Update README.md

Signed-off-by: Ruslan Senatorov <55090151+ruslansenatorov@users.noreply.github.com>

Author: ruslansenatorov

README.md

@@ -1 +1,247 @@
-# Conjugate-Gradient-stepik
+# stepik: https://stepik.org/a/260000
+
+
+# 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.
+
+---
+
+## 🔥 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
+
+---
+
+## 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
+
+```
+
+---
+
+## 📚 Mathematical Background
+
+### Linear System Problem
+
+We solve:
+
+$$
+Ax = b
+$$
+
+Where:
+
+- $$A$$ — symmetric positive definite matrix
+- $$x$$ — unknown vector
+- $$b$$ — right-hand side
+
+---
+
+## 🔵 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:
+
+$$
+\alpha_k =
+\frac{r_k^T r_k}
+{p_k^T A p_k}
+$$
+
+---
+
+### Residual Update
+
+$$
+r_k = b - Ax_k
+$$
+
+Direction update:
+
+$$
+p_{k+1} = r_{k+1} + \beta_k p_k
+$$
+
+Where:
+
+$$
+\beta_k =
+\frac{r_{k+1}^T r_{k+1}}
+{r_k^T r_k}
+$$
+
+---
+
+## ⚡ 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  
+
+---
+
+## 🧠 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.
+
+---
+
+## 🏗 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  
+
+---
+
+## 🐍 Example — Basic Conjugate Gradient Implementation
+
+```python
+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
+````
+
+---
+
+## 🚀 Installation
+
+```bash id="install-cg"
+pip install -r requirements.txt
+```
+
+Run example:
+
+```bash id="run-cg"
+python examples/demo.py
+```
+
+---
+
+## 📊 Visualization (Highly Recommended)
+
+Add:
+
+* Residual norm vs iteration
+* Convergence curve
+* Sparse matrix structure plot
+
+Example:
+
+```python id="plot-cg"
+import matplotlib.pyplot as plt
+
+plt.plot(residual_history)
+plt.xlabel("Iteration")
+plt.ylabel("Residual Norm")
+plt.title("CG Convergence")
+plt.show()
+```
+
+
+