arnoldi_iteration_posts.txt

# Social Media Posts: Arnoldi Iteration

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## 1. Twitter/X (280 chars max)

🔢 Arnoldi Iteration: Build an orthonormal basis for Krylov subspace K_k(A,b) = span{b, Ab, A²b,...}

Result? Eigenvalues of the Hessenberg matrix H_k converge to extreme eigenvalues of A!

Foundation of GMRES & eigenvalue solvers.

#Python #NumPy #LinearAlgebra #Math

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## 2. Bluesky (300 chars max)

Implemented the Arnoldi iteration algorithm in Python today.

It constructs an orthonormal basis Q_k for the Krylov subspace and produces an upper Hessenberg matrix H_k satisfying:

AQ_k = Qk₊₁H̃_k

The Ritz values (eigenvalues of H_k) rapidly approximate the extreme eigenvalues of A.

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## 3. Threads (500 chars max)

Ever wondered how iterative eigenvalue solvers work?

The Arnoldi iteration is the answer! Starting from just a matrix A and vector b, it builds:

• An orthonormal Krylov basis: span{b, Ab, A²b,...}
• An upper Hessenberg matrix H_k

The magic: eigenvalues of small matrix H_k approximate eigenvalues of huge matrix A!

In my test with a 100×100 matrix, the top 5 eigenvalues converged in just ~30 iterations. This powers GMRES, the go-to solver for large linear systems.

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## 4. Mastodon (500 chars max)

Exploring the Arnoldi iteration - the workhorse behind GMRES and implicitly restarted Arnoldi methods.

Key insight: Given A ∈ ℝⁿˣⁿ and b ∈ ℝⁿ, construct orthonormal Q_k spanning K_k(A,b) via modified Gram-Schmidt.

The Arnoldi relation AQ_k = Qk₊₁H̃_k gives us an upper Hessenberg H_k whose eigenvalues (Ritz values) converge to extreme eigenvalues of A.

Complexity: O(k²n) for orthogonalization, O(kn) storage. Sweet spot for large sparse systems!

#NumericalLinearAlgebra #Python #ScientificComputing

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## 5. Reddit

**Title:** Implemented Arnoldi Iteration from scratch - here's how eigenvalue approximation actually works

**Body:**

Hey r/learnpython!

I just built an implementation of the **Arnoldi iteration** and wanted to share what I learned.

### What is it?

The Arnoldi iteration takes a big matrix A and finds its eigenvalues without computing them directly. Instead, it:

1. Starts with a random vector b
2. Builds a sequence: b, Ab, A²b, A³b,... (called a Krylov subspace)
3. Orthogonalizes these vectors (like Gram-Schmidt)
4. Produces a small "summary" matrix H_k

The cool part? The eigenvalues of the small matrix H_k approximate the eigenvalues of the big matrix A!

### Why it matters

This is the foundation of:
- **GMRES** - solving Ax = b for huge sparse systems
- **Eigenvalue algorithms** - finding eigenvalues of matrices too big to decompose directly

### What I found

Testing on a 100×100 matrix with clustered eigenvalues:
- Top eigenvalues converged in ~30 iterations
- Orthogonality preserved to machine precision (~10⁻¹⁵)
- Arnoldi relation AQ_k = Qk₊₁H̃_k verified numerically

The visualization shows Ritz value convergence, the characteristic Hessenberg structure (zeros below subdiagonal), and eigenvalue distribution matching.

### View the full notebook

You can run and modify this notebook directly in your browser:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/arnoldiᵢteration.ipynb

Happy to answer questions about the implementation!

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## 6. Facebook (500 chars max)

Ever wonder how computers find eigenvalues of massive matrices? They don't compute them directly - that's way too expensive!

Instead, algorithms like Arnoldi iteration build a small "summary" matrix whose eigenvalues approximate the real ones.

I implemented this in Python and watched eigenvalues converge in real-time. After just 30 iterations on a 100×100 matrix, the approximations matched to 10+ decimal places!

This powers Google's PageRank and countless scientific simulations.

Check out the interactive notebook: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/arnoldiᵢteration.ipynb

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## 7. LinkedIn (1000 chars max)

**Implementing the Arnoldi Iteration: A Foundation of Modern Numerical Computing**

Just completed an implementation of the Arnoldi iteration algorithm, which serves as the backbone for solving large-scale linear systems and eigenvalue problems in scientific computing.

**Technical Approach:**

The algorithm constructs an orthonormal basis for the Krylov subspace K_k(A,b) = span{b, Ab, A²b,...,Ak⁻¹b} using modified Gram-Schmidt orthogonalization. This produces:

• Orthonormal matrix Q_k ∈ ℝⁿˣᵏ
• Upper Hessenberg matrix H_k satisfying AQ_k = Qk₊₁H̃_k

**Key Results:**

• Ritz values (eigenvalues of H_k) converged to true eigenvalues within 30 iterations
• Maintained orthogonality to machine precision (~10⁻¹⁵)
• Verified the Arnoldi relation to numerical precision

**Applications:**

This method underpins GMRES (solving Ax = b), implicitly restarted Arnoldi for eigenproblems, model order reduction, and exponential integrators for PDEs.

**Skills demonstrated:** Python, NumPy, numerical linear algebra, algorithm implementation, scientific visualization with Matplotlib.

View the complete implementation: https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/arnoldiᵢteration.ipynb

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## 8. Instagram (500 chars max)

Building eigenvalue solvers from scratch ✨

The Arnoldi iteration transforms finding eigenvalues of huge matrices into a tractable problem.

How? It builds a small matrix H_k whose eigenvalues approximate those of the original matrix A.

Swipe to see:
📊 Ritz value convergence → eigenvalues emerging
🔢 Hessenberg matrix structure → the zeros below subdiagonal
📈 Distribution matching → approximation quality

The largest eigenvalues converge fastest - exactly what we need for most applications!

#NumericalMethods
#LinearAlgebra
#Python
#DataScience
#Mathematics
#CodingLife
#ScienceVisualization