菜单
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.
用 Codex 或 Claude 帮你安装 复制这段 Prompt，粘贴到 Codex、Claude 或其他助手里，让它检查 Skill 页面并帮你完成安装。
基于 SOC 职业分类
Produce a structured summary of an academic paper from its PDF, in the workshop's literature/<bibkey>.md format. Use when the user has a new paper to add to the project's literature/ directory, asks "what does this paper say?", or asks for a structured intake of a PDF for the lab. Always pair with a verified BibTeX entry; never emit citations from memory.
Render slides or pages of Office documents (.pptx, .ppt, .odp, .docx, .doc, .odt, .xlsx) to PNG images for visual inspection, using LibreOffice headless plus poppler. Use this WHENEVER you need to actually SEE what a PowerPoint, slide deck, or Word/Excel document looks like — verifying a generated or edited .pptx, previewing specific slides, checking layout/overflow/branding, comparing before/after, or turning a deck into thumbnails. Reach for this instead of `qlmanage` (which only renders the first slide) any time the task involves eyeballing more than page one of an Office file.
Compute the dominant k singular values of a LARGE SPARSE matrix or a MATRIX-FREE linear operator (available only through matvec) using Lanczos (scipy.sparse.linalg.svds). Use when the object is sparse or is only available as an apply()/matvec — never densify it. For a small dense matrix, use the qr-iteration skill instead.
Verify that a candidate primal-dual point (x, y, z) satisfies the KKT conditions for a QP or NLP of the form min f(x) s.t. A x = b, x >= 0 Reports max residuals for stationarity, primal feasibility, dual feasibility, and complementarity. Use whenever a "candidate solution" is reported and you want to confirm it is a KKT point, or to inspect per-block residuals.
Verify that a proposed timestep dt satisfies the element-wise CFL condition dt < 2/sqrt(lambda_max) for the second-order wave equation u_tt = div(c^2 grad u) on a Firedrake mesh, where lambda_max is the largest eigenvalue of the generalized eigenproblem K_e v = lambda M_e v on each cell (M_e: local consistent mass, K_e: local c^2-weighted stiffness). Reports per-element violations with offending cell indices, their lambda_max, and the local time-step bound.
| 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. |
Use this skill when:
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.
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
--check differences (if any) live in the tail.lanczos.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.