U »|eb@sddlmZddlZddlmZddlmZmZmZm Z m Z m Z ddl m Z ddlmZddlZddlmZd ZzddlmZWnek rd ZYnXe Zd d d ddddgZGdddeZdd ZddZddd Zddddefdd ZdddZ ddZ!dddZ"dS))warnN)asarray)isspmatrix_cscisspmatrix_csr isspmatrixSparseEfficiencyWarning csc_matrix csr_matrix)is_pydata_spmatrix) LinAlgError)_superluFT use_solverspsolvespluspilu factorizedMatrixRankWarningspsolve_triangularc@s eZdZdS)rN)__name__ __module__ __qualname__rra/var/www/website-v5/atlas_env/lib/python3.8/site-packages/scipy/sparse/linalg/_dsolve/linsolve.pyrscKs6d|kr|dtd<tr2d|kr2tj|dddS)as Select default sparse direct solver to be used. Parameters ---------- useUmfpack : bool, optional Use UMFPACK [1]_, [2]_, [3]_, [4]_. over SuperLU. Has effect only if ``scikits.umfpack`` is installed. Default: True assumeSortedIndices : bool, optional Allow UMFPACK to skip the step of sorting indices for a CSR/CSC matrix. Has effect only if useUmfpack is True and ``scikits.umfpack`` is installed. Default: False Notes ----- The default sparse solver is UMFPACK when available (``scikits.umfpack`` is installed). This can be changed by passing useUmfpack = False, which then causes the always present SuperLU based solver to be used. UMFPACK requires a CSR/CSC matrix to have sorted column/row indices. If sure that the matrix fulfills this, pass ``assumeSortedIndices=True`` to gain some speed. References ---------- .. [1] T. A. Davis, Algorithm 832: UMFPACK - an unsymmetric-pattern multifrontal method with a column pre-ordering strategy, ACM Trans. on Mathematical Software, 30(2), 2004, pp. 196--199. https://dl.acm.org/doi/abs/10.1145/992200.992206 .. [2] T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. on Mathematical Software, 30(2), 2004, pp. 165--195. https://dl.acm.org/doi/abs/10.1145/992200.992205 .. [3] T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices, ACM Trans. on Mathematical Software, 25(1), 1999, pp. 1--19. https://doi.org/10.1145/305658.287640 .. [4] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Analysis and Computations, 18(1), 1997, pp. 140--158. https://doi.org/10.1137/S0895479894246905T. Examples -------- >>> import numpy as np >>> from scipy.sparse.linalg import use_solver, spsolve >>> from scipy.sparse import csc_matrix >>> R = np.random.randn(5, 5) >>> A = csc_matrix(R) >>> b = np.random.randn(5) >>> use_solver(useUmfpack=False) # enforce superLU over UMFPACK >>> x = spsolve(A, b) >>> np.allclose(A.dot(x), b) True >>> use_solver(useUmfpack=True) # reset umfPack usage to default useUmfpackassumeSortedIndices)rN)globalsrumfpack configure)kwargsrrrrs= c Cstjtjfdtjtjfdtjtjfdtjtjfdi}tj|jj}tj|jjj}z|||f}Wn8t k r}zd||f}t ||W5d}~XYnX|dd}t |}tj |j d tjd |_ tj |jd tjd |_||fS) z8Get umfpack family string given the sparse matrix dtype.diZzidlZzlzmonly float64 or complex128 matrices with int32 or int64 indices are supported! (got: matrix: %s, indices: %s)NrlF)copydtype)npfloat64int32 complex128int64 sctypeDictr$nameindicesKeyError ValueErrorr#arrayindptr)AZ _familiesZf_typeZi_typefamilyemsgZA_newrrr_get_umf_family_s.      r5c Cs<t|r|}t|s6t|s6t|}tdtt|pDt|}|sRt |}|j dkpr|j dkor|j ddk}| | }t|j|j}|j|kr||}|j|kr||}|j \}}||krtd||ff||j dkrtd|j |j df|ot}|r|r|r.|} n|} t | |jd} trRtd|jjd krhtd t|\} }t| } | jtj|| d d } n|r|r|}d }|s*t|rd} nd} t|d}tj ||j!|j"|j#|j$|| |d\} }|dkrtdt%| &tj'|r8| } nt(|}t|sXt|sXtdtt|}g}g}g}t)|j dD]v}|dd|gf}||}t*|}|j d}|+||+tj,||t-d|+tj |||jdqrt.|}t.|}t.