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1296 lines
49 KiB
Python
1296 lines
49 KiB
Python
"""Base class for sparse matrix formats using compressed storage."""
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__all__ = []
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from warnings import warn
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import operator
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import numpy as np
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from scipy._lib._util import _prune_array
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from .base import spmatrix, isspmatrix, SparseEfficiencyWarning
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from .data import _data_matrix, _minmax_mixin
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from .dia import dia_matrix
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from . import _sparsetools
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from ._sparsetools import (get_csr_submatrix, csr_sample_offsets, csr_todense,
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csr_sample_values, csr_row_index, csr_row_slice,
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csr_column_index1, csr_column_index2)
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from ._index import IndexMixin
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from .sputils import (upcast, upcast_char, to_native, isdense, isshape,
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getdtype, isscalarlike, isintlike, get_index_dtype,
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downcast_intp_index, get_sum_dtype, check_shape,
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matrix, asmatrix, is_pydata_spmatrix)
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class _cs_matrix(_data_matrix, _minmax_mixin, IndexMixin):
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"""base matrix class for compressed row- and column-oriented matrices"""
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def __init__(self, arg1, shape=None, dtype=None, copy=False):
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_data_matrix.__init__(self)
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if isspmatrix(arg1):
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if arg1.format == self.format and copy:
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arg1 = arg1.copy()
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else:
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arg1 = arg1.asformat(self.format)
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self._set_self(arg1)
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elif isinstance(arg1, tuple):
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if isshape(arg1):
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# It's a tuple of matrix dimensions (M, N)
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# create empty matrix
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self._shape = check_shape(arg1)
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M, N = self.shape
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# Select index dtype large enough to pass array and
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# scalar parameters to sparsetools
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idx_dtype = get_index_dtype(maxval=max(M, N))
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self.data = np.zeros(0, getdtype(dtype, default=float))
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self.indices = np.zeros(0, idx_dtype)
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self.indptr = np.zeros(self._swap((M, N))[0] + 1,
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dtype=idx_dtype)
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else:
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if len(arg1) == 2:
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# (data, ij) format
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from .coo import coo_matrix
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other = self.__class__(coo_matrix(arg1, shape=shape))
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self._set_self(other)
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elif len(arg1) == 3:
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# (data, indices, indptr) format
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(data, indices, indptr) = arg1
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# Select index dtype large enough to pass array and
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# scalar parameters to sparsetools
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maxval = None
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if shape is not None:
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maxval = max(shape)
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idx_dtype = get_index_dtype((indices, indptr),
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maxval=maxval,
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check_contents=True)
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self.indices = np.array(indices, copy=copy,
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dtype=idx_dtype)
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self.indptr = np.array(indptr, copy=copy, dtype=idx_dtype)
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self.data = np.array(data, copy=copy, dtype=dtype)
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else:
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raise ValueError("unrecognized {}_matrix "
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"constructor usage".format(self.format))
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else:
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# must be dense
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try:
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arg1 = np.asarray(arg1)
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except Exception as e:
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raise ValueError("unrecognized {}_matrix constructor usage"
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"".format(self.format)) from e
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from .coo import coo_matrix
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self._set_self(self.__class__(coo_matrix(arg1, dtype=dtype)))
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# Read matrix dimensions given, if any
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if shape is not None:
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self._shape = check_shape(shape)
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else:
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if self.shape is None:
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# shape not already set, try to infer dimensions
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try:
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major_dim = len(self.indptr) - 1
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minor_dim = self.indices.max() + 1
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except Exception as e:
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raise ValueError('unable to infer matrix dimensions') from e
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else:
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self._shape = check_shape(self._swap((major_dim,
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minor_dim)))
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if dtype is not None:
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self.data = self.data.astype(dtype, copy=False)
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self.check_format(full_check=False)
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def getnnz(self, axis=None):
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if axis is None:
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return int(self.indptr[-1])
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else:
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if axis < 0:
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axis += 2
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axis, _ = self._swap((axis, 1 - axis))
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_, N = self._swap(self.shape)
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if axis == 0:
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return np.bincount(downcast_intp_index(self.indices),
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minlength=N)
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elif axis == 1:
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return np.diff(self.indptr)
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raise ValueError('axis out of bounds')
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getnnz.__doc__ = spmatrix.getnnz.__doc__
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def _set_self(self, other, copy=False):
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"""take the member variables of other and assign them to self"""
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if copy:
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other = other.copy()
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self.data = other.data
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self.indices = other.indices
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self.indptr = other.indptr
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self._shape = check_shape(other.shape)
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def check_format(self, full_check=True):
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"""check whether the matrix format is valid
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Parameters
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----------
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full_check : bool, optional
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If `True`, rigorous check, O(N) operations. Otherwise
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basic check, O(1) operations (default True).
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"""
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# use _swap to determine proper bounds
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major_name, minor_name = self._swap(('row', 'column'))
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major_dim, minor_dim = self._swap(self.shape)
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# index arrays should have integer data types
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if self.indptr.dtype.kind != 'i':
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warn("indptr array has non-integer dtype ({})"
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"".format(self.indptr.dtype.name), stacklevel=3)
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if self.indices.dtype.kind != 'i':
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warn("indices array has non-integer dtype ({})"
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"".format(self.indices.dtype.name), stacklevel=3)
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idx_dtype = get_index_dtype((self.indptr, self.indices))
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self.indptr = np.asarray(self.indptr, dtype=idx_dtype)
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self.indices = np.asarray(self.indices, dtype=idx_dtype)
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self.data = to_native(self.data)
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# check array shapes
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for x in [self.data.ndim, self.indices.ndim, self.indptr.ndim]:
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if x != 1:
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raise ValueError('data, indices, and indptr should be 1-D')
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# check index pointer
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if (len(self.indptr) != major_dim + 1):
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raise ValueError("index pointer size ({}) should be ({})"
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"".format(len(self.indptr), major_dim + 1))
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if (self.indptr[0] != 0):
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raise ValueError("index pointer should start with 0")
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# check index and data arrays
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if (len(self.indices) != len(self.data)):
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raise ValueError("indices and data should have the same size")
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if (self.indptr[-1] > len(self.indices)):
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raise ValueError("Last value of index pointer should be less than "
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"the size of index and data arrays")
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self.prune()
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if full_check:
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# check format validity (more expensive)
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if self.nnz > 0:
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if self.indices.max() >= minor_dim:
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raise ValueError("{} index values must be < {}"
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"".format(minor_name, minor_dim))
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if self.indices.min() < 0:
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raise ValueError("{} index values must be >= 0"
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"".format(minor_name))
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if np.diff(self.indptr).min() < 0:
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raise ValueError("index pointer values must form a "
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"non-decreasing sequence")
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# if not self.has_sorted_indices():
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# warn('Indices were not in sorted order. Sorting indices.')
