`scipy.linalg._decomp_lu:lu_factor`

source: /scipy/linalg/_decomp_lu.py :20

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. Note that calls with zero-size batches are unsupported and will raise a ValueError.

Parameters

a : (M, N) array_like: Matrix to decompose
overwrite_a : bool, optional: Whether to overwrite data in A (may increase performance)
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

✓

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)))

✓

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)))

✓

or alternatively:

P = np.eye(4)[p]
np.allclose(A - P @ L @ U, np.zeros((4, 4)))

✓

Aliases

scipy.linalg.lu_factor

Referenced by

This package

release:1.10.0-notes