scipy.linalg.

lu#

scipy.linalg.lu(a, permute_l=False, overwrite_a=False, check_finite=True, p_indices=False)[source]#

Compute LU decomposition of a matrix with partial pivoting.

The decomposition satisfies:

A = P @ L @ U

where P is a permutation matrix, L lower triangular with unit diagonal elements, and U upper triangular. If permute_l is set to True then L is returned already permuted and hence satisfying A = L @ U.

Parameters:
a(M, N) array_like

Array to decompose

permute_lbool, optional

Perform the multiplication P*L (Default: do not permute)

overwrite_abool, optional

Whether to overwrite data in a (may improve 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.

p_indicesbool, optional

If True the permutation information is returned as row indices. The default is False for backwards-compatibility reasons.

Returns:
(If `permute_l` is ``False``)
p(…, M, M) ndarray

Permutation arrays or vectors depending on p_indices

l(…, M, K) ndarray

Lower triangular or trapezoidal array with unit diagonal. K = min(M, N)

u(…, K, N) ndarray

Upper triangular or trapezoidal array

(If `permute_l` is ``True``)
pl(…, M, K) ndarray

Permuted L matrix. K = min(M, N)

u(…, K, N) ndarray

Upper triangular or trapezoidal array

Notes

Permutation matrices are costly since they are nothing but row reorder of L and hence indices are strongly recommended to be used instead if the permutation is required. The relation in the 2D case then becomes simply A = L[P, :] @ U. In higher dimensions, it is better to use permute_l to avoid complicated indexing tricks.

In 2D case, if one has the indices however, for some reason, the permutation matrix is still needed then it can be constructed by np.eye(M)[P, :].

Examples

>>> import numpy as np
>>> from scipy.linalg import lu
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> p, l, u = lu(A)
>>> np.allclose(A, p @ l @ u)
True
>>> p  # Permutation matrix
array([[0., 1., 0., 0.],  # Row index 1
       [0., 0., 0., 1.],  # Row index 3
       [1., 0., 0., 0.],  # Row index 0
       [0., 0., 1., 0.]]) # Row index 2
>>> p, _, _ = lu(A, p_indices=True)
>>> p
array([1, 3, 0, 2], dtype=int32)  # as given by row indices above
>>> np.allclose(A, l[p, :] @ u)
True

We can also use nd-arrays, for example, a demonstration with 4D array:

>>> rng = np.random.default_rng()
>>> A = rng.uniform(low=-4, high=4, size=[3, 2, 4, 8])
>>> p, l, u = lu(A)
>>> p.shape, l.shape, u.shape
((3, 2, 4, 4), (3, 2, 4, 4), (3, 2, 4, 8))
>>> np.allclose(A, p @ l @ u)
True
>>> PL, U = lu(A, permute_l=True)
>>> np.allclose(A, PL @ U)
True