Module decompositions 

Named results of matrix decompositions (Qr, Lu, HermiteNormalForm, …) shared by Matrix, ZMatrix and QMatrix. Named results of matrix decompositions and normal forms.

Every factorisation here used to come back as a tuple of two or three matrices — (Q, R), (P, D), (H, U), (S, U, V) — where nothing but memory said which position was which, and a transposition compiled silently. These structs name each factor and document the identity it satisfies, at zero runtime cost. See CONTRIBUTING.md, “Tuples versus structs”.

The structs are generic over the matrix type so that the symbolic Matrix and the exact ZMatrix / QMatrix share one vocabulary: Qr<Matrix>, HermiteNormalForm<ZMatrix>, ….

Examples

use symplex::prelude::*;

let ctx = Context::new();
let a = matrix![ctx, [4, 2], [2, 3]];
let ldl = a.ldl().unwrap();
assert_eq!((&(&ldl.l * &ldl.d) * &ldl.l.transpose()).eval(), a);

let Qr { q, r } = matrix![ctx, [1, 1], [0, 1]].qr().unwrap();
assert_eq!((&q * &r).simplify(), matrix![ctx, [1, 1], [0, 1]]);

Structs

Diagonalization

Eigendecomposition A = P·D·P⁻¹: the columns of P are eigenvectors, D is diagonal with the eigenvalues in the same order.

HermiteNormalForm

Row-style Hermite normal form H = U·A with U unimodular (det U = ±1).

Hessenberg

Upper Hessenberg form by similarity: H = P⁻¹·A·P (equivalently A·P = P·H) with h_ij = 0 for i > j + 1.

JordanForm

Jordan normal form A = P·J·P⁻¹: J is block diagonal with Jordan blocks J_k(λ) (eigenvalue on the diagonal, ones on the superdiagonal), P holds the (generalized) eigenvectors.

Ldl

LDLᵀ decomposition A = L·D·Lᵀ of a symmetric matrix: L unit lower triangular, D diagonal.

LllReduction

LLL reduction of the lattice basis formed by the rows of A: reduced = transform·A with transform unimodular (det = ±1), so both span the same lattice.

Lu

LU decomposition with partial pivoting P·A = L·U: L unit lower triangular, U upper triangular, and perm the row permutation — row i of P·A is row perm[i] of A.

Qr

QR decomposition A = Q·R: Q has orthonormal columns, R is upper triangular.

RankDecomposition

Full-rank factorisation A = C·F with r = rank A: C (m × r) is made of the pivot columns of A, F (r × n) of the nonzero rows of rref(A).

SmithNormalForm

Smith normal form S = U·A·V with U, V unimodular (det U = det V = ±1) and S = diag(d₁, …, dᵣ, 0, …), dᵢ | dᵢ₊₁.