Numerical Methods for Linear Systems

Overview

This Python script implements several numerical methods for solving systems of linear equations. It provides implementations for the Jacobi Method, Gauss-Seidel Method, Successive Over-Relaxation (SOR) Method, and Iterative Refinement Method. Additionally, the script includes a direct solution using NumPy's linear solver for comparison purposes.

Methods Implemented

• Jacobi Method: An iterative technique for solving the diagonal entries of a matrix equation.
• Gauss-Seidel Method: An improved version of the Jacobi Method that uses the latest updated values for convergence.
• SOR Method: Extends the Gauss-Seidel method by using a relaxation factor to potentially accelerate the convergence.
• Iterative Refinement Method: Refines a given solution by iteratively correcting the residual error.
• Direct Solution (for verification and not asked in the assignment): Uses NumPy's linalg.solve function to compute the exact solution, providing a benchmark for verifying the iterative solutions.

Requirements

Requirements will be found in the requirements.txt file in the repository

Usage

To use the script, ensure that Python 3 and NumPy are installed on your system. You can run the script from the command line as follows:

python main.py

Output

The results are saved to numerical_methods_output.txt in the same directory as the script. The output file will contain the results for each method, formatted clearly for easy comparison.

Conclusion

This script is useful for educational purposes to understand different iterative methods for solving linear systems and for verifying the convergence and accuracy of these methods against a direct solution.