linalg · PLUQFactorization

9 accepted declarations. Python and Rust forms, contracts, evidence, failure boundaries, and checked usages are kept together below.

Worked example

The complete checked program Exact affine systems and canonical submodules over Z and Z/nZ uses this API directly.

# SPDX-License-Identifier: MIT
# Copyright (c) 2026 Benjamin Antieau and Deven Manam
"""Exact affine systems and canonical submodules over Z and Z/nZ."""

from __future__ import annotations

from noether import IntegerRing, Z
from noether.commalg import ResidueRing
from noether.homalg import FreeModule, ModuleElement, ModuleHomomorphism, Submodule
from noether.linalg import Matrix, Vector

operator = Matrix.from_rows(Z, [[2, 4, 0], [0, 0, 3]])
right_hand_side = Vector.from_values(Z, [6, 9])
integer_result = operator.solve_exact(right_hand_side)
assert integer_result.is_consistent
integer_solutions = integer_result.solutions
assert integer_solutions is not None
assert operator @ integer_solutions.particular_solution == right_hand_side
assert operator @ integer_solutions.homogeneous_generators == Matrix.zeros(Z, (2, 1))
integer_result.verify()

# Empty systems retain a replayable divisibility obstruction.
impossible = Matrix.from_rows(Z, [[6]]).solve_exact(Vector.from_values(Z, [9]))
assert not impossible.is_consistent
assert impossible.obstruction is not None
assert impossible.obstruction.left_gcd == 6
assert abs(impossible.obstruction.combined_rhs) == 9
impossible.obstruction.verify()

# Composite moduli are not treated as fields.  The result is an affine module,
# including all homogeneous torsion directions.
z12 = ResidueRing(12)
congruence = Matrix.from_rows(z12, [[8]]).solve_exact(Vector.from_values(z12, [4]))
assert congruence.is_consistent
congruence_solutions = congruence.solutions
assert congruence_solutions is not None
assert {
    (int(congruence_solutions.particular_solution[0]) + 3 * offset) % 12 for offset in range(4)
} == {
    2,
    5,
    8,
    11,
}
congruence.verify()

# Submodules retain a canonical normal form, so equality and lattice operations
# compare mathematics rather than the supplied generating lists.
ambient = FreeModule(Z, 2)


def element(x: int, y: int) -> ModuleElement[IntegerRing]:
    return ModuleElement(ambient, Vector.from_values(Z, [x, y]))


lattice = Submodule.from_generators(ambient, [element(2, 0), element(0, 3)])
assert lattice.finite_index() == 6
witness = lattice.membership_witness(element(6, 9))
assert witness is not None
witness.verify()
assert lattice.saturation().finite_index() == 1

homomorphism = ModuleHomomorphism(
    FreeModule(Z, 3),
    ambient,
    operator,
)
kernel_result = homomorphism.kernel()
image_result = homomorphism.image()
kernel = kernel_result.submodule
image = image_result.submodule
assert kernel.generator_count == 1
assert image.contains(element(2, 0))
kernel_result.verify()
image_result.verify()

print(
    "exact systems: integer affine rank 1; four Z/12Z solutions; "
    "index-6 lattice saturated and verified"
)

Run it with .venv/bin/python examples/38_exact_linear_systems.py.

linalg.PLUQFactorization

A checked, rank-revealing P @ L @ U @ Q == source factorization.

PythonRust

class PLUQFactorization

noether::linalg::PLUQFactorization · Direct Rust declaration

pub struct PLUQFactorization<R: Ring> { /* private fields */ }

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: validation/unit/linalg/test_exact_linear_algebra_facade.py:26 · validation/unit/linalg/test_exact_linear_algebra_facade.py:41 · validation/unit/linalg/test_exact_linear_algebra_facade.py:42 · validation/unit/linalg/test_exact_linear_algebra_facade.py:52 · validation/unit/linalg/test_exact_linear_algebra_facade.py:54

linalg.PLUQFactorization.l

Unit lower-trapezoidal factor.

Detailed contract

The unit lower-trapezoidal factor.

PythonRust

(self) -> Matrix[R]

noether::linalg::PLUQFactorization::l · Direct Rust declaration

pub fn l(&self) -> &Matrix<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: validation/unit/linalg/test_exact_linear_algebra_facade.py:47 · validation/unit/linalg/test_exact_linear_algebra_facade.py:50

linalg.PLUQFactorization.p

Left permutation matrix.

