ó î&]\c@`sÄdZddlmZmZmZddlmZddlmZm Z ddl m Z m Z ddl mZddlmZd d d gZeed „Zdeed „Zeeed„ZdS(sLU decomposition functions.i(tdivisiontprint_functiontabsolute_import(twarn(tasarraytasarray_chkfinitei(t _datacopiedt LinAlgWarning(tget_lapack_funcs(tget_flinalg_funcstlutlu_solvet lu_factorcC`sô|rt|ƒ}n t|ƒ}t|jƒdksP|jd|jdkr_tdƒ‚n|pqt||ƒ}td |fƒ\}||d|ƒ\}}}|dkrÄtd| ƒ‚n|dkrêtd|td dƒn||fS( s­ 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 iiisexpected square matrixtgetrft overwrite_as=illegal value in %d-th argument of internal getrf (lu_factor)s4Diagonal number %d is exactly zero. Singular matrix.t stacklevel(R ( RRtlentshapet ValueErrorRRRR(taRt check_finiteta1R R tpivtinfo((s5lib/python2.7/site-packages/scipy/linalg/decomp_lu.pyR s7 /   c C`sÌ|\}}|r!t|ƒ}n t|ƒ}|p?t||ƒ}|jd|jdkrktdƒ‚ntd||fƒ\}||||d|d|ƒ\} } | dkr·| Std| ƒ‚dS( sSolve 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 isincompatible dimensions.tgetrsttranst overwrite_bs5illegal value in %d-th argument of internal gesv|posvN(R(RRRRRR( t lu_and_pivtbRRRR Rtb1RtxR((s5lib/python2.7/site-packages/scipy/linalg/decomp_lu.pyR Zs.  $ c C`sÐ|rt|ƒ}n t|ƒ}t|jƒdkrEtdƒ‚n|pWt||ƒ}td|fƒ\}||d|d|ƒ\}}}} | dkr³td| ƒ‚n|rÃ||fS|||fS( sþ 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 isexpected matrixR t permute_lRis4illegal value in %d-th argument of internal lu.getrf(R (RRRRRRR ( RRRRRtflutptltuR((s5lib/python2.7/site-packages/scipy/linalg/decomp_lu.pyR ™s9 $  N(t__doc__t __future__RRRtwarningsRtnumpyRRtmiscRRtlapackRtflinalgR t__all__tFalsetTrueR R R (((s5lib/python2.7/site-packages/scipy/linalg/decomp_lu.pytsI?