scipy.linalg.lu_factor

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

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.

Parameters:

a(M, N) array_like

Matrix to decompose

overwrite_abool, optional

Whether to overwrite data in A (may increase performance)

check_finitebool, 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.

Returns:

lu(M, 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

lu

gives lu factorization in more user-friendly format

lu_solve

solve an equation system using the LU factorization of a matrix

Notes

This is a wrapper to the *GETRF routines from LAPACK. Unlike lu, it outputs the L and U factors into a single array and returns pivot indices instead of a permutation matrix.

Examples

>>> import numpy as np
>>> from scipy.linalg import lu_factor
>>> 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)))
True