Ma Trận và Cơ Sở của Tính Toán Khoa Học (Matrices and Fundamentals of Scientific Computing)

Course: MTH00051 - Toán Ứng Dụng và Thống Kê (Applied Mathematics and Statistics)

Institution: Ho Chi Minh City University of Science (HCMUS)

Author:

• Nguyễn Anh Thái
• Nguyễn Đình Tuấn
• Nguyễn Huỳnh Gia Bảo
• Vòng Sau Hậu
• Lương Nhật Tân

📌 Project Overview

This repository contains a from-scratch Python implementation of core numerical linear algebra algorithms. The project is divided into three main components:

• Part 1: Gaussian Elimination & Applications
  • Implementation of Gaussian elimination with partial pivoting.
  • Solving systems of linear equations via back-substitution.
  • Calculating matrix determinants and inverses (Gauss-Jordan).
  • Finding matrix rank and bases for column, row, and null spaces.
• Part 2: Matrix Decomposition & Visualization
  • Implementation of a chosen matrix decomposition (LU, QR, SVD, or Cholesky).
  • Matrix diagonalization (eigenvalues and eigenvectors).
  • Mathematical animation and visualization of the decomposition using Manim.
• Part 3: Performance Analysis & Numerical Stability
  • Implementation of the Gauss-Seidel iterative method.
  • Benchmarking runtime and relative error against random matrices of varying sizes ($n \in {50, 100, 200, 500, 1000}$).
  • Log-log plotting and stability analysis regarding the matrix Condition Number.

🛠️ Technology Stack

• Language: Python $\ge 3.10$
• Visualization: Manim Community v0.18.0
• Data Analysis & Plotting: Matplotlib, Jupyter Notebook
• Verification (Not used for core logic): NumPy, SciPy, SymPy

📂 Repository Structure

The codebase strictly follows the required project structure:

Group_<ID>/
|-- README.md
|-- requirements.txt
|-- report/
|   |-- report.pdf
|-- part1/
|   |-- gaussian.py
|   |-- determinant.py
|   |-- inverse.py
|   |-- rank_basis.py
|   |-- part1_demo.ipynb
|-- part2/
|   |-- decomposition.py
|   |-- diagonalization.py
|   |-- manim_scene.py
|   |-- part2_demo.ipynb
|   |-- manim_link.txt
|-- part3/
|   |-- solvers.py
|   |-- benchmark.py
|   |-- analysis.ipynb
|   |-- benchmark_results.json

🚀 Setup & Installation

1. Clone this repository.

   git clone https://github.com/ThayLoser/matrix-computing.git

2. Install the required dependencies:

   pip install -r requirements.txt

3. Navigate to the correct folder if needed.

   cd <FolderName>

4. To run the Manim visualization for Part 2:

   manim -pqh part2/manim_scene.py <SceneClassName>

5. Explore the Jupyter notebooks (part1_demo.ipynb, part2_demo.ipynb and analysis.ipynb) to see the implementations and benchmarks in action.