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Showing content from https://sparse.pydata.org/en/latest/api/COO/ below:

COO - sparse

class COO(SparseArray, NDArrayOperatorsMixin):  # lgtm [py/missing-equals]
    """
    A sparse multidimensional array.

    This is stored in COO format.  It depends on NumPy and Scipy.sparse for
    computation, but supports arrays of arbitrary dimension.

    Parameters
    ----------
    coords : numpy.ndarray (COO.ndim, COO.nnz)
        An array holding the index locations of every value
        Should have shape (number of dimensions, number of non-zeros).
    data : numpy.ndarray (COO.nnz,)
        An array of Values. A scalar can also be supplied if the data is the same across
        all coordinates. If not given, defers to [`sparse.as_coo`][].
    shape : tuple[int] (COO.ndim,)
        The shape of the array.
    has_duplicates : bool, optional
        A value indicating whether the supplied value for [`sparse.COO.coords`][] has
        duplicates. Note that setting this to `False` when `coords` does have
        duplicates may result in undefined behaviour.
    sorted : bool, optional
        A value indicating whether the values in `coords` are sorted. Note
        that setting this to `True` when [`sparse.COO.coords`][] isn't sorted may
        result in undefined behaviour.
    prune : bool, optional
        A flag indicating whether or not we should prune any fill-values present in
        `data`.
    cache : bool, optional
        Whether to enable cacheing for various operations. See
        [`sparse.COO.enable_caching`][].
    fill_value: scalar, optional
        The fill value for this array.

    Attributes
    ----------
    coords : numpy.ndarray (ndim, nnz)
        An array holding the coordinates of every nonzero element.
    data : numpy.ndarray (nnz,)
        An array holding the values corresponding to [`sparse.COO.coords`][].
    shape : tuple[int] (ndim,)
        The dimensions of this array.

    See Also
    --------
    - [`sparse.DOK`][]: A mostly write-only sparse array.
    - [`sparse.as_coo`][]: Convert any given format to [`sparse.COO`][].

    Examples
    --------
    You can create [`sparse.COO`][] objects from Numpy arrays.

    >>> x = np.eye(4, dtype=np.uint8)
    >>> x[2, 3] = 5
    >>> s = COO.from_numpy(x)
    >>> s
    <COO: shape=(4, 4), dtype=uint8, nnz=5, fill_value=0>
    >>> s.data  # doctest: +NORMALIZE_WHITESPACE
    array([1, 1, 1, 5, 1], dtype=uint8)
    >>> s.coords  # doctest: +NORMALIZE_WHITESPACE
    array([[0, 1, 2, 2, 3],
           [0, 1, 2, 3, 3]])

    [`sparse.COO`][] objects support basic arithmetic and binary operations.

    >>> x2 = np.eye(4, dtype=np.uint8)
    >>> x2[3, 2] = 5
    >>> s2 = COO.from_numpy(x2)
    >>> (s + s2).todense()  # doctest: +NORMALIZE_WHITESPACE
    array([[2, 0, 0, 0],
           [0, 2, 0, 0],
           [0, 0, 2, 5],
           [0, 0, 5, 2]], dtype=uint8)
    >>> (s * s2).todense()  # doctest: +NORMALIZE_WHITESPACE
    array([[1, 0, 0, 0],
           [0, 1, 0, 0],
           [0, 0, 1, 0],
           [0, 0, 0, 1]], dtype=uint8)

    Binary operations support broadcasting.

    >>> x3 = np.zeros((4, 1), dtype=np.uint8)
    >>> x3[2, 0] = 1
    >>> s3 = COO.from_numpy(x3)
    >>> (s * s3).todense()  # doctest: +NORMALIZE_WHITESPACE
    array([[0, 0, 0, 0],
           [0, 0, 0, 0],
           [0, 0, 1, 5],
           [0, 0, 0, 0]], dtype=uint8)

    [`sparse.COO`][] objects also support dot products and reductions.

