cholesky_op

→TensorImplPtr

int cholesky_op(const int & a, bool upper)

source edit

Python wrappers

pylucid.linalg.choleskyfunction

Compute the Cholesky factor of a symmetric positive-definite matrix.

Returns the lower-triangular $L$ (default) or upper-triangular $U$ such that $A = LL^\top$ or $A = U^\top U$ respectively. Forward dispatches to LAPACK *potrf (CPU) or mlx::core::linalg::cholesky (GPU). Batched inputs are supported: every dimension except the trailing two is treated as a batch axis and the factorisation is run independently on each $N \times N$ slice.

Math

A = LL^\top \quad\text{or}\quad A = U^\top U.

The corresponding reverse-mode rule (Murray 2016) is

\frac{\partial L}{\partial A} = \tfrac{1}{2}\, L^{-\top} \Phi\!\bigl(L^\top \tfrac{\partial L}{\partial L}\bigr) L^{-1},

where $\Phi(M) = \mathrm{tril}(M) + \mathrm{tril}(M, -1)^\top$ copies the strict lower triangle into the strict upper. This formula is realised in the Python wrapper via two triangular solves, not in this engine op.

Shape

a: (..., N, N).
return: (..., N, N).

See Also

solve_triangular_op — triangular back/forward substitution used in the Python-side Cholesky backward.
ldl_factor_op — analogous factorisation for symmetric indefinite matrices.

Parameters

aTensorImplPtr

Square symmetric positive-definite matrix of shape (..., N, N), dtype F32 or F64.

upperbool= None

If true return the upper factor

U

; otherwise (default) return the lower factor

L

Returns

TensorImplPtr

Triangular Cholesky factor with the same shape and dtype as a. The opposite triangle is filled with zeros.

Raises

LucidError

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

LucidError

If LAPACK info > 0 — a non-positive pivot was encountered, i.e. a is not positive definite (within working precision).

Notes

No autograd node is wired; the returned TensorImpl has requires_grad = false. Gradient-aware Cholesky is provided one layer up by lucid.linalg.cholesky.

cholesky_op

→TensorImplPtr

int cholesky_op(const int & a, bool upper)

source edit

Python wrappers

pylucid.linalg.choleskyfunction

Compute the Cholesky factor of a symmetric positive-definite matrix.

Math

A = LL^\top \quad\text{or}\quad A = U^\top U.

The corresponding reverse-mode rule (Murray 2016) is

\frac{\partial L}{\partial A} = \tfrac{1}{2}\, L^{-\top} \Phi\!\bigl(L^\top \tfrac{\partial L}{\partial L}\bigr) L^{-1},

Shape

a: (..., N, N).
return: (..., N, N).

See Also

solve_triangular_op — triangular back/forward substitution used in the Python-side Cholesky backward.
ldl_factor_op — analogous factorisation for symmetric indefinite matrices.

Parameters

aTensorImplPtr

Square symmetric positive-definite matrix of shape (..., N, N), dtype F32 or F64.

upperbool= None

If true return the upper factor

U

; otherwise (default) return the lower factor

L

Returns

TensorImplPtr

Triangular Cholesky factor with the same shape and dtype as a. The opposite triangle is filled with zeros.

Raises

LucidError

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

LucidError

If LAPACK info > 0 — a non-positive pivot was encountered, i.e. a is not positive definite (within working precision).

Notes

No autograd node is wired; the returned TensorImpl has requires_grad = false. Gradient-aware Cholesky is provided one layer up by lucid.linalg.cholesky.