(ph2VSrSSKJr SSKJrJr SSKrSSKJr SSK J r J r SSK J r SS KJr \R S VVs0sH>nUS R#S Vs/sHn\R%X5(dMUPM sn5_M@ snnr/S QrSSjrSSjrSSjrgs snfs snnf)zLU decomposition functions.)warn)asarrayasarray_chkfiniteN)product) _datacopied LinAlgWarning)get_lapack_funcs) lu_dispatcherAllfdFD)lulu_solve lu_factorcU(a [U5nO [U5nURS:Xa<[R"U5n[R "S[R S9nXE4$U=(d [X05n[SU45unU"X1S9upEnUS:a[SU*-5eUS:a[SU-[SS9 XE4$) 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 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? ? ax Cd J q * 7NcJUupVU(a [U5nO [U5nU=(d [Xq5nURSURS:wa&[ SURSURS35eUR S:Xah[ [R"SURS9SS/4[R"SURS95n[R"XxRS9$[SXW45un U "XVXrUS 9upU S:XaU $[ S U *-5e) aSolve an equation system, a x = b, given the LU factorization of a Parameters ---------- (lu, piv) Factorization of the coefficient matrix a, as given by lu_factor. In particular piv are 0-indexed pivot indices. b : array Right-hand side trans : {0, 1, 2}, optional Type of system to solve: ===== ========= trans system ===== ========= 0 a x = b 1 a^T x = b 2 a^H x = b ===== ========= overwrite_b : bool, optional Whether to overwrite data in b (may increase performance) check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- x : array Solution to the system See Also -------- lu_factor : LU factorize a matrix Examples -------- >>> import numpy as np >>> from scipy.linalg import lu_factor, lu_solve >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) >>> b = np.array([1, 1, 1, 1]) >>> lu, piv = lu_factor(A) >>> x = lu_solve((lu, piv), b) >>> np.allclose(A @ x - b, np.zeros((4,))) True rz Shapes of lu z and b z are incompatiblerrr)getrs)trans overwrite_bz4illegal value in %dth argument of internal gesv|posv) rrrshaperrrreyeronesrr ) lu_and_pivbr(r)r rr"b1mr'xr#s r$rrs`IR q ! QZ3R!3K xx{bhhqk!= '"((CTUVV ww!| bffQbhh/!Q8"''!177:S T}}Rww// j2( 3FEBR+FGA qy Ku r%c U(a[R"U5O[R"U5nURS:a [ S5eUR R S;aS[UR R nU(d[SUR S35eURUS5nSnURGtpxn [X5n UR R S;aS OS n [UR6S:XaU(aM[R"/UQUPU PUR S 9n [R"/UQU PU PUR S 9n X4$U(a'[R"/UQSP[RS 9O[R"/UQSPSPU S 9n[R"/UQUPU PUR S 9n[R"/UQU PU PUR S 9n XU 4$URS SS:XaU(a0[R"U5U(aU4$UR54$U(a[R "/UQUP["S 9O[R"U5nU[R"U5U(aU4$UR54$[%XP5(dU(dURSS9nUR&S(aUR&S(dURSS9nU(de[R"U[RS 9n[R "X/UR S 9n[)UUUU5 X:aUUU4OUUU4upn GO[R"/UQUP[RS 9nX:au[R "/UQU PU PUR S 9n [+URSS Vs/sHn[-U5PM sn6Hn[)UUU UUUU5 M UnOt[R "/UQU PU PUR S 9n[+URSS Vs/sHn[-U5PM sn6Hn[)UUUUUUU5 M Un U(dU(dU(ax[R "/UQUPUPU S 9n[R."UVs/sHn[R0"U5PM sn[R0"U5/-6nSU/UQUP7'UnO3[R "X/U S 9nSU[R0"U5U4'UnU(aX4$XU 4$s snfs snfs snf)a: 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 rz1The input array must be at least two-dimensional.rz The dtype z5 cannot be cast to float(32, 64) or complex(64, 128).rTfFfd)r*rrN)rrC)order C_CONTIGUOUS WRITEABLEr)rrrndimrrcharlapack_cast_dict TypeErrorastyper*minemptyr ones_likecopyzerosintrflagsr rrangeix_r)r permute_lrr p_indicesr! dtype_charndr0nk real_dcharPLUPLpur1indPand_ixs r$rrs@%1  a bjjmB ww{LMM xx}}F"%bhhmm4 j 3DDE EYYz!} % IRA A A -3J BHH~  " a  288 WW3W  HHQbhh ' HHaV288 ,b!Q * !1b!*Aq":a HHXrX1XRXX . 52q!BHH5A288CR= A=aq= ABbgqvqvyACA2q!BHH5A288CR= A=aq= ABbgqvqvyACA  +B++1+Z8BFFB7BqbiilB71FHEB{{{OA1& 3B"#Bryy|Q AA6-Q1I-1!B !B8sUU U )FT)rFT)FFTF)__doc__warningsrnumpyrrr itertoolsr_miscrr lapackr _decomp_lu_cythonr typecodesjoincan_castr=__all__rrr)r1ys00r$res!,.$, \\%020rww6G6aR[[5F6GHH02 *jZEP<@|.w H2sB%B 5B ; B% B%