    >>> s.dot(s.T).sum(axis=0).todense()  # doctest: +NORMALIZE_WHITESPACE
    array([ 1,  1, 31,  6], dtype=uint64)

    You can use Numpy `ufunc` operations on [`sparse.COO`][] arrays as well.

    >>> np.sum(s, axis=1).todense()  # doctest: +NORMALIZE_WHITESPACE
    array([1, 1, 6, 1], dtype=uint64)
    >>> np.round(np.sqrt(s, dtype=np.float64), decimals=1).todense()  # doctest: +SKIP
    array([[ 1. ,  0. ,  0. ,  0. ],
           [ 0. ,  1. ,  0. ,  0. ],
           [ 0. ,  0. ,  1. ,  2.2],
           [ 0. ,  0. ,  0. ,  1. ]])

    Operations that will result in a dense array will usually result in a different
    fill value, such as the following.

    >>> np.exp(s)
    <COO: shape=(4, 4), dtype=float16, nnz=5, fill_value=1.0>

    You can also create [`sparse.COO`][] arrays from coordinates and data.

    >>> coords = [[0, 0, 0, 1, 1], [0, 1, 2, 0, 3], [0, 3, 2, 0, 1]]
    >>> data = [1, 2, 3, 4, 5]
    >>> s4 = COO(coords, data, shape=(3, 4, 5))
    >>> s4
    <COO: shape=(3, 4, 5), dtype=int64, nnz=5, fill_value=0>

    If the data is same across all coordinates, you can also specify a scalar.

    >>> coords = [[0, 0, 0, 1, 1], [0, 1, 2, 0, 3], [0, 3, 2, 0, 1]]
    >>> data = 1
    >>> s5 = COO(coords, data, shape=(3, 4, 5))
    >>> s5
    <COO: shape=(3, 4, 5), dtype=int64, nnz=5, fill_value=0>

    Following scipy.sparse conventions you can also pass these as a tuple with
    rows and columns

    >>> rows = [0, 1, 2, 3, 4]
    >>> cols = [0, 0, 0, 1, 1]
    >>> data = [10, 20, 30, 40, 50]
    >>> z = COO((data, (rows, cols)), shape=(5, 2))
    >>> z.todense()  # doctest: +NORMALIZE_WHITESPACE
    array([[10,  0],
           [20,  0],
           [30,  0],
           [ 0, 40],
           [ 0, 50]])

    You can also pass a dictionary or iterable of index/value pairs. Repeated
    indices imply summation:

    >>> d = {(0, 0, 0): 1, (1, 2, 3): 2, (1, 1, 0): 3}
    >>> COO(d, shape=(2, 3, 4))
    <COO: shape=(2, 3, 4), dtype=int64, nnz=3, fill_value=0>
    >>> L = [((0, 0), 1), ((1, 1), 2), ((0, 0), 3)]
    >>> COO(L, shape=(2, 2)).todense()  # doctest: +NORMALIZE_WHITESPACE
    array([[4, 0],
           [0, 2]])

    You can convert [`sparse.DOK`][] arrays to [`sparse.COO`][] arrays.

    >>> from sparse import DOK
    >>> s6 = DOK((5, 5), dtype=np.int64)
    >>> s6[1:3, 1:3] = [[4, 5], [6, 7]]
    >>> s6
    <DOK: shape=(5, 5), dtype=int64, nnz=4, fill_value=0>
    >>> s7 = s6.asformat("coo")
    >>> s7
    <COO: shape=(5, 5), dtype=int64, nnz=4, fill_value=0>
    >>> s7.todense()  # doctest: +NORMALIZE_WHITESPACE
    array([[0, 0, 0, 0, 0],
           [0, 4, 5, 0, 0],
           [0, 6, 7, 0, 0],
           [0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0]])
    """

