lu_factor

→Tensor

lu_factor(A: Tensor)

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Implementing kernel

C++lu_factor_opfree fn

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

Matrix of shape (*, m, n). Need not be square.

Returns

Tensor

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

Notes

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

For a rectangular $A \in \mathbb{R}^{m \times n}$ with $k = \min(m, n)$ , $L$ is $m \times k$ unit-lower-trapezoidal and $U$ is $k \times n$ upper-trapezoidal. lu_solve still requires a square factor: only a square system has a solution.

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([[0.8], [1.4]])

>>> 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([[0.8], [1.4]])