a Ree@sddlmZddlZddlmZddlmZmZmZm Z ddl m Z ddl m Z ddlZddlmZd ZzddlmZWneyd ZYn0e Zgd ZGd d d eZddZddZddZdddZddddefddZdddZddZ d ddZ!dS)!)warnN)asarray)issparseSparseEfficiencyWarning csc_matrix csr_matrix)is_pydata_spmatrix) LinAlgError)_superluFT) use_solverspsolvespluspilu factorizedMatrixRankWarningspsolve_triangularc@s eZdZdS)rN)__name__ __module__ __qualname__rrX/var/www/sunrise/env/lib/python3.9/site-packages/scipy/sparse/linalg/_dsolve/linsolve.pyrsrcKs6d|vr|dtd<tr2d|vr2tj|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)kwargsrrrr s= r c Cstjtjfdtjtjfdtjtjfdtjtjfdi}tj|jj}tj|jjj}z|||f}Wn:t y}z"d||f}t ||WYd}~n d}~00|dd}t |}tj |j d tjd |_ tj |jd tjd |_||fS) z8Get umfpack family string given the sparse matrix dtype.ZdiZzidlZzlzmonly float64 or complex128 matrices with int32 or int64 indices are supported! (got: matrix: %s, indices: %s)NrlF)copydtype)npfloat64int32Z complex128int64Z sctypeDictr!nameindicesKeyError ValueErrorr arrayindptr)AZ _familiesZf_typeZi_typefamilyemsgZA_newrrr_get_umf_family_s&       r0cCsvttjj}|jd|kr$tdt|j|krJt|j|krJtd|jj tjdd}|jj tjdd}||fS)Nz#indptr values too large for SuperLUz$indices values too large for SuperLUFr ) r"Ziinfointcmaxr+r)shapeanyr'astype)r,Z max_valuer'r+rrr_safe_downcast_indices~sr8c Csdt|r|}t|r&|jdvs8t|}tdtt|pFt|}|sTt|}|j dkpt|j dkot|j ddk}| | }t |j|j}|j|kr||}|j|kr||}|j \}}||krtd||fd||j dkr td|j |j df|ot}|r|r|r2|} n|} t| |jd } trVtd |jjd vrltd t|\} }t| } | jtj|| d d} n|r|r|}d}|sP|jdkrd} nd} |jjt jdd}|jjt jdd}t |d}t!j"||j#|j$|||| |d\} }|dkr>tdt%| &t j'|r`| } nt(|}|jdkst|stdtt|}g}g}g}t)|j dD]v}|dd|gf}||}t *|}|j d}|+||+t j,||t-d |+t j|||jd qt .|}t .|}t .|}|j/|||ff|j |jd} t|r`|/| } | 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 )csccsrz.spsolve requires A be CSC or CSR matrix formatr z!matrix must be square (has shape )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 autoTransposeFr9r2)ColPerm)optionszMatrix is exactly singularzCspsolve is more efficient when sparse b is in the CSC matrix formatN)r5r!)0rto_scipy_sparsetocscrformatrrrrndimr5sum_duplicates _asfptyper"Z promote_typesr!r7r)rZtoarrayZravelnoScikit RuntimeErrorcharr0rUmfpackContextZlinsolve UMFPACK_Ar'r3r+dictr ZgssvnnzdatarfillnanrrangeZ flatnonzeroappendfullintZ concatenate __class__)r,b permc_specZ use_umfpackZ b_is_sparseZ b_is_vectorZ result_dtypeMNZb_vec umf_familyumfxflagr'r+rBinfoZ AfactsolveZ data_segsZrow_segsZcol_segsjZbjZxjwZsegment_lengthZ sparse_dataZ sparse_rowZ sparse_colrrrr sS "                            r c Cst|r(t|ddd}|}nt}t|r>|jdksPt|}tdt| | }|j \}}||krzt dt |\} } t||||d} |dur| || d d krd | d <tj||j|j| | |d | dS)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|SNrrdarrrcsc_construct_funcsz splu..csc_construct_funcr9&splu converted its input to CSC formatcan only factor square matrices)DiagPivotThreshrA PanelSizeRelaxNrANATURALT SymmetricModeFriZilurBrtyperCrDrrrErrrGrHr5r)r8rNupdater ZgstrfrOrP) r,rYdiag_pivot_threshrelax panel_sizerBrirZr[r'r+_optionsrrrrOs0D     rc  Cst|r(t|ddd} |}nt} t|r>|jdksPt|}tdt| | }|j \} } | | krzt dt |\} } t|||||||d}|dur|||d d krd |d <tj| |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.]) rccWs |t|Srerfrgrrrrisz!spilu..csc_construct_funcr9z'spilu converted its input to CSC formatrk)Z ILU_DropRuleZ ILU_DropTolZILU_FillFactorrlrArmrnNrAroTrprqrr)r,Zdrop_tolZ fill_factorZ drop_rulerYrurvrwrBrirZr[r'r+rxrrrrs8;    rcstrtrtr$tdtr6jdksHtt dt  j j dvrdtdt\}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.]) r=r9rjr>r?csFtjddd$jtj|dd}Wdn1s80Y|S)Nignore)divideinvalidTr@)r"ZerrstatesolverrM)rXresultr,r]rrr|Ls2zfactorized..solveN)rrCrDrrIrJrrErrrrHr!rKr)r0rrLnumericrr|)r,r\r|rr~rrs&     rcCs>t|r|}t|r&|jdks:tdtt|}n |sF|}|j d|j dkrlt d|j d| t |}|jdvrt d|j d|j d|j dkrt d |j |j t |j|t j}|rt j|j|d d r|}nt d |j|n|j|d d}|r(tt|}ntt|ddd}|D]} |j| } |j| d} |rz| d} t| | d} n| } t| d| } |s| | ks|j| | krtd| d|s|j| | krtd| |j| |j| }|j| }|| t ||j|8<|s@|| |j| <q@|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 r:z8CSR matrix format is required. Converting to CSR matrix.rr z+A must be a square matrix but its shape is .)r r;z)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 {}.Z same_kind)Zcastingz5Cannot overwrite b (dtype {}) with result of type {}.Tr2r1zA is singular: diagonal z is zero.z*A is not triangular: A[{}, {}] is nonzero.)rrCZtocsrrrErrrr r5r)rGr"Z asanyarrayrFZ result_typerPr#Zcan_castr!r7rSlenr+slicer'r dotT)r,rXlowerZ overwrite_AZ overwrite_bZ unit_diagonalZx_dtyper^Z row_indicesiZ indptr_startZ indptr_stopZA_diagonal_index_row_iZA_off_diagonal_indices_row_iZA_column_indices_in_row_iZA_values_in_row_irrrrXs:            r)NT)NNNNNNNN)TFFF)"warningsrnumpyr"rZ scipy.sparserrrrZscipy.sparse._sputilsrZ scipy.linalgr r r rIZscikits.umfpackr ImportErrorr__all__ UserWarningrr r0r8r rNrrrrrrrrs<       B C g aA