"""Functions to construct sparse matrices and arrays """ __docformat__ = "restructuredtext en" __all__ = ['spdiags', 'eye', 'identity', 'kron', 'kronsum', 'hstack', 'vstack', 'bmat', 'rand', 'random', 'diags', 'block_diag', 'diags_array', 'block_array', 'eye_array', 'random_array'] import numbers import math import numpy as np from scipy._lib._util import check_random_state, rng_integers from ._sputils import upcast, get_index_dtype, isscalarlike from ._sparsetools import csr_hstack from ._bsr import bsr_matrix, bsr_array from ._coo import coo_matrix, coo_array from ._csc import csc_matrix, csc_array from ._csr import csr_matrix, csr_array from ._dia import dia_matrix, dia_array from ._base import issparse, sparray def spdiags(data, diags, m=None, n=None, format=None): """ Return a sparse matrix from diagonals. Parameters ---------- data : array_like Matrix diagonals stored row-wise diags : sequence of int or an int Diagonals to set: * k = 0 the main diagonal * k > 0 the kth upper diagonal * k < 0 the kth lower diagonal m, n : int, tuple, optional Shape of the result. If `n` is None and `m` is a given tuple, the shape is this tuple. If omitted, the matrix is square and its shape is len(data[0]). format : str, optional Format of the result. By default (format=None) an appropriate sparse matrix format is returned. This choice is subject to change. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``diags_array`` to take advantage of the sparse array functionality. See Also -------- diags_array : more convenient form of this function diags : matrix version of diags_array dia_matrix : the sparse DIAgonal format. Examples -------- >>> import numpy as np >>> from scipy.sparse import spdiags >>> data = np.array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]) >>> diags = np.array([0, -1, 2]) >>> spdiags(data, diags, 4, 4).toarray() array([[1, 0, 3, 0], [1, 2, 0, 4], [0, 2, 3, 0], [0, 0, 3, 4]]) """ if m is None and n is None: m = n = len(data[0]) elif n is None: m, n = m return dia_matrix((data, diags), shape=(m, n)).asformat(format) def diags_array(diagonals, /, *, offsets=0, shape=None, format=None, dtype=None): """ Construct a sparse array from diagonals. Parameters ---------- diagonals : sequence of array_like Sequence of arrays containing the array diagonals, corresponding to `offsets`. offsets : sequence of int or an int, optional Diagonals to set: - k = 0 the main diagonal (default) - k > 0 the kth upper diagonal - k < 0 the kth lower diagonal shape : tuple of int, optional Shape of the result. If omitted, a square array large enough to contain the diagonals is returned. format : {"dia", "csr", "csc", "lil", ...}, optional Matrix format of the result. By default (format=None) an appropriate sparse array format is returned. This choice is subject to change. dtype : dtype, optional Data type of the array. Notes ----- The result from `diags_array` is the sparse equivalent of:: np.diag(diagonals[0], offsets[0]) + ... + np.diag(diagonals[k], offsets[k]) Repeated diagonal offsets are disallowed. .. versionadded:: 1.11 Examples -------- >>> from scipy.sparse import diags_array >>> diagonals = [[1, 2, 3, 4], [1, 2, 3], [1, 2]] >>> diags_array(diagonals, offsets=[0, -1, 2]).toarray() array([[1, 0, 1, 0], [1, 2, 0, 2], [0, 2, 3, 0], [0, 0, 3, 4]]) Broadcasting of scalars is supported (but shape needs to be specified): >>> diags_array([1, -2, 1], offsets=[-1, 0, 1], shape=(4, 4)).toarray() array([[-2., 1., 0., 0.], [ 1., -2., 1., 0.], [ 0., 1., -2., 1.], [ 0., 0., 1., -2.]]) If only one diagonal is wanted (as in `numpy.diag`), the following works as well: >>> diags_array([1, 2, 3], offsets=1).toarray() array([[ 0., 1., 0., 0.], [ 0., 0., 2., 0.], [ 0., 0., 0., 3.], [ 0., 0., 0., 0.]]) """ # if offsets is not a sequence, assume that there's only one diagonal if isscalarlike(offsets): # now check that there's actually only one diagonal if len(diagonals) == 0 or isscalarlike(diagonals[0]): diagonals = [np.atleast_1d(diagonals)] else: raise ValueError("Different number of diagonals and offsets.") else: diagonals = list(map(np.atleast_1d, diagonals)) offsets = np.atleast_1d(offsets) # Basic check if len(diagonals) != len(offsets): raise ValueError("Different number of diagonals and offsets.") # Determine shape, if omitted if shape is None: m = len(diagonals[0]) + abs(int(offsets[0])) shape = (m, m) # Determine data type, if omitted if dtype is None: dtype = np.common_type(*diagonals) # Construct data array m, n = shape M = max([min(m + offset, n - offset) + max(0, offset) for offset in offsets]) M = max(0, M) data_arr = np.zeros((len(offsets), M), dtype=dtype) K = min(m, n) for j, diagonal in enumerate(diagonals): offset = offsets[j] k = max(0, offset) length = min(m + offset, n - offset, K) if length < 0: raise ValueError("Offset %d (index %d) out of bounds" % (offset, j)) try: data_arr[j, k:k+length] = diagonal[...,:length] except ValueError as e: if len(diagonal) != length and len(diagonal) != 1: raise ValueError( "Diagonal length (index %d: %d at offset %d) does not " "agree with array size (%d, %d)." % ( j, len(diagonal), offset, m, n)) from e raise return dia_array((data_arr, offsets), shape=(m, n)).asformat(format) def diags(diagonals, offsets=0, shape=None, format=None, dtype=None): """ Construct a sparse matrix from diagonals. