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Compute pivoted LU decomposition of a matrix.
The decomposition is::
A = P L U
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_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 : (N,) ndarray
Pivot indices representing the permutation matrix P:
row i of matrix was interchanged with row piv[i].
See Also
--------
lu : gives lu factorization in more user-friendly format
lu_solve : solve an equation system using the LU factorization of a matrix
Notes
-----
This is a wrapper to the ``*GETRF`` routines from LAPACK. Unlike
:func:`lu`, it outputs the L and U factors into a single array
and returns pivot indices instead of a permutation matrix.
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
>>> piv_py = [2, 0, 3, 1]
>>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu)
>>> np.allclose(A[piv_py] - L @ U, np.zeros((4, 4)))
True
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z � � �)a Solve an equation system, a x = b, given the LU factorization of a
Parameters
----------
(lu, piv)
Factorization of the coefficient matrix a, as given by lu_factor
b : array
Right-hand side
trans : {0, 1, 2}, optional
Type of system to solve:
===== =========
trans system
===== =========
0 a x = b
1 a^T x = b
2 a^H x = b
===== =========
overwrite_b : bool, optional
Whether to overwrite data in b (may increase performance)
check_finite : bool, optional
Whether to check that the input matrices contain 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
-------
x : array
Solution to the system
See Also
--------
lu_factor : LU factorize a matrix
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import lu_factor, lu_solve
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
>>> b = np.array([1, 1, 1, 1])
>>> lu, piv = lu_factor(A)
>>> x = lu_solve((lu, piv), b)
>>> np.allclose(A @ x - b, np.zeros((4,)))
True
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d|f� � \ } ||||�� � \ }}}} | dk rt d| z � � �|r||fS |||fS )a
Compute pivoted LU decomposition of a matrix.
The decomposition is::
A = P L U
where P is a permutation matrix, L lower triangular with unit
diagonal elements, and U upper triangular.
Parameters
----------
a : (M, N) array_like
Array to decompose
permute_l : bool, optional
Perform the multiplication P*L (Default: do not permute)
overwrite_a : bool, optional
Whether to overwrite data in a (may improve 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
-------
**(If permute_l == False)**
p : (M, M) ndarray
Permutation matrix
l : (M, K) ndarray
Lower triangular or trapezoidal matrix with unit diagonal.
K = min(M, N)
u : (K, N) ndarray
Upper triangular or trapezoidal matrix
**(If permute_l == True)**
pl : (M, K) ndarray
Permuted L matrix.
K = min(M, N)
u : (K, N) ndarray
Upper triangular or trapezoidal matrix
Notes
-----
This is a LU factorization routine written for SciPy.
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, np.zeros((4, 4)))
True
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