linalg op• Data kinds: matrix → table
• Call: import fullseye as fs; fs.ledger.mat_qr(a, method='householder') (to call the implementation directly, import mathops; mathops.mat_qr(a, method='householder'); from the registry, opsmath.get("mat_qr"))
QR factorisation `A = Q @ R (Q with orthonormal columns, R` upper triangular).
`method: "householder" (default, LAPACK geqrf), "mgs"` (modified Gram–Schmidt) or
`"cgs" (classical Gram–Schmidt). All three give the same Q` in exact arithmetic; in floating
point the loss of orthogonality `max|QᵀQ − I|` grows like ε·κ² for CGS, ε·κ for MGS and stays
near ε for Householder (Björck 1967; Giraud et al. 2005) — the reason libraries use Householder.
Returns a dict: `Q, R, orthogonality_loss, residual` = max|A − QR| / max|A|,
`cond = 2-norm condition number of A`.
Every mathops op validates its input before computing (nothing slips through silently):
• **complex input raises ValueError** — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• **masked arrays with masked elements raise ValueError** — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• **NaN/Inf raises ValueError on every input** (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• blas_threads_and_memory — 行列分解が遅い理由の知識 — BLAS スレッド・キャッシュ・メモリ配置
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• numerics_tour — py -3.11 examples/numerics_tour.py
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
linalg)mat_solve · mat_lstsq · mat_svd · mat_eigh · mat_pinv · mat_cond · mat_lu · mat_cholesky
*Provenance: mathops.py — MATH operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.