README.md

https://SenatorovAI.com

Conjugate Gradient & Sparse CG Solver Course

🚀 Professional implementation and mathematical explanation of Conjugate Gradient (CG) and Sparse Conjugate Gradient methods for large-scale linear systems.

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🔥 Project Overview

This repository provides a complete course-style implementation of:

• Conjugate Gradient (CG) algorithm
• Preconditioned Conjugate Gradient
• Sparse Conjugate Gradient
• Large-scale linear system solving
• Numerical stability analysis
• Optimization perspective of CG

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Keywords

conjugate gradient
conjugate gradient method
sparse conjugate gradient
pcg solver
cg solver python
large scale linear systems
numerical linear algebra
iterative solver
preconditioned conjugate gradient
optimization solver
python cg implementation

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📚 Mathematical Background

Linear System Problem

We solve:

$$ Ax = b $$

Where:

• $$A$$ — symmetric positive definite matrix
• $$x$$ — unknown vector
• $$b$$ — right-hand side

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🔵 Conjugate Gradient Method

CG minimizes quadratic function:

$$ f(x) = \frac{1}{2}x^T A x - b^T x $$

Update rule:

$$ x_{k+1} = x_k + \alpha_k p_k $$

Where:

• $$p_k$$ — conjugate direction
• $$\alpha_k$$ — optimal step size

Step size:

$$ \alpha_k = \frac{r_k^T r_k} {p_k^T A p_k} $$

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Residual Update

$$ r_k = b - Ax_k $$

Direction update:

$$ p_{k+1} = r_{k+1} + \beta_k p_k $$

Where:

$$ \beta_k = \frac{r_{k+1}^T r_{k+1}} {r_k^T r_k} $$

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⚡ Sparse Conjugate Gradient

For sparse matrices:

• Store matrix in CSR/CSC format
• Avoid dense multiplication
• Reduce memory complexity

Advantages:

✅ Memory efficient
✅ Faster computation
✅ Scalable to large systems

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🧠 Why This Project Is Important

CG is used in:

• Finite element methods
• Physics simulations
• Machine learning
• PDE solvers
• Large sparse systems
• Scientific computing

It is one of the most important iterative solvers.

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🏗 Project Structure

conjugate-gradient-sparse-cg-solver-course/
│
├── README.md
├── LICENSE
├── CITATION.cff
├── requirements.txt
│
├── src/
│   ├── cg_solver.py
│   ├── sparse_cg.py
│   ├── preconditioner.py
│
├── examples/
│   └── demo.py
│
├── docs/
│   ├── theory.md
│   ├── convergence.md
│
├── images/
│   └── convergence_plot.png
│
└── index.html

Clean structure improves:

✔ Search ranking
✔ Professional appearance
✔ Research credibility

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🐍 Example — Basic Conjugate Gradient Implementation

import numpy as np

def conjugate_gradient(A, b, x0=None, tol=1e-8, max_iter=1000):
    n = len(b)
    x = np.zeros(n) if x0 is None else x0

    r = b - A @ x
    p = r.copy()

    for _ in range(max_iter):
        Ap = A @ p
        alpha = (r @ r) / (p @ Ap)
        x = x + alpha * p

        r_new = r - alpha * Ap

        if np.linalg.norm(r_new) < tol:
            break

        beta = (r_new @ r_new) / (r @ r)
        p = r_new + beta * p
        r = r_new

    return x

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🚀 Installation

pip install -r requirements.txt

Run example:

python examples/demo.py

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📊 Visualization (Highly Recommended)

Add:

• Residual norm vs iteration
• Convergence curve
• Sparse matrix structure plot

Example:

import matplotlib.pyplot as plt

plt.plot(residual_history)
plt.xlabel("Iteration")
plt.ylabel("Residual Norm")
plt.title("CG Convergence")
plt.show()