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# self.sort_indices()
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# assert(self.has_sorted_indices())
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# TODO check for duplicates?
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#######################
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# Boolean comparisons #
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#######################
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def _scalar_binopt(self, other, op):
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"""Scalar version of self._binopt, for cases in which no new nonzeros
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are added. Produces a new spmatrix in canonical form.
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"""
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self.sum_duplicates()
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res = self._with_data(op(self.data, other), copy=True)
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res.eliminate_zeros()
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return res
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def __eq__(self, other):
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# Scalar other.
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if isscalarlike(other):
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if np.isnan(other):
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return self.__class__(self.shape, dtype=np.bool_)
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if other == 0:
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warn("Comparing a sparse matrix with 0 using == is inefficient"
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", try using != instead.", SparseEfficiencyWarning,
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stacklevel=3)
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all_true = self.__class__(np.ones(self.shape, dtype=np.bool_))
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inv = self._scalar_binopt(other, operator.ne)
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return all_true - inv
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else:
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return self._scalar_binopt(other, operator.eq)
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# Dense other.
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elif isdense(other):
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return self.todense() == other
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# Pydata sparse other.
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elif is_pydata_spmatrix(other):
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return NotImplemented
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# Sparse other.
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elif isspmatrix(other):
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warn("Comparing sparse matrices using == is inefficient, try using"
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" != instead.", SparseEfficiencyWarning, stacklevel=3)
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# TODO sparse broadcasting
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if self.shape != other.shape:
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return False
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elif self.format != other.format:
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other = other.asformat(self.format)
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res = self._binopt(other, '_ne_')
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all_true = self.__class__(np.ones(self.shape, dtype=np.bool_))
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return all_true - res
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else:
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return False
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def __ne__(self, other):
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# Scalar other.
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if isscalarlike(other):
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if np.isnan(other):
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warn("Comparing a sparse matrix with nan using != is"
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" inefficient", SparseEfficiencyWarning, stacklevel=3)
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all_true = self.__class__(np.ones(self.shape, dtype=np.bool_))
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return all_true
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elif other != 0:
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warn("Comparing a sparse matrix with a nonzero scalar using !="
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" is inefficient, try using == instead.",
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SparseEfficiencyWarning, stacklevel=3)
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all_true = self.__class__(np.ones(self.shape), dtype=np.bool_)
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inv = self._scalar_binopt(other, operator.eq)
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return all_true - inv
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else:
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return self._scalar_binopt(other, operator.ne)
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# Dense other.
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elif isdense(other):
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return self.todense() != other
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# Pydata sparse other.
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elif is_pydata_spmatrix(other):
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return NotImplemented
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# Sparse other.
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elif isspmatrix(other):
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# TODO sparse broadcasting
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if self.shape != other.shape:
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return True
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elif self.format != other.format:
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other = other.asformat(self.format)
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return self._binopt(other, '_ne_')
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else:
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return True
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def _inequality(self, other, op, op_name, bad_scalar_msg):
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# Scalar other.
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if isscalarlike(other):
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if 0 == other and op_name in ('_le_', '_ge_'):
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raise NotImplementedError(" >= and <= don't work with 0.")
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elif op(0, other):
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warn(bad_scalar_msg, SparseEfficiencyWarning)
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other_arr = np.empty(self.shape, dtype=np.result_type(other))
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other_arr.fill(other)
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other_arr = self.__class__(other_arr)
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return self._binopt(other_arr, op_name)
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else:
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return self._scalar_binopt(other, op)
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# Dense other.
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elif isdense(other):
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return op(self.todense(), other)
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# Sparse other.
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elif isspmatrix(other):
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# TODO sparse broadcasting
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if self.shape != other.shape:
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raise ValueError("inconsistent shapes")
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elif self.format != other.format:
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other = other.asformat(self.format)
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if op_name not in ('_ge_', '_le_'):
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return self._binopt(other, op_name)
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warn("Comparing sparse matrices using >= and <= is inefficient, "
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"using <, >, or !=, instead.", SparseEfficiencyWarning)
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all_true = self.__class__(np.ones(self.shape, dtype=np.bool_))
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res = self._binopt(other, '_gt_' if op_name == '_le_' else '_lt_')
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return all_true - res
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else:
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raise ValueError("Operands could not be compared.")
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def __lt__(self, other):
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return self._inequality(other, operator.lt, '_lt_',
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"Comparing a sparse matrix with a scalar "
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"greater than zero using < is inefficient, "
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"try using >= instead.")
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def __gt__(self, other):
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return self._inequality(other, operator.gt, '_gt_',
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"Comparing a sparse matrix with a scalar "
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"less than zero using > is inefficient, "
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"try using <= instead.")
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def __le__(self, other):
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return self._inequality(other, operator.le, '_le_',
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"Comparing a sparse matrix with a scalar "
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"greater than zero using <= is inefficient, "
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"try using > instead.")
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def __ge__(self, other):
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return self._inequality(other, operator.ge, '_ge_',
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"Comparing a sparse matrix with a scalar "
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"less than zero using >= is inefficient, "
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"try using < instead.")
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#################################
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# Arithmetic operator overrides #
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#################################
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def _add_dense(self, other):
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if other.shape != self.shape:
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raise ValueError('Incompatible shapes ({} and {})'
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.format(self.shape, other.shape))
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dtype = upcast_char(self.dtype.char, other.dtype.char)
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order = self._swap('CF')[0]
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result = np.array(other, dtype=dtype, order=order, copy=True)
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M, N = self._swap(self.shape)
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y = result if result.flags.c_contiguous else result.T
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csr_todense(M, N, self.indptr, self.indices, self.data, y)
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return matrix(result, copy=False)
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def _add_sparse(self, other):
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return self._binopt(other, '_plus_')
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def _sub_sparse(self, other):
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return self._binopt(other, '_minus_')
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def multiply(self, other):
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"""Point-wise multiplication by another matrix, vector, or
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scalar.
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"""
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# Scalar multiplication.
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if isscalarlike(other):
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return self._mul_scalar(other)
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# Sparse matrix or vector.
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if isspmatrix(other):
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if self.shape == other.shape:
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other = self.__class__(other)
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return self._binopt(other, '_elmul_')
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# Single element.