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``diags_array`` to take advantage of the sparse array functionality. Parameters ---------- diagonals : sequence of array_like Sequence of arrays containing the matrix diagonals, corresponding to `offsets`. offsets : sequence of int or an int, optional Diagonals to set: - k = 0 the main diagonal (default) - k > 0 the kth upper diagonal - k < 0 the kth lower diagonal shape : tuple of int, optional Shape of the result. If omitted, a square matrix large enough to contain the diagonals is returned. format : {"dia", "csr", "csc", "lil", ...}, optional Matrix format of the result. By default (format=None) an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional Data type of the matrix. See Also -------- spdiags : construct matrix from diagonals diags_array : construct sparse array instead of sparse matrix Notes ----- This function differs from `spdiags` in the way it handles off-diagonals. The result from `diags` is the sparse equivalent of:: np.diag(diagonals[0], offsets[0]) + ... + np.diag(diagonals[k], offsets[k]) Repeated diagonal offsets are disallowed. .. versionadded:: 0.11 Examples -------- >>> from scipy.sparse import diags >>> diagonals = [[1, 2, 3, 4], [1, 2, 3], [1, 2]] >>> diags(diagonals, [0, -1, 2]).toarray() array([[1, 0, 1, 0], [1, 2, 0, 2], [0, 2, 3, 0], [0, 0, 3, 4]]) Broadcasting of scalars is supported (but shape needs to be specified): >>> diags([1, -2, 1], [-1, 0, 1], shape=(4, 4)).toarray() array([[-2., 1., 0., 0.], [ 1., -2., 1., 0.], [ 0., 1., -2., 1.], [ 0., 0., 1., -2.]]) If only one diagonal is wanted (as in `numpy.diag`), the following works as well: >>> diags([1, 2, 3], 1).toarray() array([[ 0., 1., 0., 0.], [ 0., 0., 2., 0.], [ 0., 0., 0., 3.], [ 0., 0., 0., 0.]]) """ A = diags_array(diagonals, offsets=offsets, shape=shape, dtype=dtype) return dia_matrix(A).asformat(format) def identity(n, dtype='d', format=None): """Identity matrix in sparse format Returns an identity matrix with shape (n,n) using a given sparse format and dtype. This differs from `eye_array` in that it has a square shape with ones only on the main diagonal. It is thus the multiplicative identity. `eye_array` allows rectangular shapes and the diagonal can be offset from the main one. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``eye_array`` to take advantage of the sparse array functionality. Parameters ---------- n : int Shape of the identity matrix. dtype : dtype, optional Data type of the matrix format : str, optional Sparse format of the result, e.g., format="csr", etc. Examples -------- >>> import scipy as sp >>> sp.sparse.identity(3).toarray() array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) >>> sp.sparse.identity(3, dtype='int8', format='dia') >>> sp.sparse.eye_array(3, dtype='int8', format='dia') """ return eye(n, n, dtype=dtype, format=format) def eye_array(m, n=None, *, k=0, dtype=float, format=None): """Identity matrix in sparse array format Return a sparse array with ones on diagonal. Specifically a sparse array (m x n) where the kth diagonal is all ones and everything else is zeros. Parameters ---------- m : int or tuple of ints Number of rows requested. n : int, optional Number of columns. Default: `m`. k : int, optional Diagonal to place ones on. Default: 0 (main diagonal). dtype : dtype, optional Data type of the array format : str, optional (default: "dia") Sparse format of the result, e.g., format="csr", etc. Examples -------- >>> import numpy as np >>> import scipy as sp >>> sp.sparse.eye_array(3).toarray() array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) >>> sp.sparse.eye_array(3, dtype=np.int8) """ # TODO: delete next 15 lines [combine with _eye()] once spmatrix removed return _eye(m, n, k, dtype, format) def _eye(m, n, k, dtype, format, as_sparray=True): if as_sparray: csr_sparse = csr_array csc_sparse = csc_array coo_sparse = coo_array diags_sparse = diags_array else: csr_sparse = csr_matrix csc_sparse = csc_matrix coo_sparse = coo_matrix diags_sparse = diags if n is None: n = m m, n = int(m), int(n) if m == n and k == 0: # fast branch for special formats if format in ['csr', 'csc']: idx_dtype = get_index_dtype(maxval=n) indptr = np.arange(n+1, dtype=idx_dtype) indices = np.arange(n, dtype=idx_dtype) data = np.ones(n, dtype=dtype) cls = {'csr': csr_sparse, 'csc': csc_sparse}[format] return cls((data, indices, indptr), (n, n)) elif format == 'coo': idx_dtype = get_index_dtype(maxval=n) row = np.arange(n, dtype=idx_dtype) col = np.arange(n, dtype=idx_dtype) data = np.ones(n, dtype=dtype) return coo_sparse((data, (row, col)), (n, n)) data = np.ones((1, max(0, min(m + k, n))), dtype=dtype) return diags_sparse(data, offsets=[k], shape=(m, n), dtype=dtype).asformat(format) def eye(m, n=None, k=0, dtype=float, format=None): """Sparse matrix with ones on diagonal Returns a sparse matrix (m x n) where the kth diagonal is all ones and everything else is zeros. Parameters ---------- m : int Number of rows in the matrix. n : int, optional Number of columns. Default: `m`. k : int, optional Diagonal to place ones on. Default: 0 (main diagonal). dtype : dtype, optional Data type of the matrix. format : str, optional Sparse format of the result, e.g., format="csr", etc. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``eye_array`` to take advantage of the sparse array functionality. Examples -------- >>> import numpy as np >>> import scipy as sp >>> sp.sparse.eye(3).toarray() array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]]) >>> sp.sparse.eye(3, dtype=np.int8) """ return _eye(m, n, k, dtype, format, False) def kron(A, B, format=None): """kronecker product of sparse matrices A and B Parameters ---------- A : sparse or dense matrix first matrix of the product B : sparse or dense matrix second matrix of the product format : str, optional (default: 'bsr' or 'coo') format of the result (e.g. "csr") If None, choose 'bsr' for relatively dense array and 'coo' for others Returns ------- kronecker product in a sparse format. Returns a sparse matrix unless either A or B is a sparse array in which case returns a sparse array. Examples -------- >>> import numpy as np >>> import scipy as sp >>> A = sp.sparse.csr_array(np.array([[0, 2], [5, 0]])) >>> B = sp.sparse.csr_array(np.array([[1, 2], [3, 4]])) >>> sp.sparse.kron(A, B).toarray() array([[ 0, 0, 2, 4], [ 0, 0, 6, 8], [ 5, 10, 0, 0], [15, 20, 0, 0]]) >>> sp.sparse.kron(A, [[1, 2], [3, 4]]).toarray() array([[ 0, 0, 2, 4], [ 0, 0, 6, 8], [ 5, 10, 0, 0], [15, 20, 0, 0]]) """ # TODO: delete next 10 lines and replace _sparse with _array when spmatrix removed if isinstance(A, sparray) or isinstance(B, sparray): # convert to local variables bsr_sparse = bsr_array csr_sparse = csr_array coo_sparse = coo_array else: # use spmatrix bsr_sparse = bsr_matrix csr_sparse = csr_matrix coo_sparse = coo_matrix B = coo_sparse(B) if B.ndim != 2: raise ValueError(f"kron requires 2D input arrays. `B` is {B.ndim}D.") # B is fairly dense, use BSR if (format is None or format == "bsr") and 2*B.nnz >= B.shape[0] * B.shape[1]: A = csr_sparse(A,copy=True) if A.ndim != 2: raise ValueError(f"kron requires 2D input arrays. `A` is {A.ndim}D.") output_shape = (A.shape[0]*B.shape[0], A.shape[1]*B.shape[1]) if A.nnz == 0 or B.nnz == 0: # kronecker product is the zero matrix return coo_sparse(output_shape).asformat(format) B = B.toarray() data = A.data.repeat(B.size).reshape(-1,B.shape[0],B.shape[1]) data = data * B return bsr_sparse((data,A.indices,A.indptr), shape=output_shape) else: # use COO A = coo_sparse(A) if A.ndim != 2: raise ValueError(f"kron requires 2D input arrays. `A` is {A.ndim}D.") output_shape = (A.shape[0]*B.shape[0], A.shape[1]*B.shape[1]) if A.nnz == 0 or B.nnz == 0: # kronecker product is the zero matrix return coo_sparse(output_shape).asformat(format) # expand entries of a into blocks row = A.row.repeat(B.nnz) col = A.col.repeat(B.nnz) data = A.data.repeat(B.nnz) if max(A.shape[0]*B.shape[0], A.shape[1]*B.shape[1]) > np.iinfo('int32').max: row = row.astype(np.int64) col = col.astype(np.int64) row *= B.shape[0] col *= B.shape[1] # increment block indices row,col = row.reshape(-1,B.nnz),col.reshape(-1,B.nnz) row += B.row col += B.col row,col = row.reshape(-1),col.reshape(-1) # compute block entries data = data.reshape(-1,B.nnz) * B.data data = data.reshape(-1) return coo_sparse((data,(row,col)), shape=output_shape).asformat(format) def kronsum(A, B, format=None): """kronecker sum of square sparse matrices A and B Kronecker sum of two sparse matrices is a sum of two Kronecker products kron(I_n,A) + kron(B,I_m) where A has shape (m,m) and B has shape (n,n) and I_m and I_n are identity matrices of shape (m,m) and (n,n), respectively. Parameters ---------- A square matrix B square matrix format : str format of the result (e.g. "csr") Returns ------- kronecker sum in a sparse matrix format """ # TODO: delete next 8 lines and replace _sparse with _array when spmatrix removed if isinstance(A, sparray) or isinstance(B, sparray): # convert to local variables coo_sparse = coo_array identity_sparse = eye_array else: coo_sparse = coo_matrix identity_sparse = identity A = coo_sparse(A) B = coo_sparse(B) if A.ndim != 2: raise ValueError(f"kronsum requires 2D inputs. `A` is {A.ndim}D.") if B.ndim != 2: raise ValueError(f"kronsum requires 2D inputs. `B` is {B.ndim}D.") if A.shape[0] != A.shape[1]: raise ValueError('A is not square') if B.shape[0] != B.shape[1]: raise ValueError('B is not square') dtype = upcast(A.dtype, B.dtype) I_n = identity_sparse(A.shape[0], dtype=dtype) I_m = identity_sparse(B.shape[0], dtype=dtype) L = kron(I_m, A, format='coo') R = kron(B, I_n, format='coo') return (L + R).asformat(format) def _compressed_sparse_stack(blocks, axis, return_spmatrix): """ Stacking fast path for CSR/CSC matrices or arrays (i) vstack for CSR, (ii) hstack for CSC. """ other_axis = 1 if axis == 0 else 0 data = np.concatenate([b.data for b in blocks]) constant_dim = blocks[0]._shape_as_2d[other_axis] idx_dtype = get_index_dtype(arrays=[b.indptr for b in blocks], maxval=max(data.size, constant_dim)) indices = np.empty(data.size, dtype=idx_dtype) indptr = np.empty(sum(b._shape_as_2d[axis] for b in blocks) + 1, dtype=idx_dtype) last_indptr = idx_dtype(0) sum_dim = 0 sum_indices = 0 for b in blocks: if b._shape_as_2d[other_axis] != constant_dim: raise ValueError(f'incompatible dimensions for axis {other_axis}') indices[sum_indices:sum_indices+b.indices.size] = b.indices sum_indices += b.indices.size idxs = slice(sum_dim, sum_dim + b._shape_as_2d[axis]) indptr[idxs] = b.indptr[:-1] indptr[idxs] += last_indptr sum_dim += b._shape_as_2d[axis] last_indptr += b.indptr[-1] indptr[-1] = last_indptr # TODO remove this if-structure when sparse matrices removed if return_spmatrix: if axis == 0: return csr_matrix((data, indices, indptr), shape=(sum_dim, constant_dim)) else: return csc_matrix((data, indices, indptr), shape=(constant_dim, sum_dim)) if axis == 0: return csr_array((data, indices, indptr), shape=(sum_dim, constant_dim)) else: return csc_array((data, indices, indptr), shape=(constant_dim, sum_dim)) def _stack_along_minor_axis(blocks, axis): """ Stacking fast path for CSR/CSC matrices along the minor axis (i) hstack for CSR, (ii) vstack for CSC. """ n_blocks = len(blocks) if n_blocks == 0: raise ValueError('Missing block matrices') if n_blocks == 1: return blocks[0] # check for incompatible dimensions other_axis = 1 if axis == 0 else 0 other_axis_dims = {b._shape_as_2d[other_axis] for b in blocks} if len(other_axis_dims) > 1: raise ValueError(f'Mismatching dimensions along axis {other_axis}: ' f'{other_axis_dims}') constant_dim, = other_axis_dims # Do the stacking indptr_list = [b.indptr for b in blocks] data_cat = np.concatenate([b.data for b in blocks]) # Need to check if any indices/indptr, would be too large post- # concatenation for np.int32: # - The max value of indices is the output array's stacking-axis length - 1 # - The max value in indptr is the number of non-zero entries. This is # exceedingly unlikely to require int64, but is checked out of an # abundance of caution. sum_dim = sum(b._shape_as_2d[axis] for b in blocks) nnz = sum(len(b.indices) for b in blocks) idx_dtype = get_index_dtype(maxval=max(sum_dim - 1, nnz)) stack_dim_cat = np.array([b._shape_as_2d[axis] for b in blocks], dtype=idx_dtype) if data_cat.size > 0: indptr_cat = np.concatenate(indptr_list).astype(idx_dtype) indices_cat = (np.concatenate([b.indices for b in blocks]) .astype(idx_dtype)) indptr = np.empty(constant_dim + 1, dtype=idx_dtype) indices = np.empty_like(indices_cat) data = np.empty_like(data_cat) csr_hstack(n_blocks, constant_dim, stack_dim_cat, indptr_cat, indices_cat, data_cat, indptr, indices, data) else: indptr = np.zeros(constant_dim + 1, dtype=idx_dtype) indices = np.empty(0, dtype=idx_dtype) data = np.empty(0, dtype=data_cat.dtype) if axis == 0: return blocks[0]._csc_container((data, indices, indptr), shape=(sum_dim, constant_dim)) else: return blocks[0]._csr_container((data, indices, indptr), shape=(constant_dim, sum_dim)) def hstack(blocks, format=None, dtype=None): """ Stack sparse matrices horizontally (column wise) Parameters ---------- blocks sequence of sparse matrices with compatible shapes format : str sparse format of the result (e.g., "csr") by default an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output matrix. If not given, the dtype is determined from that of `blocks`. Returns ------- new_array : sparse matrix or array If any block in blocks is a sparse array, return a sparse array. Otherwise return a sparse matrix. If you want a sparse array built from blocks that are not sparse arrays, use `block(hstack(blocks))` or convert one block e.g. `blocks[0] = csr_array(blocks[0])`. See Also -------- vstack : stack sparse matrices vertically (row wise) Examples -------- >>> from scipy.sparse import