Numerical Analysis Methods - Term Project

Yıldız Technical University
BLM1022 - Numerical Analysis
2024-2025 Spring Semester - 1st Year Term Project

---

📌 About This Project

This project is a comprehensive tool that implements fundamental numerical analysis methods using the C programming language. It's an interactive console application that can parse mathematical functions and perform calculations using 10 different numerical methods.

✨ Features

🔧 Implemented Numerical Methods

1. Bisection Method - Root finding
2. Regula-Falsi (False Position) Method - Root finding
3. Newton-Raphson Method - Root finding
4. Inverse Matrix - Linear equation systems
5. Cholesky (LU Decomposition) Method - Matrix decomposition
6. Gauss-Seidel Method - Iterative equation solving
7. Numerical Differentiation - Forward, backward, and central difference
8. Simpson's Rule - Numerical integration (1/3 and 3/8)
9. Trapezoidal Rule - Numerical integration
10. Gregory-Newton Interpolation - Polynomial interpolation

🎯 Core Capabilities

• Mathematical Expression Parser: Accepts and evaluates complex mathematical functions as strings
• Supported Functions:
  • Trigonometric: sin(x), cos(x), tan(x), cot(x)
  • Inverse Trigonometric: arcsin(x), arccos(x), arctan(x), arccot(x)
  • Logarithmic: log(x), log_n(x) (for any base n)
  • Exponential: exp(x), e^x, 2^x
  • Constants: pi, e
• Interactive User Interface: Menu-driven easy-to-use interface
• Detailed Iteration Information: Computation details at each step

🚀 Installation and Running

Requirements

• GCC or any C compiler
• math.h library support

Compilation

gcc numerical_analysis.c -o numerical_analysis -lm

Running

./numerical_analysis

For Windows:

numerical_analysis.exe

📖 Usage

When you start the program, you'll be greeted with a menu:

---------------------------------------------------------
Quit: 0
Bisection: 1
Regula-Falsi: 2
Newton-Raphson: 3
Inverse Matrix: 4
Cholesky Method: 5
Gauss-Seidal: 6
Numerical differentiation: 7
Simpson's Rule: 8
Trapezoidal Rule: 9
Gregory-Newton: 10
---------------------------------------------------------

Example Usage

Root Finding (Bisection Method):

Which operation do you want to do? 1
Please enter your function:
> x^3 - 2*x - 5
Enter a value: 2
Enter b value: 3
Enter max error: 0.0001
Enter maximum iterations: 100

Numerical Differentiation:

Which operation do you want to do? 7
Please enter your function:
> sin(x) + x^2
At what x value do you want to calculate the derivative? 1.5
What will be the h value? (recommended to be small like 0.0001) 0.0001

Simpson Integration:

Which operation do you want to do? 8
Please enter your function:
> x^2 + 2*x + 1
Lower limit of the integration? 0
Upper limit of the integration? 2
How many parts do you want to split the integral into? 10

📁 Project Files

• numerical_analysis.c - Main source code
• examples.pdf - Solved example problems
• project_report.pdf - Detailed project report and methodology
• README.md - Project documentation

🔍 Mathematical Expression Syntax Rules

• Variable: Use x
• Exponentiation: ^ operator (e.g., x^2, 2^x)
• Multiplication/Division: * and /
• Addition/Subtraction: + and -
• Parentheses: ( and ) are supported
• Trigonometric functions: Work in radians
• Special constants: pi ≈ 3.14159, e ≈ 2.71828

Example Functions:

• x^3 - 2*x + 1
• sin(x) + cos(2*x)
• log(x) + x^2
• exp(x) - 3*x
• (x+1)*(x-2)/(x+3)

🛠️ Technical Details

Data Structures

The program represents mathematical expressions using Abstract Syntax Tree (AST):

typedef enum {
    NODE_CONSTANT,    // Constant values
    NODE_VARIABLE,    // Variables (x)
    NODE_OPERATOR,    // +, -, *, /, ^
    NODE_FUNCTION     // sin, cos, log, etc.
} NodeType;

Parser Architecture

Mathematical expressions are parsed using recursive descent parsing:

• addsub_parser() - Addition/subtraction
• muldiv_parser() - Multiplication/division
• exp_parser() - Exponentiation
• parse_primary() - Primary expressions (numbers, variables, functions)

📊 Performance and Precision

• Maximum iterations: User-definable
• Error tolerance: Adjustable (e.g., 1e-6, 1e-10)
• Memory management: Dynamic memory allocation for flexible matrix sizes
• Numerical stability: Pivot selection and condition checking

🐛 Known Limitations

• Functions work only with single variable (x)
• Trigonometric functions operate in radian mode
• Matrix operations require square matrices
• Gregory-Newton requires equally spaced points

📚 References

The algorithms and methods used in this project are taken from standard numerical analysis sources:

• Numerical Analysis (Burden & Faires)
• Numerical Methods for Engineers (Chapra & Canale)
• BLM1022 Numerical Analysis Course Notes

👨‍💻 Developer

Yıldız Technical University

Aziz Eren YILDIRIM
Computer Engineering - 1st Year
2024-2025 Spring Semester

📄 License

This project is licensed under the MIT License. See the LICENSE file for details.

---

🤝 Contributing

This is a student project. Feel free to open issues or submit pull requests for suggestions and improvements.

⭐ Support

If you like this project, don't forget to give it a star! ⭐

---

Note: This project is for educational purposes and was developed as part of the Numerical Analysis course at Yıldız Technical University.