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elif other.shape == (1, 1):
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return self._mul_scalar(other.toarray()[0, 0])
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elif self.shape == (1, 1):
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return other._mul_scalar(self.toarray()[0, 0])
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# A row times a column.
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elif self.shape[1] == 1 and other.shape[0] == 1:
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return self._mul_sparse_matrix(other.tocsc())
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elif self.shape[0] == 1 and other.shape[1] == 1:
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return other._mul_sparse_matrix(self.tocsc())
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# Row vector times matrix. other is a row.
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elif other.shape[0] == 1 and self.shape[1] == other.shape[1]:
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other = dia_matrix((other.toarray().ravel(), [0]),
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shape=(other.shape[1], other.shape[1]))
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return self._mul_sparse_matrix(other)
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# self is a row.
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elif self.shape[0] == 1 and self.shape[1] == other.shape[1]:
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copy = dia_matrix((self.toarray().ravel(), [0]),
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shape=(self.shape[1], self.shape[1]))
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return other._mul_sparse_matrix(copy)
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# Column vector times matrix. other is a column.
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elif other.shape[1] == 1 and self.shape[0] == other.shape[0]:
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other = dia_matrix((other.toarray().ravel(), [0]),
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shape=(other.shape[0], other.shape[0]))
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return other._mul_sparse_matrix(self)
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# self is a column.
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elif self.shape[1] == 1 and self.shape[0] == other.shape[0]:
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copy = dia_matrix((self.toarray().ravel(), [0]),
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shape=(self.shape[0], self.shape[0]))
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return copy._mul_sparse_matrix(other)
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else:
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raise ValueError("inconsistent shapes")
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# Assume other is a dense matrix/array, which produces a single-item
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# object array if other isn't convertible to ndarray.
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other = np.atleast_2d(other)
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if other.ndim != 2:
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return np.multiply(self.toarray(), other)
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# Single element / wrapped object.
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if other.size == 1:
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return self._mul_scalar(other.flat[0])
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# Fast case for trivial sparse matrix.
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elif self.shape == (1, 1):
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return np.multiply(self.toarray()[0, 0], other)
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from .coo import coo_matrix
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ret = self.tocoo()
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# Matching shapes.
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if self.shape == other.shape:
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data = np.multiply(ret.data, other[ret.row, ret.col])
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# Sparse row vector times...
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elif self.shape[0] == 1:
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if other.shape[1] == 1: # Dense column vector.
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data = np.multiply(ret.data, other)
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elif other.shape[1] == self.shape[1]: # Dense matrix.
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data = np.multiply(ret.data, other[:, ret.col])
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else:
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raise ValueError("inconsistent shapes")
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row = np.repeat(np.arange(other.shape[0]), len(ret.row))
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col = np.tile(ret.col, other.shape[0])
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return coo_matrix((data.view(np.ndarray).ravel(), (row, col)),
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shape=(other.shape[0], self.shape[1]),
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copy=False)
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# Sparse column vector times...
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elif self.shape[1] == 1:
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if other.shape[0] == 1: # Dense row vector.
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data = np.multiply(ret.data[:, None], other)
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elif other.shape[0] == self.shape[0]: # Dense matrix.
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data = np.multiply(ret.data[:, None], other[ret.row])
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else:
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raise ValueError("inconsistent shapes")
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row = np.repeat(ret.row, other.shape[1])
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col = np.tile(np.arange(other.shape[1]), len(ret.col))
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return coo_matrix((data.view(np.ndarray).ravel(), (row, col)),
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shape=(self.shape[0], other.shape[1]),
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copy=False)
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# Sparse matrix times dense row vector.
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elif other.shape[0] == 1 and self.shape[1] == other.shape[1]:
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data = np.multiply(ret.data, other[:, ret.col].ravel())
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# Sparse matrix times dense column vector.
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elif other.shape[1] == 1 and self.shape[0] == other.shape[0]:
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data = np.multiply(ret.data, other[ret.row].ravel())
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else:
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raise ValueError("inconsistent shapes")
