| name | qr-iteration |
| origin | workshop |
| description | Compute the singular values of a DENSE matrix (a 2D numpy array) via
unshifted QR iteration. Use when the matrix is small enough to hold and
factor densely and you want its singular values. Not for sparse matrices
or matrix-free operators — use the lanczos skill for those.
|
QR-iteration skill
Use this skill when:
- The user has a dense matrix (a 2D numpy array) and wants its singular values.
- The matrix is small enough to hold in memory and factor densely.
Do not use it for a sparse matrix or a matrix-free operator — QR iteration on A^T A densifies the problem. Route those to the lanczos skill.
How to invoke
python .claude/skills/qr-iteration/qr_svd.py \
--problem problem.py \
--k 6 \
--check
The --problem argument points to a Python file exposing get_dense() that returns a 2D numpy array. --k sets how many (largest) singular values to print; --check cross-validates against numpy.linalg.svd.
Output looks like:
dense matrix (200, 200): 200 singular values via QR iteration
largest 6: [27.9731 27.5842 26.8831 26.416 26.2265 25.9241]
max |QR - numpy.svd| = 4.43e-04
Interpreting the output
- Unshifted QR iteration converges the dominant singular values first; the smallest ones converge slowest, so
--check differences (if any) live in the tail.
- If the helper aborts with "this object is sparse," you pointed a dense-only tool at a sparse matrix — use
lanczos.
Extending
Add a Wilkinson shift to accelerate convergence of the small singular values, or a --tol/--iters flag to trade accuracy for speed. Keep the "largest first" output order — it is how the result is read.
