Conjugate Gradient: Step-By-Step Method for Sparse Systems

Описание: *Unlock the power of the Conjugate Gradient method for solving large, sparse linear systems!*

In this concise, step‑by‑step tutorial you’ll master the mathematical foundation, algorithm flow, and a complete *Python implementation* using NumPy. Watch as we break down each iteration, compute step sizes, update search directions, and show why CG converges in at most n iterations for an n × n matrix.

What you’ll learn
*Theory & intuition:* how minimizing a quadratic form relates to solving \(Ax=b\)
*Algorithm walkthrough:* residual, direction, and step‑size calculations
*Python code:* clean, annotated implementation you can run on any sparse matrix
*Real example:* a practical Ax = b system and how to verify the solution
*When to use CG:* comparison with Jacobi, Gauss‑Seidel, and LU/Cholesky/QR methods
*Applications:* finite‑element analysis, computational physics, and machine‑learning optimizations

Why it matters
Conjugate Gradient shines when *efficiency* and *scalability* are critical—think massive scientific simulations, large‑scale data models, or any problem where a direct solver stalls. By the end of the video you’ll have a robust, reusable CG solver ready for your own projects.

*Ready to dive in?*
1️⃣ *Subscribe* for more numerical analysis tutorials
2️⃣ *Hit the bell* for notifications
3️⃣ *Download the Python code* from the GitHub link in the description
4️⃣ *Comment* with questions or your own CG use cases

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