`scipy.linalg._decomp_lu:lu_factor`

source: build-install/usr/lib/python3.14/site-packages/scipy/linalg/_decomp_lu.py:25

Signature

def lu_factor(a, overwrite_a = False, check_finite = True)

Summary

Compute pivoted LU decomposition of a matrix.

Extended Summary

The decomposition is

A = P L U

where P is a permutation matrix, L lower triangular with unit diagonal elements, and U upper triangular.

The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see linalg_batch for details.

Parameters

a : (M, N) array_like: Matrix to decompose
overwrite_a : bool, optional: Whether to overwrite data in a (may improve performance). Default is False. See tutorial_linalg_overwrite for details.
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.

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).

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
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
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

`scipy.linalg._decomp_lu:lu_factor`

Signature

Summary

Extended Summary

Parameters

Returns

Notes

Examples

See also

Aliases

Referenced by

This package