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ret.data = data.view(np.ndarray).ravel()
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return ret
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###########################
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# Multiplication handlers #
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###########################
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def _mul_vector(self, other):
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|
M, N = self.shape
|
|
|
|
# output array
|
|
result = np.zeros(M, dtype=upcast_char(self.dtype.char,
|
|
other.dtype.char))
|
|
|
|
# csr_matvec or csc_matvec
|
|
fn = getattr(_sparsetools, self.format + '_matvec')
|
|
fn(M, N, self.indptr, self.indices, self.data, other, result)
|
|
|
|
return result
|
|
|
|
def _mul_multivector(self, other):
|
|
M, N = self.shape
|
|
n_vecs = other.shape[1] # number of column vectors
|
|
|
|
result = np.zeros((M, n_vecs),
|
|
dtype=upcast_char(self.dtype.char, other.dtype.char))
|
|
|
|
# csr_matvecs or csc_matvecs
|
|
fn = getattr(_sparsetools, self.format + '_matvecs')
|
|
fn(M, N, n_vecs, self.indptr, self.indices, self.data,
|
|
other.ravel(), result.ravel())
|
|
|
|
return result
|
|
|
|
def _mul_sparse_matrix(self, other):
|
|
M, K1 = self.shape
|
|
K2, N = other.shape
|
|
|
|
major_axis = self._swap((M, N))[0]
|
|
other = self.__class__(other) # convert to this format
|
|
|
|
idx_dtype = get_index_dtype((self.indptr, self.indices,
|
|
other.indptr, other.indices))
|
|
|
|
fn = getattr(_sparsetools, self.format + '_matmat_maxnnz')
|
|
nnz = fn(M, N,
|
|
np.asarray(self.indptr, dtype=idx_dtype),
|
|
np.asarray(self.indices, dtype=idx_dtype),
|
|
np.asarray(other.indptr, dtype=idx_dtype),
|
|
np.asarray(other.indices, dtype=idx_dtype))
|
|
|
|
idx_dtype = get_index_dtype((self.indptr, self.indices,
|
|
other.indptr, other.indices),
|
|
maxval=nnz)
|
|
|
|
indptr = np.empty(major_axis + 1, dtype=idx_dtype)
|
|
indices = np.empty(nnz, dtype=idx_dtype)
|
|
data = np.empty(nnz, dtype=upcast(self.dtype, other.dtype))
|
|
|
|
fn = getattr(_sparsetools, self.format + '_matmat')
|
|
fn(M, N, np.asarray(self.indptr, dtype=idx_dtype),
|
|
np.asarray(self.indices, dtype=idx_dtype),
|
|
self.data,
|
|
np.asarray(other.indptr, dtype=idx_dtype),
|
|
np.asarray(other.indices, dtype=idx_dtype),
|
|
other.data,
|
|
indptr, indices, data)
|
|
|
|
return self.__class__((data, indices, indptr), shape=(M, N))
|
|
|
|
def diagonal(self, k=0):
|
|
rows, cols = self.shape
|
|
if k <= -rows or k >= cols:
|
|
return np.empty(0, dtype=self.data.dtype)
|
|
fn = getattr(_sparsetools, self.format + "_diagonal")
|
|
y = np.empty(min(rows + min(k, 0), cols - max(k, 0)),
|
|
dtype=upcast(self.dtype))
|
|
fn(k, self.shape[0], self.shape[1], self.indptr, self.indices,
|
|
self.data, y)
|
|
return y
|
|
|
|
diagonal.__doc__ = spmatrix.diagonal.__doc__
|
|
|
|
#####################
|
|
# Other binary ops #
|
|
#####################
|
|
|
|
def _maximum_minimum(self, other, npop, op_name, dense_check):
|
|
if isscalarlike(other):
|
|
if dense_check(other):
|
|
warn("Taking maximum (minimum) with > 0 (< 0) number results"
|
|
" to a dense matrix.", SparseEfficiencyWarning,
|
|
stacklevel=3)
|
|
other_arr = np.empty(self.shape, dtype=np.asarray(other).dtype)
|
|
other_arr.fill(other)
|
|
other_arr = self.__class__(other_arr)
|
|
return self._binopt(other_arr, op_name)
|
|
else:
|
|
self.sum_duplicates()
|
|
new_data = npop(self.data, np.asarray(other))
|
|
mat = self.__class__((new_data, self.indices, self.indptr),
|
|
dtype=new_data.dtype, shape=self.shape)
|
|
return mat
|
|
elif isdense(other):
|
|
return npop(self.todense(), other)
|
|
elif isspmatrix(other):
|
|
return self._binopt(other, op_name)
|
|
else:
|
|
raise ValueError("Operands not compatible.")
|
|
|
|
def maximum(self, other):
|
|
return self._maximum_minimum(other, np.maximum,
|
|
'_maximum_', lambda x: np.asarray(x) > 0)
|
|
|
|
maximum.__doc__ = spmatrix.maximum.__doc__
|
|
|
|
def minimum(self, other):
|
|
return self._maximum_minimum(other, np.minimum,
|
|
'_minimum_', lambda x: np.asarray(x) < 0)
|
|
|
|
minimum.__doc__ = spmatrix.minimum.__doc__
|
|
|
|
#####################
|
|
# Reduce operations #
|
|
#####################
|
|
|
|
def sum(self, axis=None, dtype=None, out=None):
|
|
"""Sum the matrix over the given axis. If the axis is None, sum
|
|
over both rows and columns, returning a scalar.
|
|
"""
|
|
# The spmatrix base class already does axis=0 and axis=1 efficiently
|
|
# so we only do the case axis=None here
|
|
if (not hasattr(self, 'blocksize') and
|
|
axis in self._swap(((1, -1), (0, 2)))[0]):
|
|
# faster than multiplication for large minor axis in CSC/CSR
|
|
res_dtype = get_sum_dtype(self.dtype)
|
|
ret = np.zeros(len(self.indptr) - 1, dtype=res_dtype)
|
|
|
|
major_index, value = self._minor_reduce(np.add)
|
|
ret[major_index] = value
|
|
ret = asmatrix(ret)
|
|
if axis % 2 == 1:
|
|
ret = ret.T
|
|
|
|
if out is not None and out.shape != ret.shape:
|
|
raise ValueError('dimensions do not match')
|
|
|
|
return ret.sum(axis=(), dtype=dtype, out=out)
|
|
# spmatrix will handle the remaining situations when axis
|
|
# is in {None, -1, 0, 1}
|
|
else:
|
|
return spmatrix.sum(self, axis=axis, dtype=dtype, out=out)
|
|
|
|
sum.__doc__ = spmatrix.sum.__doc__
|
|
|
|
def _minor_reduce(self, ufunc, data=None):
|
|
"""Reduce nonzeros with a ufunc over the minor axis when non-empty
|
|
|
|
Can be applied to a function of self.data by supplying data parameter.
|
|
|
|
Warning: this does not call sum_duplicates()
|
|
|
|
Returns
|
|
-------
|
|
major_index : array of ints
|
|
Major indices where nonzero
|
|
|
|
value : array of self.dtype
|
|
Reduce result for nonzeros in each major_index
|
|
"""
|
|
if data is None:
|
|
data = self.data
|
|
major_index = np.flatnonzero(np.diff(self.indptr))
|
|
value = ufunc.reduceat(data,
|
|
downcast_intp_index(self.indptr[major_index]))
|
|
return major_index, value
|
|
|
|
#######################
|
|
# Getting and Setting #
|
|
#######################
|
|
|
|
def _get_intXint(self, row, col):
|
|
M, N = self._swap(self.shape)
|
|
major, minor = self._swap((row, col))
|
|
indptr, indices, data = get_csr_submatrix(
|
|
M, N, self.indptr, self.indices, self.data,
|
|
major, major + 1, minor, minor + 1)
|
|
return data.sum(dtype=self.dtype)
|
|
|
|
def _get_sliceXslice(self, row, col):
|
|
major, minor = self._swap((row, col))
|
|
if major.step in (1, None) and minor.step in (1, None):
|
|
return self._get_submatrix(major, minor, copy=True)
|
|
return self._major_slice(major)._minor_slice(minor)
|
|
|
|
def _get_arrayXarray(self, row, col):
|
|
# inner indexing
|
|
idx_dtype = self.indices.dtype
|
|
M, N = self._swap(self.shape)
|
|
major, minor = self._swap((row, col))
|
|
major = np.asarray(major, dtype=idx_dtype)
|
|
minor = np.asarray(minor, dtype=idx_dtype)
|
|
|
|
val = np.empty(major.size, dtype=self.dtype)
|
|
csr_sample_values(M, N, self.indptr, self.indices, self.data,
|
|
major.size, major.ravel(), minor.ravel(), val)
|
|
if major.ndim == 1:
|
|
return asmatrix(val)
|
|
return self.__class__(val.reshape(major.shape))
|
|
|
|
def _get_columnXarray(self, row, col):
|
|
# outer indexing
|
|
major, minor = self._swap((row, col))
|
|
return self._major_index_fancy(major)._minor_index_fancy(minor)
|
|
|
|
def _major_index_fancy(self, idx):
|
|
"""Index along the major axis where idx is an array of ints.
