224 lines
6.7 KiB
Python
224 lines
6.7 KiB
Python
"""LU decomposition functions."""
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from __future__ import division, print_function, absolute_import
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from warnings import warn
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from numpy import asarray, asarray_chkfinite
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# Local imports
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from .misc import _datacopied, LinAlgWarning
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from .lapack import get_lapack_funcs
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from .flinalg import get_flinalg_funcs
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__all__ = ['lu', 'lu_solve', 'lu_factor']
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def lu_factor(a, overwrite_a=False, check_finite=True):
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"""
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Compute pivoted LU decomposition of a matrix.
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The decomposition is::
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A = P L U
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where P is a permutation matrix, L lower triangular with unit
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diagonal elements, and U upper triangular.
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Parameters
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----------
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a : (M, M) array_like
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Matrix to decompose
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overwrite_a : bool, optional
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Whether to overwrite data in A (may increase performance)
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check_finite : bool, optional
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Whether to check that the input matrix contains only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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lu : (N, N) ndarray
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Matrix containing U in its upper triangle, and L in its lower triangle.
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The unit diagonal elements of L are not stored.
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piv : (N,) ndarray
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Pivot indices representing the permutation matrix P:
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row i of matrix was interchanged with row piv[i].
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See also
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--------
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lu_solve : solve an equation system using the LU factorization of a matrix
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Notes
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-----
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This is a wrapper to the ``*GETRF`` routines from LAPACK.
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Examples
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--------
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>>> from scipy.linalg import lu_factor
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>>> from numpy import tril, triu, allclose, zeros, eye
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>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
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>>> lu, piv = lu_factor(A)
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>>> piv
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array([2, 2, 3, 3], dtype=int32)
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Convert LAPACK's ``piv`` array to NumPy index and test the permutation
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>>> piv_py = [2, 0, 3, 1]
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>>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu)
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>>> np.allclose(A[piv_py] - L @ U, np.zeros((4, 4)))
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True
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"""
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if check_finite:
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a1 = asarray_chkfinite(a)
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else:
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a1 = asarray(a)
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if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
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raise ValueError('expected square matrix')
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overwrite_a = overwrite_a or (_datacopied(a1, a))
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getrf, = get_lapack_funcs(('getrf',), (a1,))
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lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
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if info < 0:
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raise ValueError('illegal value in %d-th argument of '
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'internal getrf (lu_factor)' % -info)
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if info > 0:
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warn("Diagonal number %d is exactly zero. Singular matrix." % info,
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LinAlgWarning, stacklevel=2)
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return lu, piv
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def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True):
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"""Solve an equation system, a x = b, given the LU factorization of a
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Parameters
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----------
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(lu, piv)
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Factorization of the coefficient matrix a, as given by lu_factor
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b : array
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Right-hand side
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trans : {0, 1, 2}, optional
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Type of system to solve:
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===== =========
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trans system
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===== =========
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0 a x = b
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1 a^T x = b
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2 a^H x = b
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===== =========
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overwrite_b : bool, optional
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Whether to overwrite data in b (may increase performance)
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check_finite : bool, optional
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Whether to check that the input matrices contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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x : array
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Solution to the system
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See also
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--------
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lu_factor : LU factorize a matrix
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Examples
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--------
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>>> from scipy.linalg import lu_factor, lu_solve
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>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
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>>> b = np.array([1, 1, 1, 1])
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>>> lu, piv = lu_factor(A)
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>>> x = lu_solve((lu, piv), b)
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>>> np.allclose(A @ x - b, np.zeros((4,)))
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True
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"""
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(lu, piv) = lu_and_piv
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if check_finite:
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b1 = asarray_chkfinite(b)
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else:
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b1 = asarray(b)
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overwrite_b = overwrite_b or _datacopied(b1, b)
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if lu.shape[0] != b1.shape[0]:
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raise ValueError("incompatible dimensions.")
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getrs, = get_lapack_funcs(('getrs',), (lu, b1))
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x, info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b)
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if info == 0:
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return x
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raise ValueError('illegal value in %d-th argument of internal gesv|posv'
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% -info)
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def lu(a, permute_l=False, overwrite_a=False, check_finite=True):
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"""
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Compute pivoted LU decomposition of a matrix.
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The decomposition is::
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A = P L U
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where P is a permutation matrix, L lower triangular with unit
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diagonal elements, and U upper triangular.
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Parameters
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----------
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a : (M, N) array_like
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Array to decompose
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permute_l : bool, optional
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Perform the multiplication P*L (Default: do not permute)
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overwrite_a : bool, optional
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Whether to overwrite data in a (may improve performance)
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check_finite : bool, optional
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Whether to check that the input matrix contains only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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**(If permute_l == False)**
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p : (M, M) ndarray
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Permutation matrix
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l : (M, K) ndarray
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Lower triangular or trapezoidal matrix with unit diagonal.
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K = min(M, N)
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u : (K, N) ndarray
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Upper triangular or trapezoidal matrix
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**(If permute_l == True)**
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pl : (M, K) ndarray
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Permuted L matrix.
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K = min(M, N)
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u : (K, N) ndarray
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Upper triangular or trapezoidal matrix
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Notes
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-----
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This is a LU factorization routine written for Scipy.
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Examples
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--------
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>>> from scipy.linalg import lu
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>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
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>>> p, l, u = lu(A)
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>>> np.allclose(A - p @ l @ u, np.zeros((4, 4)))
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True
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"""
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if check_finite:
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a1 = asarray_chkfinite(a)
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else:
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a1 = asarray(a)
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if len(a1.shape) != 2:
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raise ValueError('expected matrix')
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overwrite_a = overwrite_a or (_datacopied(a1, a))
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flu, = get_flinalg_funcs(('lu',), (a1,))
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p, l, u, info = flu(a1, permute_l=permute_l, overwrite_a=overwrite_a)
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if info < 0:
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raise ValueError('illegal value in %d-th argument of '
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'internal lu.getrf' % -info)
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if permute_l:
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return l, u
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return p, l, u
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