from warnings import warn import numpy as np from numpy import asarray from scipy.sparse import (isspmatrix_csc, isspmatrix_csr, isspmatrix, SparseEfficiencyWarning, csc_matrix, csr_matrix) from scipy.sparse._sputils import is_pydata_spmatrix from scipy.linalg import LinAlgError import copy from . import _superlu noScikit = False try: import scikits.umfpack as umfpack except ImportError: noScikit = True useUmfpack = not noScikit __all__ = ['use_solver', 'spsolve', 'splu', 'spilu', 'factorized', 'MatrixRankWarning', 'spsolve_triangular'] class MatrixRankWarning(UserWarning): pass def use_solver(**kwargs): """ 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 """ if 'useUmfpack' in kwargs: globals()['useUmfpack'] = kwargs['useUmfpack'] if useUmfpack and 'assumeSortedIndices' in kwargs: umfpack.configure(assumeSortedIndices=kwargs['assumeSortedIndices']) def _get_umf_family(A): """Get umfpack family string given the sparse matrix dtype.""" _families = { (np.float64, np.int32): 'di', (np.complex128, np.int32): 'zi', (np.float64, np.int64): 'dl', (np.complex128, np.int64): 'zl' } f_type = np.sctypeDict[A.dtype.name] i_type = np.sctypeDict[A.indices.dtype.name] try: family = _families[(f_type, i_type)] except KeyError as e: msg = 'only float64 or complex128 matrices with int32 or int64' \ ' indices are supported! (got: matrix: %s, indices: %s)' \ % (f_type, i_type) raise ValueError(msg) from e # See gh-8278. Considered converting only if # A.shape[0]*A.shape[1] > np.iinfo(np.int32).max, # but that didn't always fix the issue. family = family[0] + "l" A_new = copy.copy(A) A_new.indptr = np.array(A.indptr, copy=False, dtype=np.int64) A_new.indices = np.array(A.indices, copy=False, dtype=np.int64) return family, A_new def spsolve(A, b, permc_spec=None, use_umfpack=True): """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 """ if is_pydata_spmatrix(A): A = A.to_scipy_sparse().tocsc() if not (isspmatrix_csc(A) or isspmatrix_csr(A)): A = csc_matrix(A) warn('spsolve requires A be CSC or CSR matrix format', SparseEfficiencyWarning) # b is a vector only if b have shape (n,) or (n, 1) b_is_sparse = isspmatrix(b) or is_pydata_spmatrix(b) if not b_is_sparse: b = asarray(b) b_is_vector = ((b.ndim == 1) or (b.ndim == 2 and b.shape[1] == 1)) # sum duplicates for non-canonical format A.sum_duplicates() A = A.asfptype() # upcast to a floating point format result_dtype = np.promote_types(A.dtype, b.dtype) if A.dtype != result_dtype: A = A.astype(result_dtype) if b.dtype != result_dtype: b = b.astype(result_dtype) # validate input shapes M, N = A.shape if (M != N): raise ValueError("matrix must be square (has shape %s)" % ((M, N),)) if M != b.shape[0]: raise ValueError("matrix - rhs dimension mismatch (%s - %s)" % (A.shape, b.shape[0])) use_umfpack = use_umfpack and useUmfpack if b_is_vector and use_umfpack: if b_is_sparse: b_vec = b.toarray() else: b_vec = b b_vec = asarray(b_vec, dtype=A.dtype).ravel() if noScikit: raise RuntimeError('Scikits.umfpack not installed.') if A.dtype.char not in 'dD': raise ValueError("convert matrix data to double, please, using" " .astype(), or set linsolve.useUmfpack = False") umf_family, A = _get_umf_family(A) umf = umfpack.UmfpackContext(umf_family) x = umf.linsolve(umfpack.UMFPACK_A, A, b_vec, autoTranspose=True) else: if b_is_vector and b_is_sparse: b = b.toarray() b_is_sparse = False if not b_is_sparse: if isspmatrix_csc(A): flag = 1 # CSC format else: flag = 0 # CSR format options = dict(ColPerm=permc_spec) x, info = _superlu.gssv(N, A.nnz, A.data, A.indices, A.indptr, b, flag, options=options) if info != 0: warn("Matrix is exactly singular", MatrixRankWarning) x.fill(np.nan) if b_is_vector: x = x.ravel() else: # b is sparse Afactsolve = factorized(A) if not (isspmatrix_csc(b) or is_pydata_spmatrix(b)): warn('spsolve is more efficient when sparse b ' 'is in the CSC matrix format', SparseEfficiencyWarning) b = csc_matrix(b) # Create a sparse output matrix by repeatedly applying # the sparse factorization to solve columns of b. data_segs = [] row_segs = [] col_segs = [] for j in range(b.shape[1]): # TODO: replace this with # bj = b[:, j].toarray().ravel() # once 1D sparse arrays are supported. # That is a slightly faster code path. bj = b[:, [j]].toarray().ravel() xj = Afactsolve(bj) w = np.flatnonzero(xj) segment_length = w.shape[0] row_segs.append(w) col_segs.append(np.full(segment_length, j, dtype=int)) data_segs.append(np.asarray(xj[w], dtype=A.dtype)) sparse_data = np.concatenate(data_segs) sparse_row = np.concatenate(row_segs) sparse_col = np.concatenate(col_segs) x = A.__class__((sparse_data, (sparse_row, sparse_col)), shape=b.shape, dtype=A.dtype) if is_pydata_spmatrix(b): x = b.__class__(x) return x def splu(A, permc_spec=None, diag_pivot_thresh=None, relax=None, panel_size=None, options=dict()): """ 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.]) """ if is_pydata_spmatrix(A): csc_construct_func = lambda *a, cls=type(A): cls(csc_matrix(*a)) A = A.to_scipy_sparse().tocsc() else: csc_construct_func = csc_matrix if not isspmatrix_csc(A): A = csc_matrix(A) warn('splu converted its input to CSC format', SparseEfficiencyWarning) # sum duplicates for non-canonical format A.sum_duplicates() A = A.asfptype() # upcast to a floating point format M, N = A.shape if (M != N): raise ValueError("can only factor square matrices") # is this true? _options = dict(DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec, PanelSize=panel_size, Relax=relax) if options is not None: _options.update(options) # Ensure that no column permutations are applied if (_options["ColPerm"] == "NATURAL"): _options["SymmetricMode"] = True return _superlu.gstrf(N, A.nnz, A.data, A.indices, A.indptr, csc_construct_func=csc_construct_func, ilu=False, options=_options) def spilu(A, drop_tol=None, fill_factor=None, drop_rule=None, permc_spec=None, diag_pivot_thresh=None, relax=None, panel_size=None, options=None): """ 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.]) """ if is_pydata_spmatrix(A): csc_construct_func = lambda *a, cls=type(A): cls(csc_matrix(*a)) A = A.to_scipy_sparse().tocsc() else: csc_construct_func = csc_matrix if not isspmatrix_csc(A): A = csc_matrix(A) warn('spilu converted its input to CSC format', SparseEfficiencyWarning) # sum duplicates for non-canonical format A.sum_duplicates() A = A.asfptype() # upcast to a floating point format M, N = A.shape if (M != N): raise ValueError("can only factor square matrices") # is this true? _options = dict(ILU_DropRule=drop_rule, ILU_DropTol=drop_tol, ILU_FillFactor=fill_factor, DiagPivotThresh=diag_pivot_thresh, ColPerm=permc_spec, PanelSize=panel_size, Relax=relax) if options is not None: _options.update(options) # Ensure that no column permutations are applied if (_options["ColPerm"] == "NATURAL"): _options["SymmetricMode"] = True