Bisection Method for Finding Roots of Equations

This project implements the Bisection Method to find roots of equations in one variable. The Bisection Method is a numerical method that repeatedly bisects an interval and selects a subinterval in which a root must lie.

Table of Contents

• Installation
• Usage
• Algorithm
• Example
• License

Installation

Clone this repository to your local machine.

git clone https://github.com/KuroshNazari/Bisection-Method.git

To run this program, you need to have Python installed on your machine. You also need to install the following libraries:

• sympy for symbolic mathematics
• matplotlib for plotting graphs
• numpy for numerical operations
• prettytable for displaying results in a table format

You can install these packages using pip:

pip install sympy prettytable matplotlib numpy

or

pip install -r requirements.txt

Usage

1. Run the script.
2. Input the equation you want to solve in terms of x (e.g., x**2 - 4, exp(x) or sin(x)).
3. Specify the interval (e.g., (0, 5)) to search for a root.
4. Confirm if you're satisfied with the plot showing the function and the specified interval.
5. The program will then perform the Bisection Method to find and print the approximated root.

Algorithm

To begin, set a1 = a and b1 = b, and let p1 be the midpoint of [a, b]; that is, m1 = (a1 + b1)/2

1. If f(m1) = 0, then m = m1, and we are done.
2. If f(m1) ≠ 0, then f(m1) has the same sign as either f(a1) or f(b1).
   • If f(m1) and f(a1) have the same sign, m ∈ ( m1, b1). Set a2 = m1 and b2 = b1.
   • If f(m1) and f(a1) have opposite signs, m ∈ (a1, m1). Set a2 = a1 and b2 = m1.
3. we can select a tolerance ε > 0 and generate m1, ... , mN until one of the following conditions is met:
   • If |m_N - m_N-1| < ε
   • If |f(m_N)| < ε
   • If |m_N - m_N-1|/|m_N| < ε and m ≠ 0

Example

To use the Bisection Method, you can create an instance of the BisectionMethod class and call the solve() method:

b = BisectionMethod(max_iteration=100, epsilon=0.00001, stopping_procedures=3)
b.solve()

Below is an example of the output:

If the bisection fails to approximate the root, the program will output a failure message.

License

This project is open source and available under the MIT License.

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Feel free to adjust the details as necessary to align with your coding style or specific project requirements!