    __array_priority__ = 12

    def __init__(
        self,
        coords,
        data=None,
        shape=None,
        has_duplicates=True,
        sorted=False,
        prune=False,
        cache=False,
        fill_value=None,
        idx_dtype=None,
    ):
        if isinstance(coords, COO):
            self._make_shallow_copy_of(coords)
            if data is not None or shape is not None:
                raise ValueError("If `coords` is `COO`, then no other arguments should be provided.")
            if fill_value is not None:
                self.fill_value = self.data.dtype.type(fill_value)
            return

        self._cache = None
        if cache:
            self.enable_caching()

        if data is None:
            arr = as_coo(coords, shape=shape, fill_value=fill_value, idx_dtype=idx_dtype)
            self._make_shallow_copy_of(arr)
            if cache:
                self.enable_caching()
            return

        self.data = np.asarray(data)
        self.coords = np.asarray(coords)

        if self.coords.ndim == 1:
            if self.coords.size == 0 and shape is not None:
                self.coords = self.coords.reshape((len(shape), len(data)))
            else:
                self.coords = self.coords[None, :]

        if self.data.ndim == 0:
            self.data = np.broadcast_to(self.data, self.coords.shape[1])

        if self.data.ndim != 1:
            raise ValueError("`data` must be a scalar or 1-dimensional.")

        if shape is None:
            raise ValueError("`shape` was not provided.")

        if not isinstance(shape, Iterable):
            shape = (shape,)

        if isinstance(shape, np.ndarray):
            shape = tuple(shape)

        if shape and not self.coords.size:
            self.coords = np.zeros((len(shape) if isinstance(shape, Iterable) else 1, 0), dtype=np.intp)
        super().__init__(shape, fill_value=fill_value)
        if idx_dtype:
            if not can_store(idx_dtype, max(shape)):
                raise ValueError(f"cannot cast array with shape {shape} to dtype {idx_dtype}.")
            self.coords = self.coords.astype(idx_dtype)

        if self.shape:
            if len(self.data) != self.coords.shape[1]:
                msg = "The data length does not match the coordinates given.\nlen(data) = {}, but {} coords specified."
                raise ValueError(msg.format(len(data), self.coords.shape[1]))
            if len(self.shape) != self.coords.shape[0]:
                msg = (
                    "Shape specified by `shape` doesn't match the "
                    "shape of `coords`; len(shape)={} != coords.shape[0]={}"
                    "(and coords.shape={})"
                )
                raise ValueError(msg.format(len(shape), self.coords.shape[0], self.coords.shape))

        from .._settings import WARN_ON_TOO_DENSE

        if WARN_ON_TOO_DENSE and self.nbytes >= self.size * self.data.itemsize:
            warnings.warn(
                "Attempting to create a sparse array that takes no less "
                "memory than than an equivalent dense array. You may want to "
                "use a dense array here instead.",
                RuntimeWarning,
                stacklevel=1,
            )

        if not sorted:
            self._sort_indices()

        if has_duplicates:
            self._sum_duplicates()

        if prune:
            self._prune()

    def __getstate__(self):
        return (self.coords, self.data, self.shape, self.fill_value)

    def __setstate__(self, state):
        self.coords, self.data, self.shape, self.fill_value = state
        self._cache = None

    def __dask_tokenize__(self):
        "Produce a deterministic, content-based hash for dask."
        from dask.base import normalize_token

        return normalize_token((type(self), self.coords, self.data, self.shape, self.fill_value))

    def copy(self, deep=True):
        """Return a copy of the array.

        Parameters
        ----------
        deep : boolean, optional
            If True (default), the internal coords and data arrays are also
            copied. Set to ``False`` to only make a shallow copy.
        """
        return _copy.deepcopy(self) if deep else _copy.copy(self)

    def enable_caching(self):
        """Enable caching of reshape, transpose, and tocsr/csc operations

        This enables efficient iterative workflows that make heavy use of
        csr/csc operations, such as tensordot.  This maintains a cache of
        recent results of reshape and transpose so that operations like
        tensordot (which uses both internally) store efficiently stored
        representations for repeated use.  This can significantly cut down on
        computational costs in common numeric algorithms.