coo_matrix, hstack >>> A = coo_matrix([[1, 2], [3, 4]]) >>> B = coo_matrix([[5], [6]]) >>> hstack([A,B]).toarray() array([[1, 2, 5], [3, 4, 6]]) """ blocks = np.asarray(blocks, dtype='object') if any(isinstance(b, sparray) for b in blocks.flat): return _block([blocks], format, dtype) else: return _block([blocks], format, dtype, return_spmatrix=True) def vstack(blocks, format=None, dtype=None): """ Stack sparse arrays vertically (row wise) Parameters ---------- blocks sequence of sparse arrays with compatible shapes format : str, optional sparse format of the result (e.g., "csr") by default an appropriate sparse array format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output array. If not given, the dtype is determined from that of `blocks`. Returns ------- new_array : sparse matrix or array If any block in blocks is a sparse array, return a sparse array. Otherwise return a sparse matrix. If you want a sparse array built from blocks that are not sparse arrays, use `block(vstack(blocks))` or convert one block e.g. `blocks[0] = csr_array(blocks[0])`. See Also -------- hstack : stack sparse matrices horizontally (column wise) Examples -------- >>> from scipy.sparse import coo_array, vstack >>> A = coo_array([[1, 2], [3, 4]]) >>> B = coo_array([[5, 6]]) >>> vstack([A, B]).toarray() array([[1, 2], [3, 4], [5, 6]]) """ blocks = np.asarray(blocks, dtype='object') if any(isinstance(b, sparray) for b in blocks.flat): return _block([[b] for b in blocks], format, dtype) else: return _block([[b] for b in blocks], format, dtype, return_spmatrix=True) def bmat(blocks, format=None, dtype=None): """ Build a sparse array or matrix from sparse sub-blocks Note: `block_array` is preferred over `bmat`. They are the same function except that `bmat` can return a deprecated sparse matrix. `bmat` returns a coo_matrix if none of the inputs are a sparse array. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``block_array`` to take advantage of the sparse array functionality. Parameters ---------- blocks : array_like Grid of sparse matrices with compatible shapes. An entry of None implies an all-zero matrix. format : {'bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil'}, optional The sparse format of the result (e.g. "csr"). By default an appropriate sparse matrix format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output matrix. If not given, the dtype is determined from that of `blocks`. Returns ------- bmat : sparse matrix or array If any block in blocks is a sparse array, return a sparse array. Otherwise return a sparse matrix. If you want a sparse array built from blocks that are not sparse arrays, use `block_array()`. See Also -------- block_array Examples -------- >>> from scipy.sparse import coo_array, bmat >>> A = coo_array([[1, 2], [3, 4]]) >>> B = coo_array([[5], [6]]) >>> C = coo_array([[7]]) >>> bmat([[A, B], [None, C]]).toarray() array([[1, 2, 5], [3, 4, 6], [0, 0, 7]]) >>> bmat([[A, None], [None, C]]).toarray() array([[1, 2, 0], [3, 4, 0], [0, 0, 7]]) """ blocks = np.asarray(blocks, dtype='object') if any(isinstance(b, sparray) for b in blocks.flat): return _block(blocks, format, dtype) else: return _block(blocks, format, dtype, return_spmatrix=True) def block_array(blocks, *, format=None, dtype=None): """ Build a sparse array from sparse sub-blocks Parameters ---------- blocks : array_like Grid of sparse arrays with compatible shapes. An entry of None implies an all-zero array. format : {'bsr', 'coo', 'csc', 'csr', 'dia', 'dok', 'lil'}, optional The sparse format of the result (e.g. "csr"). By default an appropriate sparse array format is returned. This choice is subject to change. dtype : dtype, optional The data-type of the output array. If not given, the dtype is determined from that of `blocks`. Returns ------- block : sparse array See Also -------- block_diag : specify blocks along the main diagonals diags : specify (possibly offset) diagonals Examples -------- >>> from scipy.sparse import coo_array, block_array >>> A = coo_array([[1, 2], [3, 4]]) >>> B = coo_array([[5], [6]]) >>> C = coo_array([[7]]) >>> block_array([[A, B], [None, C]]).toarray() array([[1, 2, 5], [3, 4, 6], [0, 0, 7]]) >>> block_array([[A, None], [None, C]]).toarray() array([[1, 2, 0], [3, 4, 0], [0, 0, 7]]) """ return _block(blocks, format, dtype) def _block(blocks, format, dtype, return_spmatrix=False): blocks = np.asarray(blocks, dtype='object') if blocks.ndim != 2: raise ValueError('blocks must be 2-D') M,N = blocks.shape # check for fast path cases if (format in (None, 'csr') and all(issparse(b) and b.format == 'csr' for b in blocks.flat) ): if N > 1: # stack along columns (axis 1): must have shape (M, 1) blocks = [[_stack_along_minor_axis(blocks[b, :], 1)] for b in range(M)] blocks = np.asarray(blocks, dtype='object') # stack