lu_factor — SciPy v1.15.3 Manual (original) (raw)
scipy.linalg.
scipy.linalg.lu_factor(a, overwrite_a=False, check_finite=True)[source]#
Compute pivoted LU decomposition of a matrix.
The decomposition is:
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
gives lu factorization in more user-friendly format
solve an equation system using the LU factorization of a matrix
Notes
This is a wrapper to the *GETRF routines from LAPACK. Unlikelu, 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, thepiv 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