        However, this also assumes that neither this object, nor the downstream
        objects will have their data mutated.

        Examples
        --------
        >>> s.enable_caching()  # doctest: +SKIP
        >>> csr1 = s.transpose((2, 0, 1)).reshape((100, 120)).tocsr()  # doctest: +SKIP
        >>> csr2 = s.transpose((2, 0, 1)).reshape((100, 120)).tocsr()  # doctest: +SKIP
        >>> csr1 is csr2  # doctest: +SKIP
        True
        """
        self._cache = defaultdict(lambda: deque(maxlen=3))

    @classmethod
    def from_numpy(cls, x, fill_value=None, idx_dtype=None):
        """
        Convert the given [`sparse.COO`][] object.

        Parameters
        ----------
        x : np.ndarray
            The dense array to convert.
        fill_value : scalar
            The fill value of the constructed [`sparse.COO`][] array. Zero if
            unspecified.

        Returns
        -------
        COO
            The converted COO array.

        Examples
        --------
        >>> x = np.eye(5)
        >>> s = COO.from_numpy(x)
        >>> s
        <COO: shape=(5, 5), dtype=float64, nnz=5, fill_value=0.0>

        >>> x[x == 0] = np.nan
        >>> COO.from_numpy(x, fill_value=np.nan)
        <COO: shape=(5, 5), dtype=float64, nnz=5, fill_value=nan>
        """
        x = np.asanyarray(x).view(type=np.ndarray)

        if fill_value is None:
            fill_value = _zero_of_dtype(x.dtype) if x.shape else x

        coords = np.atleast_2d(np.flatnonzero(~equivalent(x, fill_value)))
        data = x.ravel()[tuple(coords)]
        return cls(
            coords,
            data,
            shape=x.size,
            has_duplicates=False,
            sorted=True,
            fill_value=fill_value,
            idx_dtype=idx_dtype,
        ).reshape(x.shape)

    def todense(self):
        """
        Convert this [`sparse.COO`][] array to a dense [`numpy.ndarray`][]. Note that
        this may take a large amount of memory if the `COO` object's `shape`
        is large.

        Returns
        -------
        numpy.ndarray
            The converted dense array.

        See Also
        --------
        - [`sparse.DOK.todense`][] : Equivalent `DOK` array method.
        - [`scipy.sparse.coo_matrix.todense`][] : Equivalent Scipy method.

        Examples
        --------
        >>> x = np.random.randint(100, size=(7, 3))
        >>> s = COO.from_numpy(x)
        >>> x2 = s.todense()
        >>> np.array_equal(x, x2)
        True
        """
        x = np.full(self.shape, self.fill_value, self.dtype)

        coords = tuple([self.coords[i, :] for i in range(self.ndim)])
        data = self.data

        if len(coords) != 0:
            x[coords] = data
        else:
            if len(data) != 0:
                assert data.shape == (1,)
                x[...] = data[0]

        return x

    @classmethod
    def from_scipy_sparse(cls, x, /, *, fill_value=None):
        """
        Construct a [`sparse.COO`][] array from a [`scipy.sparse.spmatrix`][]

        Parameters
        ----------
        x : scipy.sparse.spmatrix
            The sparse matrix to construct the array from.
        fill_value : scalar
            The fill-value to use when converting.

        Returns
        -------
        COO
            The converted [`sparse.COO`][] object.

        Examples
        --------
        >>> import scipy.sparse
        >>> x = scipy.sparse.rand(6, 3, density=0.2)
        >>> s = COO.from_scipy_sparse(x)
        >>> np.array_equal(x.todense(), s.todense())
        True
        """
        x = x.asformat("coo")
        if not x.has_canonical_format:
            x.eliminate_zeros()
            x.sum_duplicates()

        coords = np.empty((2, x.nnz), dtype=x.row.dtype)
        coords[0, :] = x.row
        coords[1, :] = x.col
        return COO(
            coords,
            x.data,
            shape=x.shape,
            has_duplicates=not x.has_canonical_format,
            sorted=x.has_canonical_format,
            fill_value=fill_value,
        )

    @classmethod
    def from_iter(cls, x, shape, fill_value=None, dtype=None):
        """
        Converts an iterable in certain formats to a [`sparse.COO`][] array. See examples
        for details.

