Our project’s objective was to compare three classical approaches for solving linear equations: Gaussian Elimination, Matrix Inversion, and Cramer’s Rule. We implemented these methods in Python, leveraging its libraries. Specifically we used NumPy for matrix operations, Matplotlib for visualization, time for execution time measurement, and psutil for monitoring RAM usage.

To systematically assess each method’s performance, we first developed a random matrix generator that created square matrices and corresponding solution vectors with customizable parameters. This allowed us to test the methods under different conditions and analyze their efficiency. For Gaussian Elimination, we leveraged NumPy’s built-in function to ensure optimal performance. Similarly, for Matrix Inversion, we used NumPy’s functions to first invert the coefficient matrix and then multiply it with the solution vector. Finally, for Cramer’s Rule, we relied on NumPy’s determinant function to repeatedly compute determinants for each variable in the system.

To evaluate the effectiveness of these methods, we conducted extensive tests focusing on accuracy, numerical stability, runtime, and resource usage. We used NumPy’s Gaussian Elimination as a benchmark to compare the accuracy of each approach. Execution time was measured using Python’s time library, while psutil was used to track computational efficiency. Our use of these tools permitted us to provide insights into the strengths and limitations of each method.

We were a team of four, which allowed us to split the tasks evenly and efficiently, however it was important to go back and make sure everything is linked properly. It was fundamental for the scripts to be aligned and to generate some global input parameters that would then be used along by each of us. Also, as mentioned previously, a fresh perspective can really only benefit a project. Delegating the work according to each one’s strengths and inclinations was also a beneficial factor.