BANGLADESH UNIVERSITY OF BUSINESS AND
TECHNOLOGY (BUBT)
Assignment
On
Root finding of non-linear equation using
the Fixed Point Iteration Method
COURSE CODE: CSE 224
COURSE TITLE: Numerical Analysis Lab
SUBMITTED TO
SUBMITTED BY
Name : Bokul Islam
Id:21221203075
Intake:40
Section:1
Programm:B sc in CSE.
Name : Akib Mohammed Khan
Designation : Lecturer
Email: akibmohammed@bubt.edu.bd
Submission Date: 21 , July, 2023
th
Introduction : The task of finding the roots of a non-linear equation is a fundamental
problem in mathematics and has wide applications in various fields, including engineering,
physics, and computer science. The fixed-point iteration method is one of the numerical
techniques employed to approximate the solutions of such equations. This report aims to
provide an overview of the fixed-point iteration method, its theoretical foundations, an
example to illustrate its application, a sample implementation in code, and a discussion of its
strengths and limitations.
Theory : The fixed-point iteration method, also known as the fixed-point algorithm or the
iterative method, is based on the concept of fixed points. Given a function f(x), a fixed point
is a value c such that f(c) = c. The method utilizes this property to iteratively generate a
sequence of approximations that converge to the root of the equation.
The general iterative formula used in the fixed-point iteration method is:
x_(n+1) = g(x_n),
where x_n represents the nth approximation and g(x) is a chosen function that transforms x
into the next approximation. To ensure convergence, g(x) must satisfy two conditions:
1. g(x) must be continuous on an interval containing the root.
2. The absolute value of the derivative of g(x) must be less than 1 on the same interval.
By starting with an initial guess, x_0, and applying the iterative formula repeatedly, the
sequence of approximations will converge to the root of the equation, provided that the
conditions for convergence are met.
Example : x^2 - x - 1 = 0
Consider the non-linear equation x^2 - x - 1 = 0. We aim to find the root of this equation
using the Fixed Point Iteration Method. To apply the method, we rearrange the equation as x
= x^2 - 1 and choose the iterative function g(x) = x^2 - 1.
Starting with an initial guess of x_0 = 1, we can apply the iterative formula to find subsequent
approximations until convergence is achieved.
Using the Fixed Point Iteration formula, the first few approximations are as follows:
x_1 = g(x_0) = 1^2 - 1 = 0
x_2 = g(x_1) = 0^2 - 1 = -1
x_3 = g(x_2) = (-1)^2 - 1 = 0
Implementation (Code):
f = @(x) x^2 - x - 1;
g = @(x) x^2 - 1;
x0 = 1;
tolerance = 0.0001;
maxIterations = 100;
x = x0;
for i = 1:maxIterations
xNext = g(x);
if abs(xNext - x) < tolerance
break;
end
x = xNext;
end
if i == maxIterations
disp('No root found within the specified tolerance.');
else
disp(['Approximate root: ', num2str(xNext)]);
end
Discussion : The fixed-point iteration method is a relatively simple and intuitive approach
for finding the roots of non-linear equations. It has several advantages, such as ease of
implementation and conceptual understanding. However, it also has certain limitations. The
convergence of the method depends on the choice of the iterative function g(x) and the initial
guess x_0. If g(x) does not satisfy the convergence conditions or x_0 is too far from the root,
the method may fail to converge or converge slowly. Additionally, the fixed-point iteration
method may encounter difficulties when dealing with equations that have multiple roots or
exhibit chaotic behavior.
It is important to carefully analyze the properties of the equation and select an appropriate
iterative function to ensure successful convergence. In some cases, modifications such as
combining the fixed-point iteration method with other root-finding techniques or using
advanced convergence acceleration methods may be necessary.