lu — linalg

lu

→Tensor

lu(A: Tensor, pivot: bool = True)

source

Full LU decomposition with explicit factors $(P, L, U)$ .

Decomposes a square matrix $A$ as

A \,=\, P\,L\,U,

where $P$ is a row permutation, $L$ is unit-lower triangular (ones on the diagonal), and $U$ is upper triangular. Unlike lu_factor (which returns a packed factor + integer pivots), this routine returns the three factors as explicit dense tensors — convenient for inspection or for re-using $P,L,U$ in downstream linear-algebra expressions.

Parameters

ATensor

Square matrix of shape (n, n). Batched inputs are not yet exposed through the Python wrapper (will raise).

pivot(bool, keyword - only)= True

Must be True (the default). False would request an unpivoted LU; Lucid does not currently ship that kernel and raises NotImplementedError.

Returns

Tensor

Permutation matrix of shape (n, n).

Notes

Implemented as a Python composite over lu_factor followed by explicit triangular masking and pivot-to-matrix conversion. Cost is $O(\tfrac{2}{3} n^3)$ , dominated by the underlying $LU$ factorization.

Examples

>>> import lucid
>>> from lucid.linalg import lu
>>> A = lucid.tensor([[2.0, 1.0], [4.0, 7.0]])
>>> P, L, U = lu(A)
>>> (P @ L @ U).numpy()
array([[2., 1.],
       [4., 7.]], dtype=float32)