# scipy/scipy

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 """LU decomposition functions.""" from __future__ import division, print_function, absolute_import from warnings import warn from numpy import asarray, asarray_chkfinite # Local imports from .misc import _datacopied, LinAlgWarning from .lapack import get_lapack_funcs from .flinalg import get_flinalg_funcs __all__ = ['lu', 'lu_solve', 'lu_factor'] def lu_factor(a, overwrite_a=False, check_finite=True): """ 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, M) 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 : (N, 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_solve : solve an equation system using the LU factorization of a matrix Notes ----- This is a wrapper to the ``*GETRF`` routines from LAPACK. Examples -------- >>> from scipy.linalg import lu_factor >>> from numpy import tril, triu, allclose, zeros, eye >>> 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 """ if check_finite: a1 = asarray_chkfinite(a) else: a1 = asarray(a) if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]): raise ValueError('expected square matrix') overwrite_a = overwrite_a or (_datacopied(a1, a)) getrf, = get_lapack_funcs(('getrf',), (a1,)) lu, piv, info = getrf(a1, overwrite_a=overwrite_a) if info < 0: raise ValueError('illegal value in %d-th argument of ' 'internal getrf (lu_factor)' % -info) if info > 0: warn("Diagonal number %d is exactly zero. Singular matrix." % info, LinAlgWarning, stacklevel=2) return lu, piv def lu_solve(lu_and_piv, b, trans=0, overwrite_b=False, check_finite=True): """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 -------- >>> 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 """ (lu, piv) = lu_and_piv if check_finite: b1 = asarray_chkfinite(b) else: b1 = asarray(b) overwrite_b = overwrite_b or _datacopied(b1, b) if lu.shape[0] != b1.shape[0]: raise ValueError("incompatible dimensions.") getrs, = get_lapack_funcs(('getrs',), (lu, b1)) x, info = getrs(lu, piv, b1, trans=trans, overwrite_b=overwrite_b) if info == 0: return x raise ValueError('illegal value in %d-th argument of internal gesv|posv' % -info) def lu(a, permute_l=False, overwrite_a=False, check_finite=True): """ 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 -------- >>> 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 """ if check_finite: a1 = asarray_chkfinite(a) else: a1 = asarray(a) if len(a1.shape) != 2: raise ValueError('expected matrix') overwrite_a = overwrite_a or (_datacopied(a1, a)) flu, = get_flinalg_funcs(('lu',), (a1,)) p, l, u, info = flu(a1, permute_l=permute_l, overwrite_a=overwrite_a) if info < 0: raise ValueError('illegal value in %d-th argument of ' 'internal lu.getrf' % -info) if permute_l: return l, u return p, l, u