Calculator Input
Example Data Table
Use this sample set to test the calculator quickly.
| Field | Example Value | Reason |
|---|---|---|
| Matrix A | 4, 1, 0 1, 3, 1 0, 1, 2 |
Small symmetric matrix for clean verification. |
| Starting Vector b | 1, 0, 1 | Nonzero seed that spans more than one direction. |
| Iterations | 3 | Enough steps to observe Krylov growth. |
| Tolerance | 0.0000000001 | Useful for detecting near breakdown behavior. |
| Decimal Places | 6 | Keeps tables readable without hiding detail. |
Formula Used
The calculator builds an orthonormal Krylov basis with modified Gram-Schmidt orthogonalization. It starts with q₁ = b / ||b||.
For each step k, it forms v = A q_k, then computes hj,k = qjT v for all previous basis vectors.
It updates the work vector by v = v - Σ hj,k qj, then measures hk+1,k = ||v||.
If hk+1,k is larger than the tolerance, the next basis vector is qk+1 = v / hk+1,k.
The projected relation is A Qm = Qm Hm + hm+1,m qm+1 emT. Residual norms and Rayleigh estimates help interpret convergence and stability.
How to Use This Calculator
- Enter a square matrix with one row per line.
- Enter a nonzero starting vector with matching length.
- Choose the number of Arnoldi iterations.
- Set the tolerance used for breakdown detection.
- Select the number of decimal places for display.
- Press Solve Arnoldi Iteration to generate the basis, Hessenberg matrix, history table, and graph.
- Download the CSV or PDF file for reporting or study notes.
Frequently Asked Questions
1. What does this calculator compute?
It computes Arnoldi basis vectors, the projected Hessenberg matrix, residual norms, and Rayleigh estimates from your matrix and starting vector.
2. Why must the matrix be square?
Arnoldi iteration builds a Krylov subspace for a linear operator acting in one vector space. A square matrix keeps domain and codomain dimensions consistent.
3. What is a happy breakdown?
It means the new residual norm reached the tolerance threshold, so the current Krylov subspace already captures an invariant subspace of the matrix.
4. What does the residual norm show?
It tracks the size of the component not represented by the current basis. Smaller values often indicate stronger projection quality.
5. What is the Rayleigh estimate here?
The calculator reports qᵀAq for the current normalized basis vector. It offers a quick scalar estimate related to local eigen behavior.
6. How many iterations should I choose?
Start with a small number, such as 3 to 8, then increase it when you want a richer Krylov basis or slower convergence analysis.
7. Why does orthogonality error matter?
It measures how close the computed basis is to an ideal orthonormal set. Large values can signal numerical instability or ill conditioning.
8. Can I use decimals and negative entries?
Yes. The parser accepts decimal values, negative values, commas, spaces, and line breaks for matrix and vector input.