h ndZddlmZddlmZmZddlmZmZddl m Z ddl m Z gdZ dd Zdd Zdd ZdS)zLU decomposition functions.)warn)asarrayasarray_chkfinite) _datacopied LinAlgWarning)get_lapack_funcs)get_flinalg_funcs)lulu_solve lu_factorFTc*|rt|}nt|}|pt||}td|f\}|||\}}}|dkrt d| z|dkrt d|zt d||fS)aJ 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 )getrf) overwrite_arz?? ? axx Cd J q * * * * s7Nc|\}}|rt|}nt|}|pt||}|jd|jdkr-t d|j|jt d||f\}||||||\} } | dkr| St d| z)aSolve 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 rz)Shapes of lu {} and b {} are incompatible)getrs)trans overwrite_bz4illegal value in %dth argument of internal gesv|posv)rrrshaperformatr ) lu_and_pivbrrrr rb1rxrs rr r Ys^IR q ! ! QZZ3R!3!3K x{bhqk!!D &284466 6j2r( 3 3FEeBRu+FFFGAt qyy Ku  rcJ|rt|}nt|}t|jdkrt d|pt ||}t d|f\}||||\}}}} | dkrt 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 rzexpected matrix)r ) permute_lrrz3illegal value in %dth argument of internal lu.getrf)rrlenrrrr ) rr&rrrfluplurs rr r st q ! ! QZZ 28}}*+++5+b!"4"4K Wre , ,DCCi[IIIMAq!T axx-04u566 6!t a7NrN)FT)rFT)FFT)__doc__warningsrnumpyrr_miscrrlapackr _flinalg_pyr __all__r r r rrr4s!!,,,,,,,,.-------$$$$$$****** ) ) )GGGGT>>>>BHHHHHHr