|
|
"""
|
|
idx_dtype = self.indices.dtype
|
|
indices = np.asarray(idx, dtype=idx_dtype).ravel()
|
|
|
|
_, N = self._swap(self.shape)
|
|
M = len(indices)
|
|
new_shape = self._swap((M, N))
|
|
if M == 0:
|
|
return self.__class__(new_shape)
|
|
|
|
row_nnz = np.diff(self.indptr)
|
|
idx_dtype = self.indices.dtype
|
|
res_indptr = np.zeros(M+1, dtype=idx_dtype)
|
|
np.cumsum(row_nnz[idx], out=res_indptr[1:])
|
|
|
|
nnz = res_indptr[-1]
|
|
res_indices = np.empty(nnz, dtype=idx_dtype)
|
|
res_data = np.empty(nnz, dtype=self.dtype)
|
|
csr_row_index(M, indices, self.indptr, self.indices, self.data,
|
|
res_indices, res_data)
|
|
|
|
return self.__class__((res_data, res_indices, res_indptr),
|
|
shape=new_shape, copy=False)
|
|
|
|
def _major_slice(self, idx, copy=False):
|
|
"""Index along the major axis where idx is a slice object.
|
|
"""
|
|
if idx == slice(None):
|
|
return self.copy() if copy else self
|
|
|
|
M, N = self._swap(self.shape)
|
|
start, stop, step = idx.indices(M)
|
|
M = len(range(start, stop, step))
|
|
new_shape = self._swap((M, N))
|
|
if M == 0:
|
|
return self.__class__(new_shape)
|
|
|
|
row_nnz = np.diff(self.indptr)
|
|
idx_dtype = self.indices.dtype
|
|
res_indptr = np.zeros(M+1, dtype=idx_dtype)
|
|
np.cumsum(row_nnz[idx], out=res_indptr[1:])
|
|
|
|
if step == 1:
|
|
all_idx = slice(self.indptr[start], self.indptr[stop])
|
|
res_indices = np.array(self.indices[all_idx], copy=copy)
|
|
res_data = np.array(self.data[all_idx], copy=copy)
|
|
else:
|
|
nnz = res_indptr[-1]
|
|
res_indices = np.empty(nnz, dtype=idx_dtype)
|
|
res_data = np.empty(nnz, dtype=self.dtype)
|
|
csr_row_slice(start, stop, step, self.indptr, self.indices,
|
|
self.data, res_indices, res_data)
|
|
|
|
return self.__class__((res_data, res_indices, res_indptr),
|
|
shape=new_shape, copy=False)
|
|
|
|
def _minor_index_fancy(self, idx):
|
|
"""Index along the minor axis where idx is an array of ints.
|
|
"""
|
|
idx_dtype = self.indices.dtype
|
|
idx = np.asarray(idx, dtype=idx_dtype).ravel()
|
|
|
|
M, N = self._swap(self.shape)
|
|
k = len(idx)
|
|
new_shape = self._swap((M, k))
|
|
if k == 0:
|
|
return self.__class__(new_shape)
|
|
|
|
# pass 1: count idx entries and compute new indptr
|
|
col_offsets = np.zeros(N, dtype=idx_dtype)
|
|
res_indptr = np.empty_like(self.indptr)
|
|
csr_column_index1(k, idx, M, N, self.indptr, self.indices,
|
|
col_offsets, res_indptr)
|
|
|
|
# pass 2: copy indices/data for selected idxs
|
|
col_order = np.argsort(idx).astype(idx_dtype, copy=False)
|
|
nnz = res_indptr[-1]
|
|
res_indices = np.empty(nnz, dtype=idx_dtype)
|
|
res_data = np.empty(nnz, dtype=self.dtype)
|
|
csr_column_index2(col_order, col_offsets, len(self.indices),
|
|
self.indices, self.data, res_indices, res_data)
|
|
return self.__class__((res_data, res_indices, res_indptr),
|
|
shape=new_shape, copy=False)
|
|
|
|
def _minor_slice(self, idx, copy=False):
|
|
"""Index along the minor axis where idx is a slice object.
|
|
"""
|
|
if idx == slice(None):
|
|
return self.copy() if copy else self
|
|
|
|
M, N = self._swap(self.shape)
|
|
start, stop, step = idx.indices(N)
|
|
N = len(range(start, stop, step))
|
|
if N == 0:
|
|
return self.__class__(self._swap((M, N)))
|
|
if step == 1:
|
|
return self._get_submatrix(minor=idx, copy=copy)
|
|
# TODO: don't fall back to fancy indexing here
|
|
return self._minor_index_fancy(np.arange(start, stop, step))
|
|
|
|
def _get_submatrix(self, major=None, minor=None, copy=False):
|
|
"""Return a submatrix of this matrix.