Detailed contract

The left permutation matrix.

PythonRust

(self) -> Matrix[R]

noether::linalg::PLUQFactorization::p · Direct Rust declaration

pub fn p(&self) -> &Matrix<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: validation/unit/combinatorics/test_boundary.py:21 · validation/unit/combinatorics/test_boundary.py:22 · validation/unit/combinatorics/test_boundary.py:23 · validation/unit/combinatorics/test_partition_types.py:63 · validation/unit/combinatorics/test_partition_types.py:64 · validation/unit/infra/test_docs_site_generator.py:103

linalg.PLUQFactorization.q

Right permutation matrix.

Detailed contract

The right permutation matrix.

PythonRust

(self) -> Matrix[R]

noether::linalg::PLUQFactorization::q · Direct Rust declaration

pub fn q(&self) -> &Matrix<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: validation/unit/infra/test_macaulay2_benchmark_protocol.py:1005 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:1041 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:1068 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:1090 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:181 · validation/unit/infra/test_macaulay2_benchmark_protocol.py:226

linalg.PLUQFactorization.rank

Exact matrix rank.

Detailed contract

The exact rank of the source matrix.

PythonRust

(self) -> int

noether::linalg::PLUQFactorization::rank · Direct Rust declaration

pub fn rank(&self) -> usize

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: examples/38_exact_linear_systems.py:78 · examples/distributed/f2_linear_algebra.py:34 · examples/distributed/f2_linear_algebra.py:54 · examples/distributed/f2_linear_algebra.py:55 · examples/distributed/f2_linear_algebra.py:60 · validation/unit/combin/test_acceptance_workload.py:74

linalg.PLUQFactorization.ring

Coefficient field.

Detailed contract

The coefficient field.

PythonRust

(self) -> R

noether::linalg::PLUQFactorization::ring · Direct Rust declaration

pub fn ring(&self) -> &R

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: examples/11_qadic_witt_vectors.py:3 · examples/13_chinese_remainder_theorem.py:10 · examples/13_chinese_remainder_theorem.py:12 · examples/13_chinese_remainder_theorem.py:9 · examples/14_weighted_polynomials.py:10 · examples/14_weighted_polynomials.py:11

linalg.PLUQFactorization.source

Matrix being factored.

Detailed contract

The matrix being factored.

PythonRust

(self) -> Matrix[R]

noether::linalg::PLUQFactorization::source · Direct Rust declaration

pub fn source(&self) -> &Matrix<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: examples/09_finite_field_isomorphisms.py:11 · examples/09_finite_field_isomorphisms.py:13 · examples/09_finite_field_isomorphisms.py:9 · examples/30_modules_and_resolutions.py:34 · examples/30_modules_and_resolutions.py:37 · validation/unit/category/test_acceptance_workload.py:25

linalg.PLUQFactorization.u

Upper-trapezoidal factor.

Detailed contract

The upper-trapezoidal factor.

PythonRust

(self) -> Matrix[R]

noether::linalg::PLUQFactorization::u · Direct Rust declaration

pub fn u(&self) -> &Matrix<R>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: validation/unit/commalg/test_affine_normalization_facade.py:151 · validation/unit/commalg/test_affine_normalization_facade.py:152 · validation/unit/commalg/test_affine_normalization_facade.py:153 · validation/unit/commalg/test_affine_normalization_facade.py:156 · validation/unit/commalg/test_base_change.py:116 · validation/unit/commalg/test_base_change.py:117

linalg.PLUQFactorization.verify

Recheck factor structure, rank, and the reconstruction identity.

Detailed contract

Recheck shapes, factor structure, rank, and P * L * U * Q = source.

PythonRust

(self) -> None

noether::linalg::PLUQFactorization::verify · Direct Rust declaration

pub fn verify(&self) -> Result<(), VerificationError>

Contract	Value
Guarantee	checked deterministic result
Determinism	deterministic
Availability	default CPU build
Domain implementation crates	noether, flint-core, m4rst, m4rst-distributed, m4rste, mzs, pwvla, spar-core, sparla, sparb, lattice-enum
GIL	attached / bounded
Semantic ID	noether/linalg/PLUQFactorization@1

Executable usage: examples/01_bezout_certificates.py:10 · examples/01_bezout_certificates.py:3 · examples/03_exact_linear_systems.py:13 · examples/03_exact_linear_systems.py:3 · examples/04_smith_normal_form.py:12 · examples/06_shortest_lattice_vectors.py:12