^Mh2dZddlmZddlmZmZddlZddlmZddl m Z m Z ddl m Z dd lmZd ejd DZgd ZddZddZ ddZdS)zLU decomposition functions.)warn)asarrayasarray_chkfiniteN)product) _datacopied LinAlgWarning)get_lapack_funcs) lu_dispatchercTi|]$dfddD%S)c>g|]}tj||S)npcan_cast).0yxs W/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/linalg/_decomp_lu.py z.s*GGGaR[A5F5FGGGGfdFD)joinrrs @r rsL222rwwGGGG6GGGHH222rAll)lulu_solve lu_factorFTc|rt|}nt|}|jdkr8tj|}tjdtj}||fS|pt||}td|f\}|||\}}}|dkrtd| z|dkrtd|ztd||fS) al 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 : (K,) ndarray Pivot indices representing the permutation matrix P: row i of matrix was interchanged with row piv[i]. Of shape ``(K,)``, with ``K = min(M, N)``. 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. While the underlying ``*GETRF`` routines return 1-based pivot indices, the ``piv`` array returned by ``lu_factor`` contains 0-based indices. 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 >>> def pivot_to_permutation(piv): ... perm = np.arange(len(piv)) ... for i in range(len(piv)): ... perm[i], perm[piv[i]] = perm[piv[i]], perm[i] ... return perm ... >>> p_inv = pivot_to_permutation(piv) >>> p_inv array([2, 0, 3, 1]) >>> L, U = np.tril(lu, k=-1) + np.eye(4), np.triu(lu) >>> np.allclose(A[p_inv] - L @ U, np.zeros((4, 4))) True The P matrix in P L U is defined by the inverse permutation and can be recovered using argsort: >>> p = np.argsort(p_inv) >>> p array([1, 3, 0, 2]) >>> np.allclose(A - L[p] @ U, np.zeros((4, 4))) True or alternatively: >>> P = np.eye(4)[p] >>> np.allclose(A - P @ L @ U, np.zeros((4, 4))) True rdtype)getrf) overwrite_az?? ? axx Cd J q * * * * s7NrcF|\}}|rt|}nt|}|pt||}|jd|jdkr t d|jd|jd|jdkrat tjd|j ddgftj d|j }tj ||j Std||f\} | ||||| \} } | dkr| St d | z) aUSolve 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. In particular piv are 0-indexed pivot indices. 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 z and b z are incompatibler%r!r)getrs)trans overwrite_bz4illegal value in %dth argument of internal gesv|posv) rrrshaper+r'rreyer"onesr(r ) lu_and_pivbr3r4r-rr/b1mr2rr0s rrrs?`IR q ! ! QZZ3R!3!3K x{bhqk!!UUU"(UUUVVV w!|| bfQbh///!Q8"'!17:S:S:S T T}Rqw//// j2r( 3 3FEeBRu+FFFGAt qyy Ku  rcX |rtj|ntj|}|jdkrt d|jjdvrNt|jj}|std|jd| |d}d}|j ^}}} t|| } |jjdvrd nd } t|j dkr|rFtj g||| |j } tj g|| | |j } | | fS|r$tj g|dtj ntj g|dd| }tj g||| |j }tj g|| | |j } ||| fS|j d ddkr|r,tj||r|n|fS|rtjg||t" ntj|}|tj||r|n|fSt%||s|s|d}|jdr |jds|d}|sdtj |tj }tj| | g|j }t)|||||| kr|||fn|||f\}}} ntj g||tj }|| krktjg|| | |j } t+d|j dd DD]&}t)||| ||||'|}njtjg|| | |j }t+d|j dd DD]&}t)|||||||'|} |s|s|rWtjg|||| }tjd|Dtj|gz}d|g||R<|}n3tj||g| }d|tj||f<|}|r|| fn||| fS)ar Compute LU decomposition of a matrix with partial pivoting. The decomposition satisfies:: A = P @ L @ U where ``P`` is a permutation matrix, ``L`` lower triangular with unit diagonal elements, and ``U`` upper triangular. If `permute_l` is set to ``True`` then ``L`` is returned already permuted and hence satisfying ``A = L @ U``. 