Linear System Solver

This library contains a benchmarking tool for comparing different iterative linear algebra solvers on sparse matrix systems. The iterative methods analyzed are:

• Jacobi
• Gauss-Seidel
• Gradient
• Conjugate Gradient

Features

• Flexible Input Formats: Support for .mtx matrices and various vector formats (.npy, .txt, .csv)
• Comprehensive Analysis: Performance comparison across different tolerance levels
• Visual Output: Automated generation of comparison plots and sparsity pattern visualizations
• Detailed Results: JSON output with convergence data, iterations, execution time, and relative errors

---

Installation

1. Clone the repository:

   git clone https://github.com/Mattia-Ingrassia/Linear-System-Solver.git
   cd Linear-System-Solver

2. Create a virtual environment:

   python -m venv venv

3. Activate the virtual environment:

   • On Windows:

     venv\Scripts\activate

   • On macOS and Linux:

     source venv/bin/activate

4. Install the required packages:

   pip install -r requirements.txt

---

Project Structure

├── cli_interface.py           # Main CLI application
├── plot_generator.py          # Visualization utilities
├── solvers/
│   ├── base_solver.py         # Base solver interface
│   ├── jacobi.py              # Jacobi method implementation
│   ├── gauss_seidel.py        # Gauss-Seidel method implementation
│   ├── gradient.py            # Gradient method implementation
│   └── conjugate_gradient.py  # Conjugate Gradient implementation
└── results/                   # Generated results directory

Usage

Basic Usage

python cli_interface.py --a_path "data/spa1.mtx"

Advanced Usage

python cli_interface.py \
    --a_path matrix.mtx \
    --b_path rhs_vector.txt \
    --x_path solution_vector.txt \
    --tol 1e-4 --tol 1e-6 --tol 1e-8 \
    --verbose True

Command Line Options

Option	Type	Required	Description
--a_path	Path	Yes	Path to coefficient matrix in .mtx format
--b_path	Path	No	Path to right-hand side vector (.npy, .txt, .csv)
--x_path	Path	No	Path to true solution vector (.npy, .txt, .csv)
--tol	Float	No	Tolerance values (can be specified multiple times)
--verbose	Bool	No	Enable verbose output (default: True)

Default Behavior

• Missing b vector: Calculated as b = A * x
• Missing x vector: Initialized as a vector of ones
• Missing tolerances: Uses default values [10e-4, 10e-6, 10e-8, 10e-10]

Supported File Formats

Matrix Format

• Input: .mtx (Matrix Market format)
• Processing: Converted to dense format for computation

Vector Formats

• .npy: NumPy binary format
• .txt: Plain text (space-separated values)
• .csv: Comma-separated values

Output

Directory Structure

results/
└── matrix_name/
    ├── results.json             
    ├── matrix_name_sparsity_pattern.png
    └── images/
        ├── matrix_name_iterations_comparison.png
        ├── matrix_name_time_comparison.png
        └── matrix_name_error_comparison.png

Results JSON Format

[
    {
        "tolerance": 10e-4,
        "solutions": [
            {
                "solution" : [],
                "relative_error": 0.01758105433208519,
                "iterations": 82,
                "time_spent": 0.027469000007840805,
                "solver_name": "Jacobi",
                "converged": true
            }
        ]
    }
]

Generated plots

1. Iterations Comparison: Number of iterations vs tolerance for each solver
2. Time Comparison: Execution time vs tolerance for each solver
3. Error Comparison: Relative error vs tolerance for each solver
4. Sparsity Pattern: Visual representation of matrix structure and density