dpnp.scipy.linalg.lu

dpnp.scipy.linalg.lu(a, permute_l=False, overwrite_a=False, check_finite=True, p_indices=False)

Compute LU decomposition of a matrix with partial pivoting.

The decomposition satisfies:

A = P @ L @ U

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

For full documentation refer to scipy.linalg.lu.

Parameters:

• a ((..., M, N) {dpnp.ndarray, usm_ndarray}) -- Input array to decompose.

• permute_l (bool, optional) --

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

  Default: False.

• overwrite_a (bool, optional) --

  Whether to overwrite data in a (may increase performance).

  Default: False.

• 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.

  Default: True.

• p_indices (bool, optional) --

  If True the permutation information is returned as row indices instead of a permutation matrix.

  Default: False.

Returns:

The tuple (p, l, u) is returned if permute_l is False (default), else the tuple (pl, u) is returned, where

• p ((..., M, M) dpnp.ndarray or (..., M) dpnp.ndarray) -- Permutation matrix or permutation indices. If p_indices is False (default), a permutation matrix. The permutation matrix always has a real-valued floating-point dtype even when a is complex, since it only contains 0s and 1s. If p_indices is True, a 1-D (or batched) array of row permutation indices such that A = L[p] @ U.

• l ((..., M, K) dpnp.ndarray) -- Lower triangular or trapezoidal matrix with unit diagonal. K = min(M, N).

• pl ((..., M, K) dpnp.ndarray) -- Permuted L matrix: pl = P @ L. K = min(M, N).

• u ((..., K, N) dpnp.ndarray) -- Upper triangular or trapezoidal matrix.

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

Warning

This function synchronizes in order to validate array elements when check_finite=True, and also synchronizes to compute the permutation from LAPACK pivot indices.

See also

dpnp.scipy.linalg.lu_factor()

LU factorize a matrix (compact representation).

dpnp.scipy.linalg.lu_solve()

Solve an equation system using the LU factorization of a matrix.

Examples

>>> import dpnp as np
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8],
...               [7, 5, 6, 6], [5, 4, 4, 8]])
>>> p, l, u = np.scipy.linalg.lu(A)
>>> np.allclose(A, p @ l @ u)
array(True)

Retrieve the permutation as row indices with p_indices=True:

>>> p, l, u = np.scipy.linalg.lu(A, p_indices=True)
>>> p
array([1, 3, 0, 2])
>>> np.allclose(A, l[p] @ u)
array(True)

Return the permuted L directly with permute_l=True:

>>> pl, u = np.scipy.linalg.lu(A, permute_l=True)
>>> np.allclose(A, pl @ u)
array(True)

Non-square matrices are supported:

>>> B = np.array([[1, 2, 3], [4, 5, 6]])
>>> p, l, u = np.scipy.linalg.lu(B)
>>> np.allclose(B, p @ l @ u)
array(True)

Batched input:

>>> C = np.random.randn(3, 2, 4, 4)
>>> p, l, u = np.scipy.linalg.lu(C)
>>> np.allclose(C, p @ l @ u)
array(True)