ldl_factor

→Tensor

ldl_factor(A: Tensor, hermitian: bool = True)

source

LDL factorization of a symmetric (or Hermitian) matrix.

For a real symmetric matrix $A$ (possibly indefinite), computes a Bunch-Kaufman block factorization

A \,=\, L\,D\,L^\top,

where $L$ is unit-lower-triangular and $D$ is block-diagonal with $1 \times 1$ or $2 \times 2$ blocks. Unlike cholesky, this factorization exists for indefinite symmetric matrices (e.g., saddle-point systems).

Parameters

ATensor

Symmetric / Hermitian matrix of shape (*, n, n).

hermitianbool= True

If True (default), treat A as Hermitian (conjugate symmetric in the complex case).

Returns

Tensor

Packed factor of shape (*, n, n). The strict lower triangle holds $L$ ; the diagonal holds $D$ 's entries ( $2 \times 2$ blocks are stored in the sub-diagonal).

Notes

Backed by LAPACK sytrf. Cost is $O(n^3 / 3)$ . Pair with ldl_solve for solving symmetric indefinite systems.

Examples

>>> import lucid
>>> from lucid.linalg import ldl_factor
>>> A = lucid.tensor([[1.0, 2.0], [2.0, 3.0]])
>>> LD, piv = ldl_factor(A)