lu_factor

→Tensor

lu_factor(A: Tensor)

source

LU factorization with partial pivoting (packed form).

Computes the packed LU factorization

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

where $P$ is a row-permutation matrix, $L$ is unit-lower-triangular, and $U$ is upper-triangular. The result is returned in LAPACK's packed format suitable for repeated solves via lu_solve.

Parameters

ATensor

Square matrix of shape (*, n, n).

Returns

Tensor

Packed factorization of shape (*, n, n). $U$ occupies the diagonal and above; $L$ (without its unit diagonal) occupies the strict lower triangle.

Notes

Backed by LAPACK getrf. Cost is $O(\tfrac{2}{3} n^3)$ .

Use lu_factor + lu_solve instead of solve when the same coefficient matrix $A$ is reused with many right-hand sides — factorization is shared. For the explicit $(P, L, U)$ triple see lu.

Examples

>>> import lucid
>>> from lucid.linalg import lu_factor, lu_solve
>>> A = lucid.tensor([[2.0, 1.0], [4.0, 7.0]])
>>> LU, piv = lu_factor(A)
>>> b = lucid.tensor([[3.0], [13.0]])
>>> lu_solve(LU, piv, b)
Tensor([[0.8000],
        [1.4000]])