along rows (axis 0): A = _compressed_sparse_stack(blocks[:, 0], 0, return_spmatrix) if dtype is not None: A = A.astype(dtype) return A elif (format in (None, 'csc') and all(issparse(b) and b.format == 'csc' for b in blocks.flat) ): if M > 1: # stack along rows (axis 0): must have shape (1, N) blocks = [[_stack_along_minor_axis(blocks[:, b], 0) for b in range(N)]] blocks = np.asarray(blocks, dtype='object') # stack along columns (axis 1): A = _compressed_sparse_stack(blocks[0, :], 1, return_spmatrix) if dtype is not None: A = A.astype(dtype) return A block_mask = np.zeros(blocks.shape, dtype=bool) brow_lengths = np.zeros(M, dtype=np.int64) bcol_lengths = np.zeros(N, dtype=np.int64) # convert everything to COO format for i in range(M): for j in range(N): if blocks[i,j] is not None: A = coo_array(blocks[i,j]) blocks[i,j] = A block_mask[i,j] = True if brow_lengths[i] == 0: brow_lengths[i] = A._shape_as_2d[0] elif brow_lengths[i] != A._shape_as_2d[0]: msg = (f'blocks[{i},:] has incompatible row dimensions. ' f'Got blocks[{i},{j}].shape[0] == {A._shape_as_2d[0]}, ' f'expected {brow_lengths[i]}.') raise ValueError(msg) if bcol_lengths[j] == 0: bcol_lengths[j] = A._shape_as_2d[1] elif bcol_lengths[j] != A._shape_as_2d[1]: msg = (f'blocks[:,{j}] has incompatible column ' f'dimensions. ' f'Got blocks[{i},{j}].shape[1] == {A._shape_as_2d[1]}, ' f'expected {bcol_lengths[j]}.') raise ValueError(msg) nnz = sum(block.nnz for block in blocks[block_mask]) if dtype is None: all_dtypes = [blk.dtype for blk in blocks[block_mask]] dtype = upcast(*all_dtypes) if all_dtypes else None row_offsets = np.append(0, np.cumsum(brow_lengths)) col_offsets = np.append(0, np.cumsum(bcol_lengths)) shape = (row_offsets[-1], col_offsets[-1]) data = np.empty(nnz, dtype=dtype) idx_dtype = get_index_dtype(maxval=max(shape)) row = np.empty(nnz, dtype=idx_dtype) col = np.empty(nnz, dtype=idx_dtype) nnz = 0 ii, jj = np.nonzero(block_mask) for i, j in zip(ii, jj): B = blocks[i, j] idx = slice(nnz, nnz + B.nnz) data[idx] = B.data np.add(B.row, row_offsets[i], out=row[idx], dtype=idx_dtype) np.add(B.col, col_offsets[j], out=col[idx], dtype=idx_dtype) nnz += B.nnz if return_spmatrix: return coo_matrix((data, (row, col)), shape=shape).asformat(format) return coo_array((data, (row, col)), shape=shape).asformat(format) def block_diag(mats, format=None, dtype=None): """ Build a block diagonal sparse matrix or array from provided matrices. Parameters ---------- mats : sequence of matrices or arrays Input matrices or arrays. format : str, optional The sparse format of the result (e.g., "csr"). If not given, the result is returned in "coo" format. dtype : dtype specifier, optional The data-type of the output. If not given, the dtype is determined from that of `blocks`. Returns ------- res : sparse matrix or array If at least one input is a sparse array, the output is a sparse array. Otherwise the output is a sparse matrix. Notes ----- .. versionadded:: 0.11.0 See Also -------- block_array diags_array Examples -------- >>> from scipy.sparse import coo_array, block_diag >>> A = coo_array([[1, 2], [3, 4]]) >>> B = coo_array([[5], [6]]) >>> C = coo_array([[7]]) >>> block_diag((A, B, C)).toarray() array([[1, 2, 0, 0], [3, 4, 0, 0], [0, 0, 5, 0], [0, 0, 6, 0], [0, 0, 0, 7]]) """ if any(isinstance(a, sparray) for a in mats): container = coo_array else: container = coo_matrix row = [] col = [] data = [] r_idx = 0 c_idx = 0 for a in mats: if isinstance(a, (list, numbers.Number)): a = coo_array(np.atleast_2d(a)) if issparse(a): a = a.tocoo() nrows, ncols = a._shape_as_2d row.append(a.row + r_idx) col.append(a.col + c_idx) data.append(a.data) else: nrows, ncols = a.shape a_row, a_col = np.divmod(np.arange(nrows*ncols), ncols) row.append(a_row + r_idx) col.append(a_col + c_idx) data.append(a.ravel()) r_idx += nrows c_idx += ncols row = np.concatenate(row) col = np.concatenate(col) data = np.concatenate(data) return container((data, (row, col)), shape=(r_idx, c_idx), dtype=dtype).asformat(format) def random_array(shape, *, density=0.01, format='coo', dtype=None, random_state=None, data_sampler=None): """Return a sparse array of uniformly random numbers in [0, 1) Returns a sparse array with the given shape and density where values are generated uniformly randomly in the range [0, 1). .. warning:: Since numpy 1.17, passing a ``np.random.Generator`` (e.g. ``np.random.default_rng``) for ``random_state`` will lead to much faster execution times. A much slower implementation is used by default for backwards compatibility. Parameters ---------- shape : int or tuple of ints shape of the array density : real, optional (default: 0.01) density of the generated matrix: density equal to one means a full matrix, density of 0 means a matrix with no non-zero items. format : str, optional (default: 'coo') sparse matrix format. dtype : dtype, optional (default: np.float64) type of the returned matrix values. random_state : {None, int, `Generator`, `RandomState`}, optional A random number generator to determine nonzero structure. We recommend using a `numpy.random.Generator` manually provided for every call as it is much faster than RandomState. - If `None` (or `np.random`), the `numpy.random.RandomState` singleton is used. - If an int, a new ``Generator`` instance is used, seeded with the int. - If a ``Generator`` or ``RandomState`` instance then that instance is used. This random state will be used for sampling `indices` (the sparsity structure), and by default for the data values too (see `data_sampler`). data_sampler : callable, optional (default depends on dtype) Sampler of random data values with keyword arg `size`. This function should take a single keyword argument `size` specifying the length of its returned ndarray. It is used to generate the nonzero values in the matrix after the locations of those values are chosen. By default, uniform [0, 1) random values are used unless `dtype` is an integer (default uniform integers from that dtype) or complex (default uniform over the unit square in the complex plane). For these, the `random_state` rng is used e.g. `rng.uniform(size=size)`. Returns ------- res : sparse array Examples -------- Passing a ``np.random.Generator`` instance for better performance: >>> import numpy as np >>> import scipy as sp >>> rng = np.random.default_rng() Default sampling uniformly from [0, 1): >>> S = sp.sparse.random_array((3, 4), density=0.25, random_state=rng) Providing a sampler for the values: >>> rvs = sp.stats.poisson(25, loc=10).rvs >>> S = sp.sparse.random_array((3, 4), density=0.25, ... random_state=rng, data_sampler=rvs) >>> S.toarray() array([[ 36., 0., 33., 0.], # random [ 0., 0., 0., 0.], [ 0., 0., 36., 0.]]) Building a custom distribution. This example builds a squared normal from np.random: >>> def np_normal_squared(size=None, random_state=rng): ... return random_state.standard_normal(size) ** 2 >>> S = sp.sparse.random_array((3, 4), density=0.25, random_state=rng, ... data_sampler=np_normal_squared) Or we can build it from sp.stats style rvs functions: >>> def sp_stats_normal_squared(size=None, random_state=rng): ... std_normal = sp.stats.distributions.norm_gen().rvs ... return std_normal(size=size, random_state=random_state) ** 2 >>> S = sp.sparse.random_array((3, 4), density=0.25, random_state=rng, ... data_sampler=sp_stats_normal_squared) Or we can subclass sp.stats rv_continous or rv_discrete: >>> class NormalSquared(sp.stats.rv_continuous): ... def _rvs(self, size=None, random_state=rng): ... return random_state.standard_normal(size) ** 2 >>> X = NormalSquared() >>> Y = X().rvs >>> S = sp.sparse.random_array((3, 4), density=0.25, ... random_state=rng, data_sampler=Y) """ # Use the more efficient RNG by default. if random_state is None: random_state = np.random.default_rng() data, ind = _random(shape, density, format, dtype, random_state, data_sampler) return coo_array((data, ind), shape=shape).asformat(format) def _random(shape, density=0.01, format=None, dtype=None, random_state=None, data_sampler=None): if density < 0 or density > 1: raise ValueError("density expected to be 0 <= density <= 1") tot_prod = math.prod(shape) # use `math` for when prod is >= 2**64 # Number of non zero values size = int(round(density * tot_prod)) rng = check_random_state(random_state) if data_sampler is None: if np.issubdtype(dtype, np.integer): def data_sampler(size): return rng_integers(rng, np.iinfo(dtype).min, np.iinfo(dtype).max, size, dtype=dtype) elif np.issubdtype(dtype, np.complexfloating): def data_sampler(size): return (rng.uniform(size=size) + rng.uniform(size=size) * 1j) else: data_sampler = rng.uniform # rng.choice uses int64 if first arg is an int if tot_prod < np.iinfo(np.int64).max: raveled_ind = rng.choice(tot_prod, size=size, replace=False) ind = np.unravel_index(raveled_ind, shape=shape, order='F') else: # for ravel indices bigger than dtype max, use sets to remove duplicates ndim = len(shape) seen = set() while len(seen) < size: dsize = size - len(seen) seen.update(map(tuple, rng_integers(rng, shape, size=(dsize, ndim)))) ind = tuple(np.array(list(seen)).T) # size kwarg allows eg data_sampler=partial(np.random.poisson, lam=5) vals = data_sampler(size=size).astype(dtype, copy=False) return vals, ind def random(m, n, density=0.01, format='coo', dtype=None, random_state=None, data_rvs=None): """Generate a sparse matrix of the given shape and density with randomly distributed values. .. warning:: Since numpy 1.17, passing a ``np.random.Generator`` (e.g. ``np.random.default_rng``) for ``random_state`` will lead to much faster execution times. A much slower implementation is used by default for backwards compatibility. