ldl_factor

→Tensor

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

source edit

Implementing kernel

C++ldl_factor_opfree fn

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).

hermitian(bool, keyword - only)= False

Whether a complex A is Hermitian rather than symmetric. A real A is both, so either value factors it the same. Default False, as the reference's.

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). The strict upper triangle is zero.

Raises

LinAlgError

If a

1 \times 1

block of

D

is exactly zero — the factorization completes, but

D

is singular. ldl_factor_ex returns it with a non-zero info instead.

Notes

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

sytrf writes the factor into the lower triangle and leaves the upper one as it found it — the input's own entries. Those are cleared here, so LD holds the factorization and nothing else.

LD is differentiable in A, to any order, wherever the pivots do not change — an addition over the reference framework, which defines no derivative for this factorization and raises on backward. The gradient lands on A's lower triangle, the part sytrf reads, as cholesky's does.

Examples

>>> import lucid
>>> from lucid.linalg import ldl_factor
>>> A = lucid.tensor([[4.0, 1.0], [1.0, 3.0]])
>>> LD, piv = ldl_factor(A)
>>> LD
tensor([[4., 0.], [0.25, 2.75]])
>>> piv
tensor([1, 2], dtype=lucid.int32)

Used by 1

lucid.linalg

>>> import lucid >>> from lucid.linalg import ldl_factor >>> A = lucid.tensor([[4.0, 1.0], [1.0, 3.0]]) >>> LD, piv = ldl_factor(A) >>> LD tensor([[4., 0.], [0.25, 2.75]]) >>> piv tensor([1, 2], dtype=lucid.int32)