iteration” program in Matlab[See the codeof it in Appendix Section], we can get the result as,
Hence, the root of the given equation corrected upto 2 decimalplaces is 0.68
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Appendix
Matlab Code for the figure(example 1)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Matlab Code for the figure(example 1)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Matlab Code for the figure(example 1)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Matlab Code for the figure(example 1)
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Appendix
Matlab Code for ”Fixed Point Iteration Method”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Matlab Code for ”Fixed Point Iteration Method”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
Matlab Code for ”Fixed Point Iteration Method”
Damodar Rajbhandari Fixed point iteration method
Table of contentsIntroduction
Fixed Point Iteration MethodCondition for Convergence
ApplicationAppendix
You can find the MATLAB files of;1. Plotting code at Click Me!2. Program code at Click Me!
Damodar Rajbhandari Fixed point iteration method