ldl_factor_op

→std::vector<TensorImplPtr> of size 2

int ldl_factor_op(const int & a)

source edit

Python wrappers

pylucid.linalg.ldl_factorfunction

Compute the LDL $^\top$ factorisation of a symmetric matrix.

Returns the packed factor and the pivot index tensor. Both follow the LAPACK *sytrf storage convention so they are consumable by lucid.linalg.ldl_solve (which composes triangular solves). Batched inputs are supported: leading dimensions are treated as independent matrices and the factorisation is run on each $N \times N$ slice.

Math

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

where $P$ is the permutation implied by the pivots, $L$ is unit lower-triangular, and $D$ is block-diagonal with 1 $\times$ 1 and 2 $\times$ 2 blocks. The 2 $\times$ 2 blocks are necessary when no scalar pivot can be chosen without losing numerical stability — this is what allows the factorisation to handle indefinite $A$ .

Shape

a : (..., N, N).
return [0] : (..., N, N).
return [1] : (..., N).

See Also

cholesky_op — cheaper alternative when $A$ is SPD.
solve_triangular_op — building block for ldl_solve.

Parameters

aTensorImplPtr

Square symmetric matrix of shape (..., N, N) with dtype F32 or F64. The matrix need not be positive definite; only the lower triangle is read.

Returns

std::vector<TensorImplPtr> of size 2

[0] packed factor LD of shape (..., N, N), same dtype as a. Strict lower triangle stores $L$ (unit diagonal implicit); diagonal and sub-diagonal entries jointly encode $D$ including 2 $\times$ 2 pivot blocks. - [1] pivot tensor of shape (..., N), dtype I32, following the LAPACK ipiv convention (positive: 1 $\times$ 1 pivot; pair of equal negatives: 2 $\times$ 2 pivot block).

Raises

LucidError

If a is not at least 2-D, not square, or has a non-float dtype.

LucidError

If LAPACK info > 0 —

D

is exactly singular, so

A

is rank deficient and the factorisation cannot be completed.

Notes

No autograd node is registered; the outputs are leaves.
The current ldl_solve wrapper in lucid.linalg only supports 1 $\times$ 1 pivots (no 2 $\times$ 2 Bunch-Kaufman block solving yet).

ldl_factor_op

→std::vector<TensorImplPtr> of size 2

int ldl_factor_op(const int & a)

source edit

Python wrappers

pylucid.linalg.ldl_factorfunction

Compute the LDL $^\top$ factorisation of a symmetric matrix.

Math

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

Shape

a : (..., N, N).
return [0] : (..., N, N).
return [1] : (..., N).

See Also

cholesky_op — cheaper alternative when $A$ is SPD.
solve_triangular_op — building block for ldl_solve.

Parameters

aTensorImplPtr

Square symmetric matrix of shape (..., N, N) with dtype F32 or F64. The matrix need not be positive definite; only the lower triangle is read.

Returns

std::vector<TensorImplPtr> of size 2

[0] packed factor LD of shape (..., N, N), same dtype as a. Strict lower triangle stores $L$ (unit diagonal implicit); diagonal and sub-diagonal entries jointly encode $D$ including 2 $\times$ 2 pivot blocks. - [1] pivot tensor of shape (..., N), dtype I32, following the LAPACK ipiv convention (positive: 1 $\times$ 1 pivot; pair of equal negatives: 2 $\times$ 2 pivot block).

Raises

LucidError

If a is not at least 2-D, not square, or has a non-float dtype.

LucidError

If LAPACK info > 0 —

D

is exactly singular, so

A

is rank deficient and the factorisation cannot be completed.

Notes

No autograd node is registered; the outputs are leaves.
The current ldl_solve wrapper in lucid.linalg only supports 1 $\times$ 1 pivots (no 2 $\times$ 2 Bunch-Kaufman block solving yet).