|
|
|
|
major, minor: None, int, or slice with step 1
|
|
"""
|
|
M, N = self._swap(self.shape)
|
|
i0, i1 = _process_slice(major, M)
|
|
j0, j1 = _process_slice(minor, N)
|
|
|
|
if i0 == 0 and j0 == 0 and i1 == M and j1 == N:
|
|
return self.copy() if copy else self
|
|
|
|
indptr, indices, data = get_csr_submatrix(
|
|
M, N, self.indptr, self.indices, self.data, i0, i1, j0, j1)
|
|
|
|
shape = self._swap((i1 - i0, j1 - j0))
|
|
return self.__class__((data, indices, indptr), shape=shape,
|
|
dtype=self.dtype, copy=False)
|
|
|
|
def _set_intXint(self, row, col, x):
|
|
i, j = self._swap((row, col))
|
|
self._set_many(i, j, x)
|
|
|
|
def _set_arrayXarray(self, row, col, x):
|
|
i, j = self._swap((row, col))
|
|
self._set_many(i, j, x)
|
|
|
|
def _set_arrayXarray_sparse(self, row, col, x):
|
|
# clear entries that will be overwritten
|
|
self._zero_many(*self._swap((row, col)))
|
|
|
|
M, N = row.shape # matches col.shape
|
|
broadcast_row = M != 1 and x.shape[0] == 1
|
|
broadcast_col = N != 1 and x.shape[1] == 1
|
|
r, c = x.row, x.col
|
|
|
|
x = np.asarray(x.data, dtype=self.dtype)
|
|
if x.size == 0:
|
|
return
|
|
|
|
if broadcast_row:
|
|
r = np.repeat(np.arange(M), len(r))
|
|
c = np.tile(c, M)
|
|
x = np.tile(x, M)
|
|
if broadcast_col:
|
|
r = np.repeat(r, N)
|
|
c = np.tile(np.arange(N), len(c))
|
|
x = np.repeat(x, N)
|
|
# only assign entries in the new sparsity structure
|
|
i, j = self._swap((row[r, c], col[r, c]))
|
|
self._set_many(i, j, x)
|
|
|
|
def _setdiag(self, values, k):
|
|
if 0 in self.shape:
|
|
return
|
|
|
|
M, N = self.shape
|
|
broadcast = (values.ndim == 0)
|
|
|
|
if k < 0:
|
|
if broadcast:
|
|
max_index = min(M + k, N)
|
|
else:
|
|
max_index = min(M + k, N, len(values))
|
|
i = np.arange(max_index, dtype=self.indices.dtype)
|
|
j = np.arange(max_index, dtype=self.indices.dtype)
|
|
i -= k
|
|
|
|
else:
|
|
if broadcast:
|
|
max_index = min(M, N - k)
|
|
else:
|
|
max_index = min(M, N - k, len(values))
|
|
i = np.arange(max_index, dtype=self.indices.dtype)
|
|
j = np.arange(max_index, dtype=self.indices.dtype)
|
|
j += k
|
|
|
|
if not broadcast:
|
|
values = values[:len(i)]
|
|
|
|
self[i, j] = values
|
|
|
|
def _prepare_indices(self, i, j):
|
|
M, N = self._swap(self.shape)
|
|
|
|
def check_bounds(indices, bound):
|
|
idx = indices.max()
|
|
if idx >= bound:
|
|
raise IndexError('index (%d) out of range (>= %d)' %
|
|
(idx, bound))
|
|
idx = indices.min()
|
|
if idx < -bound:
|
|
raise IndexError('index (%d) out of range (< -%d)' %
|
|
(idx, bound))
|
|
|
|
i = np.array(i, dtype=self.indices.dtype, copy=False, ndmin=1).ravel()
|
|
j = np.array(j, dtype=self.indices.dtype, copy=False, ndmin=1).ravel()
|
|
check_bounds(i, M)
|
|
check_bounds(j, N)
|
|
return i, j, M, N
|
|
|
|
def _set_many(self, i, j, x):
|
|
"""Sets value at each (i, j) to x
|
|
|
|
Here (i,j) index major and minor respectively, and must not contain
|
|
duplicate entries.
|
|
"""
|
|
i, j, M, N = self._prepare_indices(i, j)
|
|
x = np.array(x, dtype=self.dtype, copy=False, ndmin=1).ravel()
|
|
|
|
n_samples = x.size
|
|
offsets = np.empty(n_samples, dtype=self.indices.dtype)
|
|
ret = csr_sample_offsets(M, N, self.indptr, self.indices, n_samples,
|
|
i, j, offsets)
|
|
if ret == 1:
|
|
# rinse and repeat
|
|
self.sum_duplicates()
|
|
csr_sample_offsets(M, N, self.indptr, self.indices, n_samples,
|
|
i, j, offsets)
|
|
|
|
if -1 not in offsets:
|
|
# only affects existing non-zero cells
|
|
self.data[offsets] = x
|
|
return
|
|
|
|
else:
|
|
warn("Changing the sparsity structure of a {}_matrix is expensive."
|
|
" lil_matrix is more efficient.".format(self.format),
|
|
SparseEfficiencyWarning, stacklevel=3)
|
|
# replace where possible
|
|
mask = offsets > -1
|
|
self.data[offsets[mask]] = x[mask]
|
|
# only insertions remain
|
|
mask = ~mask
|
|
i = i[mask]
|
|
i[i < 0] += M
|
|
j = j[mask]
|
|
j[j < 0] += N
|
|
self._insert_many(i, j, x[mask])
|
|
|
|
def _zero_many(self, i, j):
|
|
"""Sets value at each (i, j) to zero, preserving sparsity structure.
|
|
|
|
Here (i,j) index major and minor respectively.
|
|
"""
|
|
i, j, M, N = self._prepare_indices(i, j)
|
|
|
|
n_samples = len(i)
|
|
offsets = np.empty(n_samples, dtype=self.indices.dtype)
|
|
ret = csr_sample_offsets(M, N, self.indptr, self.indices, n_samples,
|
|
i, j, offsets)
|
|
if ret == 1:
|
|
# rinse and repeat
|
|
self.sum_duplicates()
|
|
csr_sample_offsets(M, N, self.indptr, self.indices, n_samples,
|
|
i, j, offsets)
|
|
|
|
# only assign zeros to the existing sparsity structure
|
|
self.data[offsets[offsets > -1]] = 0
|
|
|
|
def _insert_many(self, i, j, x):
|
|
"""Inserts new nonzero at each (i, j) with value x
|
|
|
|
Here (i,j) index major and minor respectively.
|
|
i, j and x must be non-empty, 1d arrays.
|
|
Inserts each major group (e.g. all entries per row) at a time.
|
|
Maintains has_sorted_indices property.
|
|
Modifies i, j, x in place.
