scipy.linalg.lu_factor#

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.

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(K,) ndarray: Pivot indices representing the permutation matrix P: row i of matrix was interchanged with row piv[i]. Of shape (K,), with K = min(M, N).

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.

While the underlying *GETRF routines return 1-based pivot indices, the piv array returned by lu_factor contains 0-based indices.

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

>>> def pivot_to_permutation(piv):
...     perm = np.arange(len(piv))
...     for i in range(len(piv)):
...         perm[i], perm[piv[i]] = perm[piv[i]], perm[i]
...     return perm
...
>>> p_inv = pivot_to_permutation(piv)
>>> p_inv
array([2, 0, 3, 1])
>>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu)
>>> np.allclose(A[p_inv] - L @ U, np.zeros((4, 4)))
True

The P matrix in P L U is defined by the inverse permutation and can be recovered using argsort:

>>> p = np.argsort(p_inv)
>>> p
array([1, 3, 0, 2])
>>> np.allclose(A - L[p] @ U, np.zeros((4, 4)))
True

or alternatively:

>>> P = np.eye(4)[p]
>>> np.allclose(A - P @ L @ U, np.zeros((4, 4)))
True