Math 221 — Numerical Analysis Project¶
Prince Sultan University¶
Course: Math 221: Numerical Analysis
Instructor: Dr. Nahid Fatima
Submission Date: 16 November 2025
Developed by¶
| Name | Student ID | Role |
|---|---|---|
| Shoug Fawaz Alomran | 223410392 | Implementation, documentation, and digital report development |
| Shahad Abunayan | 223410189 | Mathematical handwritten derivations and theoretical explanation |
| Manar Altuwaim | 220410529 | Handwritten calculations and verification of numerical results |
Collaboration Note:
This project was completed collaboratively. Shoug Alomran led the implementation, documentation, and site publishing, while Shahad Abunayan and Manar Altuwaim contributed to the handwritten derivations, theoretical analysis, and manual calculations of all numerical methods.
Project Title¶
Comparison of Numerical Methods for Solving Nonlinear Equations
(Bisection, Newton–Raphson, and Secant Methods)
Abstract¶
This project studies three numerical methods — Bisection, Newton–Raphson, and Secant — to find the root of nonlinear equations that cannot be solved exactly.
Each method was implemented using GNU Octave, and the results were compared based on convergence speed, accuracy, and stability.
All three methods reached the same correct root (x = 2).
The Newton–Raphson Method converged the fastest, followed by Secant, while Bisection remained the most reliable though slower.
Objectives¶
- Apply and compare classical numerical root-finding methods.
- Study their convergence rate, accuracy, and computational efficiency.
- Display results through tables and graphs.
- Strengthen understanding of iterative numerical analysis through practical implementation.
Methods Used¶
| Method | Description | Characteristics |
|---|---|---|
| Bisection Method | Divides an interval repeatedly to locate the root. | Slow but guaranteed convergence. |
| Newton–Raphson Method | Uses tangent lines based on derivatives to find the root. | Fast but requires a good initial guess. |
| Secant Method | Approximates the derivative using two points. | Moderate speed and simple to implement. |
Results Summary¶
| Method | Approximate Root | Iterations |
|---|---|---|
| Bisection | 2.000000 | 1 |
| Newton–Raphson | 2.000000 | 5 |
| Secant | 2.000000 | 6 |
All methods converged to the same root (x = 2).
The Newton–Raphson Method was the fastest, confirming the theory discussed in class.
Applications¶
Numerical root-finding methods are widely used in:
- Engineering and physics modeling
- Optimization and computer algorithms
- Financial and economic calculations
- Simulation and control systems
Tools Used¶
- GNU Octave 10.3.0 (compatible with MATLAB)
- Visual Studio Code / macOS Terminal
- MkDocs Material for creating this digital report
- Git & GitHub for version control and deployment
References¶
See the complete list in the References section.