|
|
"""
|
|
order = np.argsort(i, kind='mergesort') # stable for duplicates
|
|
i = i.take(order, mode='clip')
|
|
j = j.take(order, mode='clip')
|
|
x = x.take(order, mode='clip')
|
|
|
|
do_sort = self.has_sorted_indices
|
|
|
|
# Update index data type
|
|
idx_dtype = get_index_dtype((self.indices, self.indptr),
|
|
maxval=(self.indptr[-1] + x.size))
|
|
self.indptr = np.asarray(self.indptr, dtype=idx_dtype)
|
|
self.indices = np.asarray(self.indices, dtype=idx_dtype)
|
|
i = np.asarray(i, dtype=idx_dtype)
|
|
j = np.asarray(j, dtype=idx_dtype)
|
|
|
|
# Collate old and new in chunks by major index
|
|
indices_parts = []
|
|
data_parts = []
|
|
ui, ui_indptr = np.unique(i, return_index=True)
|
|
ui_indptr = np.append(ui_indptr, len(j))
|
|
new_nnzs = np.diff(ui_indptr)
|
|
prev = 0
|
|
for c, (ii, js, je) in enumerate(zip(ui, ui_indptr, ui_indptr[1:])):
|
|
# old entries
|
|
start = self.indptr[prev]
|
|
stop = self.indptr[ii]
|
|
indices_parts.append(self.indices[start:stop])
|
|
data_parts.append(self.data[start:stop])
|
|
|
|
# handle duplicate j: keep last setting
|
|
uj, uj_indptr = np.unique(j[js:je][::-1], return_index=True)
|
|
if len(uj) == je - js:
|
|
indices_parts.append(j[js:je])
|
|
data_parts.append(x[js:je])
|
|
else:
|
|
indices_parts.append(j[js:je][::-1][uj_indptr])
|
|
data_parts.append(x[js:je][::-1][uj_indptr])
|
|
new_nnzs[c] = len(uj)
|
|
|
|
prev = ii
|
|
|
|
# remaining old entries
|
|
start = self.indptr[ii]
|
|
indices_parts.append(self.indices[start:])
|
|
data_parts.append(self.data[start:])
|
|
|
|
# update attributes
|
|
self.indices = np.concatenate(indices_parts)
|
|
self.data = np.concatenate(data_parts)
|
|
nnzs = np.empty(self.indptr.shape, dtype=idx_dtype)
|
|
nnzs[0] = idx_dtype(0)
|
|
indptr_diff = np.diff(self.indptr)
|
|
indptr_diff[ui] += new_nnzs
|
|
nnzs[1:] = indptr_diff
|
|
self.indptr = np.cumsum(nnzs, out=nnzs)
|
|
|
|
if do_sort:
|
|
# TODO: only sort where necessary
|
|
self.has_sorted_indices = False
|
|
self.sort_indices()
|
|
|
|
self.check_format(full_check=False)
|
|
|
|
######################
|
|
# Conversion methods #
|
|
######################
|
|
|
|
def tocoo(self, copy=True):
|
|
major_dim, minor_dim = self._swap(self.shape)
|
|
minor_indices = self.indices
|
|
major_indices = np.empty(len(minor_indices), dtype=self.indices.dtype)
|
|
_sparsetools.expandptr(major_dim, self.indptr, major_indices)
|
|
row, col = self._swap((major_indices, minor_indices))
|
|
|
|
from .coo import coo_matrix
|
|
return coo_matrix((self.data, (row, col)), self.shape, copy=copy,
|
|
dtype=self.dtype)
|
|
|
|
tocoo.__doc__ = spmatrix.tocoo.__doc__
|
|
|
|
def toarray(self, order=None, out=None):
|
|
if out is None and order is None:
|
|
order = self._swap('cf')[0]
|
|
out = self._process_toarray_args(order, out)
|
|
if not (out.flags.c_contiguous or out.flags.f_contiguous):
|
|
raise ValueError('Output array must be C or F contiguous')
|
|
# align ideal order with output array order
|
|
if out.flags.c_contiguous:
|
|
x = self.tocsr()
|
|
y = out
|
|
else:
|
|
x = self.tocsc()
|
|
y = out.T
|
|
M, N = x._swap(x.shape)
|
|
csr_todense(M, N, x.indptr, x.indices, x.data, y)
|
|
return out
|
|
|
|
toarray.__doc__ = spmatrix.toarray.__doc__
|
|
|
|
##############################################################
|
|
# methods that examine or modify the internal data structure #
|
|
##############################################################
|
|
|
|
def eliminate_zeros(self):
|
|
"""Remove zero entries from the matrix
|
|
|
|
This is an *in place* operation.
|
|
"""
|
|
M, N = self._swap(self.shape)
|
|
_sparsetools.csr_eliminate_zeros(M, N, self.indptr, self.indices,
|
|
self.data)
|
|
self.prune() # nnz may have changed
|
|
|
|
def __get_has_canonical_format(self):
|
|
"""Determine whether the matrix has sorted indices and no duplicates
|
|
|
|
Returns
|
|
- True: if the above applies
|
|
- False: otherwise
|
|
|
|
has_canonical_format implies has_sorted_indices, so if the latter flag
|
|
is False, so will the former be; if the former is found True, the
|
|
latter flag is also set.
|
|
"""
|
|
|
|
# first check to see if result was cached
|
|
if not getattr(self, '_has_sorted_indices', True):
|
|
# not sorted => not canonical
|
|
self._has_canonical_format = False
|
|
elif not hasattr(self, '_has_canonical_format'):
|
|
self.has_canonical_format = _sparsetools.csr_has_canonical_format(
|
|
len(self.indptr) - 1, self.indptr, self.indices)
|
|
return self._has_canonical_format
|
|
|
|
def __set_has_canonical_format(self, val):
|
|
self._has_canonical_format = bool(val)
|
|
if val:
|
|
self.has_sorted_indices = True
|
|
|
|
has_canonical_format = property(fget=__get_has_canonical_format,
|
|
fset=__set_has_canonical_format)
|
|
|
|
def sum_duplicates(self):
|
|
"""Eliminate duplicate matrix entries by adding them together
|
|
|
|
This is an *in place* operation.
|
|
"""
|
|
if self.has_canonical_format:
|
|
return
|
|
self.sort_indices()
|
|
|
|
M, N = self._swap(self.shape)
|
|
_sparsetools.csr_sum_duplicates(M, N, self.indptr, self.indices,
|
|
self.data)
|
|
|
|
self.prune() # nnz may have changed
|
|
self.has_canonical_format = True
|
|
|
|
def __get_sorted(self):
|
|
"""Determine whether the matrix has sorted indices
|
|
|
|
Returns
|
|
- True: if the indices of the matrix are in sorted order
|
|
- False: otherwise
|
|
|
|
"""
|
|
|
|
# first check to see if result was cached
|
|
if not hasattr(self, '_has_sorted_indices'):
|
|
self._has_sorted_indices = _sparsetools.csr_has_sorted_indices(
|
|
len(self.indptr) - 1, self.indptr, self.indices)
|
|
return self._has_sorted_indices
|
|
|
|
def __set_sorted(self, val):
|
|
self._has_sorted_indices = bool(val)
|
|
|
|
has_sorted_indices = property(fget=__get_sorted, fset=__set_sorted)
|
|
|
|
def sorted_indices(self):
|
|
"""Return a copy of this matrix with sorted indices
|
|
"""
|
|
A = self.copy()
|
|
A.sort_indices()
|
|
return A
|
|
|
|
# an alternative that has linear complexity is the following
|
|
# although the previous option is typically faster
|
|
# return self.toother().toother()
|
|
|
|
def sort_indices(self):
|
|
"""Sort the indices of this matrix *in place*
|
|
"""
|
|
|
|
if not self.has_sorted_indices:
|
|
_sparsetools.csr_sort_indices(len(self.indptr) - 1, self.indptr,
|
|
self.indices, self.data)
|
|
self.has_sorted_indices = True
|
|
|
|
def prune(self):
|
|
"""Remove empty space after all non-zero elements.