|}|j/|||ff|j |jd} t|r8|/| } | S)a Solve the sparse linear system Ax=b, where b may be a vector or a matrix. Parameters ---------- A : ndarray or sparse matrix The square matrix A will be converted into CSC or CSR form b : ndarray or sparse matrix The matrix or vector representing the right hand side of the equation. If a vector, b.shape must be (n,) or (n, 1). permc_spec : str, optional How to permute the columns of the matrix for sparsity preservation. (default: 'COLAMD') - ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering [1]_, [2]_. use_umfpack : bool, optional if True (default) then use UMFPACK for the solution [3]_, [4]_, [5]_, [6]_ . This is only referenced if b is a vector and ``scikits.umfpack`` is installed. Returns ------- x : ndarray or sparse matrix the solution of the sparse linear equation. If b is a vector, then x is a vector of size A.shape[1] If b is a matrix, then x is a matrix of size (A.shape[1], b.shape[1]) Notes ----- For solving the matrix expression AX = B, this solver assumes the resulting matrix X is sparse, as is often the case for very sparse inputs. If the resulting X is dense, the construction of this sparse result will be relatively expensive. In that case, consider converting A to a dense matrix and using scipy.linalg.solve or its variants. References ---------- .. [1] T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, Algorithm 836: COLAMD, an approximate column minimum degree ordering algorithm, ACM Trans. on Mathematical Software, 30(3), 2004, pp. 377--380. :doi:`10.1145/1024074.1024080` .. [2] T. A. Davis, J. R. Gilbert, S. Larimore, E. Ng, A column approximate minimum degree ordering algorithm, ACM Trans. on Mathematical Software, 30(3), 2004, pp. 353--376. :doi:`10.1145/1024074.1024079` .. [3] T. A. Davis, Algorithm 832: UMFPACK - an unsymmetric-pattern multifrontal method with a column pre-ordering strategy, ACM Trans. on Mathematical Software, 30(2), 2004, pp. 196--199. https://dl.acm.org/doi/abs/10.1145/992200.992206 .. [4] T. A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. on Mathematical Software, 30(2), 2004, pp. 165--195. https://dl.acm.org/doi/abs/10.1145/992200.992205 .. [5] T. A. Davis and I. S. Duff, A combined unifrontal/multifrontal method for unsymmetric sparse matrices, ACM Trans. on Mathematical Software, 25(1), 1999, pp. 1--19. https://doi.org/10.1145/305658.287640 .. [6] T. A. Davis and I. S. Duff, An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Analysis and Computations, 18(1), 1997, pp. 140--158. https://doi.org/10.1137/S0895479894246905T. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import spsolve >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> B = csc_matrix([[2, 0], [-1, 0], [2, 0]], dtype=float) >>> x = spsolve(A, B) >>> np.allclose(A.dot(x).toarray(), B.toarray()) True z.spsolve requires A be CSC or CSR matrix formatr z$matrix must be square (has shape %s)rz)matrix - rhs dimension mismatch (%s - %s))r$Scikits.umfpack not installed.dDZconvert matrix data to double, please, using .astype(), or set linsolve.useUmfpack = FalseTZ autoTransposeF)ColPerm)optionszMatrix is exactly singularzCspsolve is more efficient when sparse b is in the CSC matrix formatN)shaper$)0r to_scipy_sparsetocscrrrrrrrndimr=sum_duplicatesasfptyper% promote_typesr$astyper.rtoarrayravelnoScikit RuntimeErrorcharr5rUmfpackContextlinsolve UMFPACK_Adictr Zgssvnnzdatar,r0rfillnanrrange flatnonzeroappendfullint concatenate __class__)r1b permc_specZ use_umfpackZ b_is_sparseZ b_is_vector result_dtypeMNZb_vec umf_familyumfxflagr<infoZ AfactsolveZ data_segsZrow_segsZcol_segsjbjZxjwZsegment_lengthZ sparse_dataZ sparse_rowZ sparse_colrrrr~sS "                            c Cst|r(t|ddd}|}nt}t|sFt|}tdt|| }|j \}}||krpt dt ||||d} |dk r| || dd krd | d <tj||j|j|j|j|d | d S)a Compute the LU decomposition of a sparse, square matrix. Parameters ---------- A : sparse matrix Sparse matrix to factorize. Most efficient when provided in CSC format. Other formats will be