v0.3.0 — LU / QR / Cholesky factorizations + least squares

v0.3.0 — factorizations and least squares

Two new tools, both carrying the residual they actually measured.

matrix_decompose (kind: lu | qr | cholesky)

• lu — LU with partial pivoting: P·A = L·U (unit lower L, upper U). Returns the row permutation p (row i of P·A is row p[i] of A), the swap count, the determinant sign, the determinant implied by U, and max |P·A − L·U|. Singular matrices are still factored and flagged with singular: true instead of silently returning wrong factors.
• qr — thin QR with a positive diagonal in R, which makes the factorization unique: A = Q·R, Q has orthonormal columns, plus max |A − Q·R| and max |QᵀQ − I|. Linearly dependent columns are an error naming the column — no orthonormal completion is invented. Rows must be ≥ columns.
• cholesky — A = L·Lᵀ for symmetric positive definite input, with max |A − L·Lᵀ| and the determinant implied by L. An asymmetric matrix is rejected with the offending entry and the asymmetry size; an indefinite one with the failing pivot.

least_squares

For tall systems (m ≥ n), returns the x minimizing ‖A·x − b‖, the minimum residual norm itself, and max |Aᵀ(A·x − b)| — zero confirms the answer really is the minimizer. Solved through the thin QR factorization rather than the normal equations (which square the condition number). Rank-deficient or underdetermined input is an error: the minimizer is not unique, so none is invented.

least_squares(a=[[1,1],[1,2],[1,3]], b=[1,2,2])
→ x1 = 0.6666666667, x2 = 0.5
  min ‖A·x − b‖ = 0.4082482905, max |Aᵀ(A·x − b)| = 0

Quality

• 128 → 156 tests; line coverage 94.48% → 96.08% (baseline ratcheted up).
• The oracle test/oracle/anchors.py gained a section D that re-derives every anchor independently: exact rational Gaussian elimination for LU / Cholesky / least squares (via the normal equations) and classical Gram–Schmidt for QR — a different algorithm from the implementation's modified Gram–Schmidt. Its --check asserts L·U = P·A, QᵀQ = I, L·Lᵀ = A, that the LU determinant equals the exact determinant, and that the least-squares residual is orthogonal to the columns.
• Writing those self-checks caught one wrong expectation of mine immediately (the sign convention of the residual vector), which is exactly what the oracle is for.

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中文摘要：新增 matrix_decompose（lu：P·A = L·U，奇异矩阵如实标记；qr：薄分解 A = Q·R，R 对角为正；cholesky：A = L·Lᵀ，仅对称正定）与 least_squares（超定最小二乘：解、最小残差范数、max |Aᵀ(A·x − b)|；秩亏或欠定直接报错）。每种分解都回报实测重构残差，不编造正交补、不挑一个解充数。测试 128 → 156，行覆盖 94.48% → 96.08%，随仓 oracle 增 D 区（精确有理数实现 + 经典 Gram–Schmidt，与实现算法不同）。