|
|
"""
|
|
major_dim = self._swap(self.shape)[0]
|
|
|
|
if len(self.indptr) != major_dim + 1:
|
|
raise ValueError('index pointer has invalid length')
|
|
if len(self.indices) < self.nnz:
|
|
raise ValueError('indices array has fewer than nnz elements')
|
|
if len(self.data) < self.nnz:
|
|
raise ValueError('data array has fewer than nnz elements')
|
|
|
|
self.indices = _prune_array(self.indices[:self.nnz])
|
|
self.data = _prune_array(self.data[:self.nnz])
|
|
|
|
def resize(self, *shape):
|
|
shape = check_shape(shape)
|
|
if hasattr(self, 'blocksize'):
|
|
bm, bn = self.blocksize
|
|
new_M, rm = divmod(shape[0], bm)
|
|
new_N, rn = divmod(shape[1], bn)
|
|
if rm or rn:
|
|
raise ValueError("shape must be divisible into %s blocks. "
|
|
"Got %s" % (self.blocksize, shape))
|
|
M, N = self.shape[0] // bm, self.shape[1] // bn
|
|
else:
|
|
new_M, new_N = self._swap(shape)
|
|
M, N = self._swap(self.shape)
|
|
|
|
if new_M < M:
|
|
self.indices = self.indices[:self.indptr[new_M]]
|
|
self.data = self.data[:self.indptr[new_M]]
|
|
self.indptr = self.indptr[:new_M + 1]
|
|
elif new_M > M:
|
|
self.indptr = np.resize(self.indptr, new_M + 1)
|
|
self.indptr[M + 1:].fill(self.indptr[M])
|
|
|
|
if new_N < N:
|
|
mask = self.indices < new_N
|
|
if not np.all(mask):
|
|
self.indices = self.indices[mask]
|
|
self.data = self.data[mask]
|
|
major_index, val = self._minor_reduce(np.add, mask)
|
|
self.indptr.fill(0)
|
|
self.indptr[1:][major_index] = val
|
|
np.cumsum(self.indptr, out=self.indptr)
|
|
|
|
self._shape = shape
|
|
|
|
resize.__doc__ = spmatrix.resize.__doc__
|
|
|
|
###################
|
|
# utility methods #
|
|
###################
|
|
|
|
# needed by _data_matrix
|
|
def _with_data(self, data, copy=True):
|
|
"""Returns a matrix with the same sparsity structure as self,
|
|
but with different data. By default the structure arrays
|
|
(i.e. .indptr and .indices) are copied.
|
|
"""
|
|
if copy:
|
|
return self.__class__((data, self.indices.copy(),
|
|
self.indptr.copy()),
|
|
shape=self.shape,
|
|
dtype=data.dtype)
|
|
else:
|
|
return self.__class__((data, self.indices, self.indptr),
|
|
shape=self.shape, dtype=data.dtype)
|
|
|
|
def _binopt(self, other, op):
|
|
"""apply the binary operation fn to two sparse matrices."""
|
|
other = self.__class__(other)
|
|
|
|
# e.g. csr_plus_csr, csr_minus_csr, etc.
|
|
fn = getattr(_sparsetools, self.format + op + self.format)
|
|
|
|
maxnnz = self.nnz + other.nnz
|
|
idx_dtype = get_index_dtype((self.indptr, self.indices,
|
|
other.indptr, other.indices),
|
|
maxval=maxnnz)
|
|
indptr = np.empty(self.indptr.shape, dtype=idx_dtype)
|
|
indices = np.empty(maxnnz, dtype=idx_dtype)
|
|
|
|
bool_ops = ['_ne_', '_lt_', '_gt_', '_le_', '_ge_']
|
|
if op in bool_ops:
|
|
data = np.empty(maxnnz, dtype=np.bool_)
|
|
else:
|
|
data = np.empty(maxnnz, dtype=upcast(self.dtype, other.dtype))
|
|
|
|
fn(self.shape[0], self.shape[1],
|
|
np.asarray(self.indptr, dtype=idx_dtype),
|
|
np.asarray(self.indices, dtype=idx_dtype),
|
|
self.data,
|
|
np.asarray(other.indptr, dtype=idx_dtype),
|
|
np.asarray(other.indices, dtype=idx_dtype),
|
|
other.data,
|
|
indptr, indices, data)
|
|
|
|
A = self.__class__((data, indices, indptr), shape=self.shape)
|
|
A.prune()
|
|
|
|
return A
|
|
|
|
def _divide_sparse(self, other):
|
|
"""
|
|
Divide this matrix by a second sparse matrix.
|
|
"""
|
|
if other.shape != self.shape:
|
|
raise ValueError('inconsistent shapes')
|
|
|
|
r = self._binopt(other, '_eldiv_')
|
|
|
|
if np.issubdtype(r.dtype, np.inexact):
|
|
# Eldiv leaves entries outside the combined sparsity
|
|
# pattern empty, so they must be filled manually.
|
|
# Everything outside of other's sparsity is NaN, and everything
|
|
# inside it is either zero or defined by eldiv.
|
|
out = np.empty(self.shape, dtype=self.dtype)
|
|
out.fill(np.nan)
|
|
row, col = other.nonzero()
|
|
out[row, col] = 0
|
|
r = r.tocoo()
|
|
out[r.row, r.col] = r.data
|
|
out = matrix(out)
|
|
else:
|
|
# integers types go with nan <-> 0
|
|
out = r
|
|
|
|
return out
|
|
|
|
|
|
def _process_slice(sl, num):
|
|
if sl is None:
|
|
i0, i1 = 0, num
|
|
elif isinstance(sl, slice):
|
|
i0, i1, stride = sl.indices(num)
|
|
if stride != 1:
|
|
raise ValueError('slicing with step != 1 not supported')
|
|
i0 = min(i0, i1) # give an empty slice when i0 > i1
|
|
elif isintlike(sl):
|
|
if sl < 0:
|
|
sl += num
|
|
i0, i1 = sl, sl + 1
|
|
if i0 < 0 or i1 > num:
|
|
raise IndexError('index out of bounds: 0 <= %d < %d <= %d' %
|
|
(i0, i1, num))
|
|
else:
|
|
raise TypeError('expected slice or scalar')
|
|
|
|
return i0, i1
|