converted to CSC before factorization. permc_spec : str, optional How to permute the columns of the matrix for sparsity preservation. (default: 'COLAMD') - ``NATURAL``: natural ordering. - ``MMD_ATA``: minimum degree ordering on the structure of A^T A. - ``MMD_AT_PLUS_A``: minimum degree ordering on the structure of A^T+A. - ``COLAMD``: approximate minimum degree column ordering diag_pivot_thresh : float, optional Threshold used for a diagonal entry to be an acceptable pivot. See SuperLU user's guide for details [1]_ relax : int, optional Expert option for customizing the degree of relaxing supernodes. See SuperLU user's guide for details [1]_ panel_size : int, optional Expert option for customizing the panel size. See SuperLU user's guide for details [1]_ options : dict, optional Dictionary containing additional expert options to SuperLU. See SuperLU user guide [1]_ (section 2.4 on the 'Options' argument) for more details. For example, you can specify ``options=dict(Equil=False, IterRefine='SINGLE'))`` to turn equilibration off and perform a single iterative refinement. Returns ------- invA : scipy.sparse.linalg.SuperLU Object, which has a ``solve`` method. See also -------- spilu : incomplete LU decomposition Notes ----- This function uses the SuperLU library. References ---------- .. [1] SuperLU https://portal.nersc.gov/project/sparse/superlu/ Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import splu >>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float) >>> B = splu(A) >>> x = np.array([1., 2., 3.], dtype=float) >>> B.solve(x) array([ 1. , -3. , -1.5]) >>> A.dot(B.solve(x)) array([ 1., 2., 3.]) >>> B.solve(A.dot(x)) array([ 1., 2., 3.]) clscWs |t|SNrrgarrrzsplu..&splu converted its input to CSC formatcan only factor square matrices)DiagPivotThreshr; PanelSizeRelaxNr;NATURALT SymmetricModeFcsc_construct_funcZilur<r typer>r?rrrrrArBr=r.rMupdater ZgstrfrNrOr,r0) r1rZdiag_pivot_threshrelax panel_sizer<rvr\r]_optionsrrrr>s2D    c Cst|r(t|ddd} |}nt} t|sFt|}tdt|| }|j \} } | | krpt dt |||||||d} |dk r| || dd krd | d <tj| |j|j|j|j| d | d S) a Compute an incomplete LU decomposition for a sparse, square matrix. The resulting object is an approximation to the inverse of `A`. Parameters ---------- A : (N, N) array_like Sparse matrix to factorize. Most efficient when provided in CSC format. Other formats will be converted to CSC before factorization. drop_tol : float, optional Drop tolerance (0 <= tol <= 1) for an incomplete LU decomposition. (default: 1e-4) fill_factor : float, optional Specifies the fill ratio upper bound (>= 1.0) for ILU. (default: 10) drop_rule : str, optional Comma-separated string of drop rules to use. Available rules: ``basic``, ``prows``, ``column``, ``area``, ``secondary``, ``dynamic``, ``interp``. (Default: ``basic,area``) See SuperLU documentation for details. Remaining other options Same as for `splu` Returns ------- invA_approx : scipy.sparse.linalg.SuperLU Object, which has a ``solve`` method. See also -------- splu : complete LU decomposition Notes ----- To improve the better approximation to the inverse, you may need to increase `fill_factor` AND decrease `drop_tol`. This function uses the SuperLU library. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import spilu >>> A = csc_matrix([[1., 0., 0.], [5., 0., 2.], [0., -1., 0.]], dtype=float) >>> B = spilu(A) >>> x = np.array([1., 2., 3.], dtype=float) >>> B.solve(x) array([ 1. , -3. , -1.5]) >>> A.dot(B.solve(x)) array([ 1., 2., 3.]) >>> B.solve(A.dot(x)) array([ 1., 2., 3.]) rfcWs |t|Srhrirjrrrrlrmzspilu..z'spilu converted its input to CSC formatro)Z ILU_DropRuleZ ILU_DropTolZILU_FillFactorrpr;rqrrNr;rsTrtrurw) r1Zdrop_tolZ fill_factorZ drop_rulerZrzr{r|r<rvr\r]r}rrrrs<;   cstrtrtr$tdts>ttdt  j j dkrZt dt\}t|fdd}|StjSdS)aK Return a function for solving a sparse linear system, with A pre-factorized. Parameters ---------- A : (N, N) array_like Input. A in CSC format is most efficient. A CSR format matrix will be converted to CSC before factorization. Returns ------- solve : callable To solve the linear system of equations given in `A`, the `solve` callable should be passed an ndarray of shape (N,). Examples -------- >>> import numpy as np >>> from scipy.sparse.linalg import factorized >>> A = np.array([[ 3. , 2. , -1. ], ... [ 2. , -2. , 4. ], ... [-1. , 0.5, -1. ]]) >>> solve = factorized(A) # Makes LU decomposition. >>> rhs1 = np.array([1, -2, 0]) >>> solve(rhs1) # Uses the LU factors. array([ 1., -2., -2.]) r7rnr8r9c s2tjdddjtj|dd}W5QRX|S)Nignore)divideinvalidTr:)r%errstatesolverrL)rYresultr1r_rrr5szfactorized..solveN)r r>r?rrGrHrrrrrBr$rIr.r5rrJnumericrr)r1r^rrrrrs&     cCs,t|r|}t|s0tdtt|}n |s<|}|jd|jdkr`t d |j| t |}|jdkrt d |j|jd|jdkrt d |j|jt |j|t j}|rt j|j|dd r|}nt d |j|n|j|d d }|rtt|}ntt|dd d }|D]} |j| } |j| d} |rj| d} t| | d} n| } t| d| } |s| | ks|j| | krtd | |s|j| | krtd | |j| |j| }|j| }|| t ||j|8<|s0|| |j| <q0|S)a Solve the equation ``A x = b`` for `x`, assuming A is a triangular matrix. Parameters ---------- A : (M, M) sparse matrix A sparse square triangular matrix. Should be in CSR format. b : (M,) or (M, N) array_like Right-hand side matrix in ``A x = b`` lower : bool, optional Whether `A` is a lower or upper triangular matrix. Default is lower triangular matrix. overwrite_A : bool, optional Allow changing `A`. The indices of `A` are going to be sorted and zero entries are going to be removed. Enabling gives a performance gain. Default is False. overwrite_b : bool, optional Allow overwriting data in `b`. Enabling gives a performance gain. Default is False. If `overwrite_b` is True, it should be ensured that `b` has an appropriate dtype to be able to store the result. unit_diagonal : bool, optional If True, diagonal elements of `a` are assumed to be 1 and will not be referenced. .. versionadded:: 1.4.0 Returns ------- x : (M,) or (M, N) ndarray Solution to the system ``A x = b``. Shape of return matches shape of `b`. Raises ------ LinAlgError If `A` is singular or not triangular. ValueError If shape of `A` or shape of `b` do not match the requirements. Notes ----- .. versionadded:: 0.19.0 Examples -------- >>> import numpy as np >>> from scipy.sparse import csr_matrix >>> from scipy.sparse.linalg import spsolve_triangular >>> A = csr_matrix([[3, 0, 0], [1, -1, 0], [2, 0, 1]], dtype=float) >>> B = np.array([[2, 0], [-1, 0], [2, 0]], dtype=float) >>> x = spsolve_triangular(A, B) >>> np.allclose(A.dot(x), B) True z8CSR matrix format is required. Converting to CSR matrix.rr z.A must be a square matrix but its shape is {}.)r r6z,b must have 1 or 2 dims but its shape is {}.zThe size of the dimensions of A must be equal to the size of the first dimension of b but the shape of A is {} and the shape of b is {}. same_kind)castingz5Cannot overwrite b (dtype {}) with result of type {}.T)r#z#A is singular: diagonal {} is zero.z*A is not triangular: A[{}, {}] is nonzero.)r r>tocsrrrrr r#r=r.formatrAr% asanyarrayr@ result_typerOr&can_castr$rDrRlenr0slicer,r dotT)r1rYlowerZ overwrite_A overwrite_b unit_diagonalZx_dtyper` row_indicesiZ indptr_startZ indptr_stopZA_diagonal_index_row_iZA_off_diagonal_indices_row_iZA_column_indices_in_row_iZA_values_in_row_irrrrAs:          )NT)NNNNNNNN)TFFF)#warningsrnumpyr%r scipy.sparserrrrrr Zscipy.sparse._sputilsr scipy.linalgr r#r rGZscikits.umfpackr ImportErrorr__all__ UserWarningrrr5rrMrrrrrrrrsJ        B A d ^A