return _superlu.gstrf(N, A.nnz, A.data, A.indices, A.indptr, csc_construct_func=csc_construct_func, ilu=True, options=_options) def factorized(A): """ 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.]) """ if is_pydata_spmatrix(A): A = A.to_scipy_sparse().tocsc() if useUmfpack: if noScikit: raise RuntimeError('Scikits.umfpack not installed.') if not isspmatrix_csc(A): A = csc_matrix(A) warn('splu converted its input to CSC format', SparseEfficiencyWarning) A = A.asfptype() # upcast to a floating point format if A.dtype.char not in 'dD': raise ValueError("convert matrix data to double, please, using" " .astype(), or set linsolve.useUmfpack = False") umf_family, A = _get_umf_family(A) umf = umfpack.UmfpackContext(umf_family) # Make LU decomposition. umf.numeric(A) def solve(b): with np.errstate(divide="ignore", invalid="ignore"): # Ignoring warnings with numpy >= 1.23.0, see gh-16523 result = umf.solve(umfpack.UMFPACK_A, A, b, autoTranspose=True) return result return solve else: return splu(A).solve def spsolve_triangular(A, b, lower=True, overwrite_A=False, overwrite_b=False, unit_diagonal=False): """ 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 """ if is_pydata_spmatrix(A): A = A.to_scipy_sparse().tocsr() # Check the input for correct type and format. if not isspmatrix_csr(A): warn('CSR matrix format is required. Converting to CSR matrix.', SparseEfficiencyWarning) A = csr_matrix(A) elif not overwrite_A: A = A.copy() if A.shape[0] != A.shape[1]: raise ValueError( 'A must be a square matrix but its shape is {}.'.format(A.shape)) # sum duplicates for non-canonical format A.sum_duplicates() b = np.asanyarray(b) if b.ndim not in [1, 2]: raise ValueError( 'b must have 1 or 2 dims but its shape is {}.'.format(b.shape)) if A.shape[0] != b.shape[0]: raise ValueError( 'The 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 {}.'.format(A.shape, b.shape)) # Init x as (a copy of) b. x_dtype = np.result_type(A.data, b, np.float64) if overwrite_b: if np.can_cast(b.dtype, x_dtype, casting='same_kind'): x = b else: raise ValueError( 'Cannot overwrite b (dtype {}) with result ' 'of type {}.'.format(b.dtype, x_dtype)) else: x = b.astype(x_dtype, copy=True) # Choose forward or backward order. if lower: row_indices = range(len(b)) else: row_indices = range(len(b) - 1, -1, -1) # Fill x iteratively. for i in row_indices: # Get indices for i-th row. indptr_start = A.indptr[i] indptr_stop = A.indptr[i + 1] if lower: A_diagonal_index_row_i = indptr_stop - 1 A_off_diagonal_indices_row_i = slice(indptr_start, indptr_stop - 1) else: A_diagonal_index_row_i = indptr_start A_off_diagonal_indices_row_i = slice(indptr_start + 1, indptr_stop) # Check regularity and triangularity of A. if not unit_diagonal and (indptr_stop <= indptr_start or A.indices[A_diagonal_index_row_i] < i): raise LinAlgError( 'A is singular: diagonal {} is zero.'.format(i)) if not unit_diagonal and A.indices[A_diagonal_index_row_i] > i: raise LinAlgError( 'A is not triangular: A[{}, {}] is nonzero.' ''.format(i, A.indices[A_diagonal_index_row_i])) # Incorporate off-diagonal entries. A_column_indices_in_row_i = A.indices[A_off_diagonal_indices_row_i] A_values_in_row_i = A.data[A_off_diagonal_indices_row_i] x[i] -= np.dot(x[A_column_indices_in_row_i].T, A_values_in_row_i) # Compute i-th entry of x. if not unit_diagonal: x[i] /= A.data[A_diagonal_index_row_i] return x