scipy.linalg.lu_factor(a, overwrite_a=False, check_finite=True)[source]

Compute pivoted LU decomposition of a matrix.

The decomposition is:

A = P L U

where P is a permutation matrix, L lower triangular with unit diagonal elements, and U upper triangular.


a : (M, M) array_like

Matrix to decompose

overwrite_a : bool, optional

Whether to overwrite data in A (may increase performance)

check_finite : bool, optional

Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.


lu : (N, N) ndarray

Matrix containing U in its upper triangle, and L in its lower triangle. The unit diagonal elements of L are not stored.

piv : (N,) ndarray

Pivot indices representing the permutation matrix P: row i of matrix was interchanged with row piv[i].

See also

solve an equation system using the LU factorization of a matrix


This is a wrapper to the *GETRF routines from LAPACK.


>>> from scipy.linalg import lu_factor
>>> from numpy import tril, triu, allclose, zeros, eye
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> lu, piv = lu_factor(A)
>>> piv
array([2, 2, 3, 3], dtype=int32)

Convert LAPACK’s piv array to NumPy index and test the permutation

>>> piv_py = [2, 0, 3, 1]
>>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu)
>>> np.allclose(A[piv_py] - L @ U, np.zeros((4, 4)))

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