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. p_indices : bool, optional If ``True`` the permutation information is returned as row indices. The default is ``False`` for backwards-compatibility reasons. Returns ------- **(If `permute_l` is ``False``)** p : (..., M, M) ndarray Permutation arrays or vectors depending on `p_indices` l : (..., M, K) ndarray Lower triangular or trapezoidal array with unit diagonal. ``K = min(M, N)`` u : (..., K, N) ndarray Upper triangular or trapezoidal array **(If `permute_l` is ``True``)** pl : (..., M, K) ndarray Permuted L matrix. ``K = min(M, N)`` u : (..., K, N) ndarray Upper triangular or trapezoidal array Notes ----- Permutation matrices are costly since they are nothing but row reorder of ``L`` and hence indices are strongly recommended to be used instead if the permutation is required. The relation in the 2D case then becomes simply ``A = L[P, :] @ U``. In higher dimensions, it is better to use `permute_l` to avoid complicated indexing tricks. In 2D case, if one has the indices however, for some reason, the permutation matrix is still needed then it can be constructed by ``np.eye(M)[P, :]``. 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) True >>> p # Permutation matrix array([[0., 1., 0., 0.], # Row index 1 [0., 0., 0., 1.], # Row index 3 [1., 0., 0., 0.], # Row index 0 [0., 0., 1., 0.]]) # Row index 2 >>> p, _, _ = lu(A, p_indices=True) >>> p array([1, 3, 0, 2], dtype=int32) # as given by row indices above >>> np.allclose(A, l[p, :] @ u) True We can also use nd-arrays, for example, a demonstration with 4D array: >>> rng = np.random.default_rng() >>> A = rng.uniform(low=-4, high=4, size=[3, 2, 4, 8]) >>> p, l, u = lu(A) >>> p.shape, l.shape, u.shape ((3, 2, 4, 4), (3, 2, 4, 4), (3, 2, 4, 8)) >>> np.allclose(A, p @ l @ u) True >>> PL, U = lu(A, permute_l=True) >>> np.allclose(A, PL @ U) True r%z1The input array must be at least two-dimensional.rz The dtype z5 cannot be cast to float(32, 64) or complex(64, 128).rTfFfd)r5r"r!N)rrC)order C_CONTIGUOUS WRITEABLEc,g|]}t|Srrangers rrzlu..m A A Aaq A A Arc,g|]}t|SrrFrs rrzlu..srHrc6g|]}tj|Sr)rr)rs rrzlu..}s 777qbill777rr)rrrndimr+r"charlapack_cast_dict TypeErrorastyper5minemptyr* ones_likecopyzerosintrflagsr rix_r))r, permute_lr$r- p_indicesr. dtype_charndr;nk real_dcharPLUPLpuindPand_ixs rrrsE@%1 C a bjmmB w{{LMMM x}F""%bhm4  EDDDDEE EYYz!} % % IRA Aq A --3J BH~   " a  28<< WW3W    HQbh ' ' ' HaV28 , , ,b!Q *** !A1b!**Aq":1aa HXrX1XRX . . . q552q!BH555A A A28CRC= A A AB B BbgqvqvyAAAAAA2q!BH555A A A28CRC= A A AB B BbgqvqvyAAAAA    +B++1+Z888BF77B7771FHEB{{{{OAA1a& 333B"#Bry||Q A -Aq66Q1I-r)FT)rFT)FFTF)__doc__warningsrnumpyrrr itertoolsr_miscrr lapackr _decomp_lu_cythonr typecodesrM__all__rrrrrrrqs#!!,,,,,,,,.-------$$$$$$,,,,,,22\%0222 * ) )jjjjZEEEEP<@|.|.|.|.|.|.r