Polynomial Integral Approximation

A Java console application that implements numerical methods for polynomial interpolation and integral approximation. Currently features Lagrange polynomial interpolation with planned support for trapezoid and Simpson integration methods.

Features

• Lagrange Polynomial Interpolation: Calculate polynomial values using Lagrange interpolation method
• Interactive console-based user interface
• Input validation and error handling
• Detailed calculation tables with formatted output
• Support for custom data point sets

Mathematical Background

Lagrange Polynomial Interpolation

The Lagrange interpolation method constructs a polynomial that passes through a given set of points. For a set of n+1 data points (x₀, y₀), (x₁, y₁), ..., (xₙ, yₙ), the Lagrange polynomial is defined as:

P(x) = Σ(i=0 to n) yi * Li(x)

Where Li(x) is the i-th Lagrange basis polynomial:

Li(x) = Π(j=0 to n, j≠i) (x - xj) / (xi - xj)

Getting Started

Prerequisites

• Java Development Kit (JDK) 8 or higher
• Any Java IDE or text editor (VS Code recommended)

Installation

1. Clone this repository:

   git clone https://github.com/AphilK/poly-integral-approx.git
   cd poly-integral-approx

2. Compile the Java source code:

   javac -d bin src/PolynomialIntegralApproximation.java

3. Run the application:

   java -cp bin PolynomialIntegralApproximation

Usage

Running the Application

When you start the program, you'll see a menu with the following options:

Please choose an option:
1 - Find the lagrange polynomial
2 - Solve an integral using the trapezoid method (Coming Soon)
3 - Solve an integral using the simpson method (Coming Soon)
4 - Leave

Lagrange Polynomial Interpolation

1. Select option 1 from the main menu
2. Enter your x-values (independent variables), type 'e' when finished
3. Enter your corresponding y-values (function values), type 'e' when finished
4. Enter the value of t where you want to evaluate the polynomial
5. The program will display a detailed calculation table and the result

Example Usage

Please enter the values of x. Type 'e' to exit.
1.0
2.0
3.0
e

Now enter the values of f(x). type 'e' to exit.
2.0
8.0
18.0
e

Please enter the value of t to calculate the lagrange polynomial: 1.5

The program will output a detailed table showing the calculation process and the final interpolated value.

Input Validation

• The program ensures that the number of x-values matches the number of y-values
• The interpolation point t must be within the range of provided x-values
• Invalid inputs are handled gracefully with error messages

Project Structure

poly-integral-approx/
├── README.md                           # This file
├── src/
│   └── PolynomialIntegralApproximation.java  # Main application source
├── bin/                                # Compiled class files
│   └── PolynomialIntegralApproximation.class
└── lib/                                # Dependencies (currently empty)

Technical Details

Code Features

• Modular Design: Clean separation between user interface and mathematical calculations
• Error Handling: Comprehensive input validation using try-catch blocks
• Formatted Output: Results displayed with 10 decimal places precision
• Dynamic Data Structures: Uses ArrayList for flexible data point management
• Interactive Console: User-friendly menu-driven interface

Key Methods

• main(String[] args): Application entry point and main menu loop
• lagrange(ArrayList<Double> x, ArrayList<Double> y, double t): Implements Lagrange interpolation algorithm

Future Enhancements

• Implement trapezoid rule for numerical integration
• Implement Simpson's rule for numerical integration
• Add support for reading data points from files
• Implement additional interpolation methods (Newton, Spline)
• Add graphical visualization of polynomials
• Support for higher-order derivatives

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

Development Setup

1. Fork the repository
2. Create your feature branch (git checkout -b feature/AmazingFeature)
3. Commit your changes (git commit -m 'Add some AmazingFeature')
4. Push to the branch (git push origin feature/AmazingFeature)
5. Open a Pull Request

License

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

Author

AphilK - GitHub Profile

Acknowledgments

• Lagrange interpolation method based on classical numerical analysis techniques
• Inspired by numerical methods courses in computational mathematics