README.md

https://SenatorovAI.com

LSQR Solver — Sparse Linear System Course

🚀 Professional implementation and mathematical explanation of the LSQR algorithm for solving large-scale sparse linear systems.

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🔥 Project Overview

This repository provides a complete course-style implementation of the LSQR algorithm, including:

• Mathematical derivation
• Connection to Conjugate Gradient
• Sparse matrix optimization
• Numerical stability analysis
• Python implementation from scratch
• Applications in large-scale optimization

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Keywords

lsqr solver
lsqr algorithm
sparse linear solver
iterative least squares
large scale linear systems
lsqr python implementation
numerical linear algebra
sparse matrix solver
least squares solver
cg based least squares

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📚 Mathematical Background

LSQR solves the least squares problem:

$$ \min_x ||Ax - b||_2 $$

Where:

• $$A \in \mathbb{R}^{m \times n}$$ (sparse or large matrix)
• $$b \in \mathbb{R}^m$$
• $$x$$ — unknown vector

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🔵 Algorithm Foundation

LSQR is based on:

• Lanczos bidiagonalization
• Krylov subspace methods
• Connection to Conjugate Gradient on normal equations

Instead of solving:

$$ A^T A x = A^T b $$

Directly (which is unstable),

LSQR solves it iteratively in a numerically stable way.

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⚡ Core Idea

LSQR constructs Krylov subspaces:

$$ \mathcal{K}_k(A^T A, A^T b) $$

and minimizes the least squares residual inside this subspace.

It avoids explicitly forming:

$$ A^T A $$

Which improves:

✅ Stability
✅ Memory efficiency
✅ Performance for sparse matrices

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🧠 Why This Project Is Important

LSQR is used in:

• Large-scale data fitting
• Image reconstruction
• Tomography
• Inverse problems
• Machine learning regularization
• Scientific computing

It is one of the most important industrial-grade solvers.

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🏗 Project Structure

lsqr-solver-course/
│
├── README.md
├── LICENSE
├── CITATION.cff
├── requirements.txt
│
├── src/
│   ├── lsqr_solver.py
│   ├── bidiagonalization.py
│   ├── operators.py
│
├── examples/
│   └── demo.py
│
├── docs/
│   ├── theory.md
│   ├── convergence.md
│
├── images/
│   └── convergence_plot.png
│
└── index.html

Clean structure improves:

✔ Discoverability
✔ Professional appearance
✔ Research credibility

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🐍 Example — Simple LSQR Skeleton

import numpy as np

def lsqr(A, b, tol=1e-8, max_iter=1000):
    m, n = A.shape
    x = np.zeros(n)

    r = b - A @ x
    beta = np.linalg.norm(r)
    u = r / beta

    for _ in range(max_iter):
        v = A.T @ u
        alpha = np.linalg.norm(v)
        v = v / alpha

        # Update solution (simplified step)
        x += (beta / alpha) * v

        r = b - A @ x
        beta = np.linalg.norm(r)

        if beta < tol:
            break

    return x

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🚀 Installation

pip install -r requirements.txt

Run example:

python examples/demo.py

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📊 Visualization (Recommended)

Add:

• Residual norm vs iteration
• Convergence curve
• Krylov subspace dimension growth

Example:

import matplotlib.pyplot as plt

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