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``random_array`` to take advantage of the sparse array functionality. Parameters ---------- m, n : int shape of the matrix density : real, optional density of the generated matrix: density equal to one means a full matrix, density of 0 means a matrix with no non-zero items. format : str, optional sparse matrix format. dtype : dtype, optional type of the returned matrix values. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional - If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. - If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. - If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. This random state will be used for sampling the sparsity structure, but not necessarily for sampling the values of the structurally nonzero entries of the matrix. data_rvs : callable, optional Samples a requested number of random values. This function should take a single argument specifying the length of the ndarray that it will return. The structurally nonzero entries of the sparse random matrix will be taken from the array sampled by this function. By default, uniform [0, 1) random values will be sampled using the same random state as is used for sampling the sparsity structure. Returns ------- res : sparse matrix See Also -------- random_array : constructs sparse arrays instead of sparse matrices Examples -------- Passing a ``np.random.Generator`` instance for better performance: >>> import scipy as sp >>> import numpy as np >>> rng = np.random.default_rng() >>> S = sp.sparse.random(3, 4, density=0.25, random_state=rng) Providing a sampler for the values: >>> rvs = sp.stats.poisson(25, loc=10).rvs >>> S = sp.sparse.random(3, 4, density=0.25, random_state=rng, data_rvs=rvs) >>> S.toarray() array([[ 36., 0., 33., 0.], # random [ 0., 0., 0., 0.], [ 0., 0., 36., 0.]]) Building a custom distribution. This example builds a squared normal from np.random: >>> def np_normal_squared(size=None, random_state=rng): ... return random_state.standard_normal(size) ** 2 >>> S = sp.sparse.random(3, 4, density=0.25, random_state=rng, ... data_rvs=np_normal_squared) Or we can build it from sp.stats style rvs functions: >>> def sp_stats_normal_squared(size=None, random_state=rng): ... std_normal = sp.stats.distributions.norm_gen().rvs ... return std_normal(size=size, random_state=random_state) ** 2 >>> S = sp.sparse.random(3, 4, density=0.25, random_state=rng, ... data_rvs=sp_stats_normal_squared) Or we can subclass sp.stats rv_continous or rv_discrete: >>> class NormalSquared(sp.stats.rv_continuous): ... def _rvs(self, size=None, random_state=rng): ... return random_state.standard_normal(size) ** 2 >>> X = NormalSquared() >>> Y = X() # get a frozen version of the distribution >>> S = sp.sparse.random(3, 4, density=0.25, random_state=rng, data_rvs=Y.rvs) """ if n is None: n = m m, n = int(m), int(n) # make keyword syntax work for data_rvs e.g. data_rvs(size=7) if data_rvs is not None: def data_rvs_kw(size): return data_rvs(size) else: data_rvs_kw = None vals, ind = _random((m, n), density, format, dtype, random_state, data_rvs_kw) return coo_matrix((vals, ind), shape=(m, n)).asformat(format) def rand(m, n, density=0.01, format="coo", dtype=None, random_state=None): """Generate a sparse matrix of the given shape and density with uniformly distributed values. .. warning:: This function returns a sparse matrix -- not a sparse array. You are encouraged to use ``random_array`` to take advantage of the sparse array functionality. Parameters ---------- m, n : int shape of the matrix density : real, optional density of the generated matrix: density equal to one means a full matrix, density of 0 means a matrix with no non-zero items. format : str, optional sparse matrix format. dtype : dtype, optional type of the returned matrix values. random_state : {None, int, `numpy.random.Generator`, `numpy.random.RandomState`}, optional If `seed` is None (or `np.random`), the `numpy.random.RandomState` singleton is used. If `seed` is an int, a new ``RandomState`` instance is used, seeded with `seed`. If `seed` is already a ``Generator`` or ``RandomState`` instance then that instance is used. Returns ------- res : sparse matrix Notes ----- Only float types are supported for now. See Also -------- random : Similar function allowing a custom random data sampler random_array : Similar to random() but returns a sparse array Examples -------- >>> from scipy.sparse import rand >>> matrix = rand(3, 4, density=0.25, format="csr", random_state=42) >>> matrix >>> matrix.toarray() array([[0.05641158, 0. , 0. , 0.65088847], # random [0. , 0. , 0. , 0.14286682], [0. , 0. , 0. , 0. ]]) """ return random(m, n, density, format, dtype, random_state)