        Parameters
        ----------
        x : Iterable or Iterator
            The iterable to convert to [`sparse.COO`][].
        shape : tuple[int]
            The shape of the array.
        fill_value : scalar
            The fill value for this array.
        dtype : numpy.dtype
            The dtype of the input array. Inferred from the input if not given.

        Returns
        -------
        out : COO
            The output [`sparse.COO`][] array.

        Examples
        --------
        You can convert items of the format [`sparse.COO`][].
        Here, the first part represents the coordinate and the second part represents the value.

        >>> x = [((0, 0), 1), ((1, 1), 1)]
        >>> s = COO.from_iter(x, shape=(2, 2))
        >>> s.todense()
        array([[1, 0],
               [0, 1]])

        You can also have a similar format with a dictionary.

        >>> x = {(0, 0): 1, (1, 1): 1}
        >>> s = COO.from_iter(x, shape=(2, 2))
        >>> s.todense()
        array([[1, 0],
               [0, 1]])

        The third supported format is ``(data, (..., row, col))``.

        >>> x = ([1, 1], ([0, 1], [0, 1]))
        >>> s = COO.from_iter(x, shape=(2, 2))
        >>> s.todense()
        array([[1, 0],
               [0, 1]])

        You can also pass in a [`collections.abc.Iterator`][] object.

        >>> x = [((0, 0), 1), ((1, 1), 1)].__iter__()
        >>> s = COO.from_iter(x, shape=(2, 2))
        >>> s.todense()
        array([[1, 0],
               [0, 1]])
        """
        if isinstance(x, dict):
            x = list(x.items())

        if not isinstance(x, Sized):
            x = list(x)

        if len(x) != 2 and not all(len(item) == 2 for item in x):
            raise ValueError("Invalid iterable to convert to COO.")

        if not x:
            ndim = 0 if shape is None else len(shape)
            coords = np.empty((ndim, 0), dtype=np.uint8)
            data = np.empty((0,), dtype=dtype)
            shape = () if shape is None else shape

        elif not isinstance(x[0][0], Iterable):
            coords = np.stack(x[1], axis=0)
            data = np.asarray(x[0], dtype=dtype)
        else:
            coords = np.array([item[0] for item in x]).T
            data = np.array([item[1] for item in x], dtype=dtype)

        if not (
            coords.ndim == 2 and data.ndim == 1 and np.issubdtype(coords.dtype, np.integer) and np.all(coords >= 0)
        ):
            raise ValueError("Invalid iterable to convert to COO.")

        return COO(coords, data, shape=shape, fill_value=fill_value)

    @property
    def dtype(self):
        """
        The datatype of this array.

        Returns
        -------
        numpy.dtype
            The datatype of this array.

        See Also
        --------
        - [`numpy.ndarray.dtype`][] : Numpy equivalent property.
        - [`scipy.sparse.coo_matrix.dtype`][] : Scipy equivalent property.

        Examples
        --------
        >>> x = (200 * np.random.rand(5, 4)).astype(np.int32)
        >>> s = COO.from_numpy(x)
        >>> s.dtype
        dtype('int32')
        >>> x.dtype == s.dtype
        True
        """
        return self.data.dtype

    @property
    def nnz(self):
        """
        The number of nonzero elements in this array. Note that any duplicates in
        `coords` are counted multiple times.

        Returns
        -------
        int
            The number of nonzero elements in this array.

        See Also
        --------
        - [`sparse.DOK.nnz`][] : Equivalent [`sparse.DOK`][] array property.
        - [`numpy.count_nonzero`][] : A similar Numpy function.
        - [`scipy.sparse.coo_matrix.nnz`][] : The Scipy equivalent property.

