# 📈 System Visualization: Solving Square Systems (GPP vs. G-S vs. Kaczmarz)

## 🌟 Project Context and Goal

This project was developed collaboratively with a colleague at the specific request of our professor. The primary goal is to provide a set of visualization tools to be used for teaching and demonstrating the behavior, steps, and convergence rates of various numerical methods for solving square linear systems.

## 🛠️ Technology Stack

The entire suite is built on Python, leveraging the following libraries for core functionality and visualization:

* **Core Language:** Python 3.x
* **Numerical Computing:** **NumPy** (`numpy`) for matrix operations, vector handling, and core algorithm implementation (GPP, G-S, Kaczmarz).
* **GUI:** **PyQt6** (`PyQt6.QtWidgets`, `PyQt6.QtCore`) for building the interactive desktop interfaces.
* **Plotting & Visualization:** **Matplotlib** (`matplotlib.pyplot`, `matplotlib.backends`) for integrating dynamic graphs and visualizing matrix elements within the PyQt GUI.
* **Data Handling:** Standard Python `csv` module for managing performance data.
### Running the Scripts

1.  **Dependency Installation:** Install all required libraries using the provided `requirements.txt` file:
    ```bash
    pip install -r ".\requirements.txt"
    ```
2.  **Execution:**
    * Navigate your terminal to the specific directory of the application you wish to run.
    * Execute the main Python script for that application.

### ⚠️ Important Note on Directories

Please be aware of the file naming conventions:

* Directories intended for **comparison visualizations** often contain two distinct Python files:
    * `Time_To_CSV.py`: Used for generating new timing data (running the algorithms and saving execution times to a CSV file).
    * `CSV_To_Plot.py`: Used for plotting the existing or newly generated data from the CSV file.

## 🔍 Application Descriptions

### 1. Gaussian Elimination with Partial Pivoting (GPP)

* **Interface:** Allows users to specify the matrix and vector size (between 2 and 10).
* **Visualization:** After manual or random data entry, the application represents the step-by-step modifications occurring in the matrix and vector.
* **Final Output:** The last step clearly shows the matrix transformed into its upper triangular form and the modified vector **b**.
* **Bonus:** The directory includes a presentation video showcasing the process.

### 2. Gauss-Seidel Method (G-S)

* **Interface:** Allows users to specify the matrix dimension.
* **Matrix Selection:** Choose the desired matrix type: **Random**, **Positive Semi-Definite**, or **Diagonally Dominant**.
* **Visualization:** Starting the algorithm generates a plot that describes the **norm of the solution vector (x)** at each iteration, clearly illustrating the algorithm's different behaviors across the three matrix cases.

### 3. Kaczmarz and Randomized Block Kaczmarz (Kaczmarz and RBK)

* **Note:** For detailed explanations of these algorithms, we highly recommend consulting the provided **`.ipynb` (Jupyter Notebook) file**.
* **Interface:** Similar to the Gauss-Seidel interface.
* **Configuration:** Set the matrix dimension and choose whether to enforce the **Diagonally Dominant** condition.
* **Convergence Plot:** Select the desired visualization type for convergence: **Normal Scale** or **Logarithmic Scale**.

### 4. GPP vs. G-S vs. Kaczmarz (Comparison)

* **Comparison Sets:** We provide two dedicated comparison directories:
    * One comparing **all three algorithms (GPP, G-S, Kaczmarz)**, requiring the diagonally dominant condition to ensure G-S convergence.
    * One comparing **GPP and Kaczmarz** without the diagonal dominance condition.
* **Plotting:** A **logarithmic scale** is used for the plot axes to clearly emphasize the differences in performance and convergence rates.
* **Time Generation:** The scripts for calculating new execution times can be run if desired. (**Warning:** Running this process can take a significant amount of time!)
* **Visual Aid:** The directory includes pre-generated images of the final comparison plots.

## 📧 Contact

For inquiries regarding the project's implementation or its use in teaching, feel free to reach out:

badescuandu2@gmail.com or https://www.linkedin.com/in/andu-badescu-80384a203/
tudor.popanica@stud.acs.upb.ro or https://www.linkedin.com/in/tudor-popanica-răboj-56149b2b3/