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
Pis a permutation matrix,Llower triangular with unit diagonal elements, andUupper triangular. If permute_l is set toTruethenLis returned already permuted and hence satisfyingA = 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
Truethe permutation information is returned as row indices. The default isFalsefor 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
Land hence indices are strongly recommended to be used instead if the permutation is required. The relation in the 2D case then becomes simplyA = 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