        Examples
        --------
        >>> x = np.array([0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0])
        >>> np.count_nonzero(x)
        6
        >>> s = COO.from_numpy(x)
        >>> s.nnz
        6
        >>> np.count_nonzero(x) == s.nnz
        True
        """
        return self.coords.shape[1]

    @property
    def format(self):
        """
        The storage format of this array.
        Returns
        -------
        str
            The storage format of this array.
        See Also
        --------
        [`scipy.sparse.dok_matrix.format`][] : The Scipy equivalent property.
        Examples
        -------
        >>> import sparse
        >>> s = sparse.random((5, 5), density=0.2, format="dok")
        >>> s.format
        'dok'
        >>> t = sparse.random((5, 5), density=0.2, format="coo")
        >>> t.format
        'coo'
        """
        return "coo"

    @property
    def nbytes(self):
        """
        The number of bytes taken up by this object. Note that for small arrays,
        this may undercount the number of bytes due to the large constant overhead.

        Returns
        -------
        int
            The approximate bytes of memory taken by this object.

        See Also
        --------
        [`numpy.ndarray.nbytes`][] : The equivalent Numpy property.

        Examples
        --------
        >>> data = np.arange(6, dtype=np.uint8)
        >>> coords = np.random.randint(1000, size=(3, 6), dtype=np.uint16)
        >>> s = COO(coords, data, shape=(1000, 1000, 1000))
        >>> s.nbytes
        42
        """
        return self.data.nbytes + self.coords.nbytes

    def __len__(self):
        """
        Get "length" of array, which is by definition the size of the first
        dimension.

        Returns
        -------
        int
            The size of the first dimension.

        See Also
        --------
        numpy.ndarray.__len__ : Numpy equivalent property.

        Examples
        --------
        >>> x = np.zeros((10, 10))
        >>> s = COO.from_numpy(x)
        >>> len(s)
        10
        """
        return self.shape[0]

    def __sizeof__(self):
        return self.nbytes

    __getitem__ = getitem

    def __str__(self):
        summary = f"<COO: shape={self.shape!s}, dtype={self.dtype!s}, nnz={self.nnz:d}, fill_value={self.fill_value!s}>"
        return self._str_impl(summary)

    __repr__ = __str__

    def _reduce_calc(self, method, axis, keepdims=False, **kwargs):
        if axis == (None,):
            axis = tuple(range(self.ndim))
        axis = tuple(a if a >= 0 else a + self.ndim for a in axis)
        neg_axis = tuple(ax for ax in range(self.ndim) if ax not in set(axis))
        a = self.transpose(neg_axis + axis)
        a = a.reshape(
            (
                np.prod([self.shape[d] for d in neg_axis], dtype=np.intp),
                np.prod([self.shape[d] for d in axis], dtype=np.intp),
            )
        )
        data, inv_idx, counts = _grouped_reduce(a.data, a.coords[0], method, **kwargs)
        n_cols = a.shape[1]
        arr_attrs = (a, neg_axis, inv_idx)
        return (data, counts, axis, n_cols, arr_attrs)

    def _reduce_return(self, data, arr_attrs, result_fill_value):
        a, neg_axis, inv_idx = arr_attrs
        coords = a.coords[0:1, inv_idx]
        out = COO(
            coords,
            data,
            shape=(a.shape[0],),
            has_duplicates=False,
            sorted=True,
            prune=True,
            fill_value=result_fill_value,
        )

        return out.reshape(tuple(self.shape[d] for d in neg_axis))

    def transpose(self, axes=None):
        """
        Returns a new array which has the order of the axes switched.

        Parameters
        ----------
        axes : Iterable[int], optional
            The new order of the axes compared to the previous one. Reverses the axes
            by default.

        Returns
        -------
        COO
            The new array with the axes in the desired order.

        See Also
        --------
        - [`sparse.COO.T`][] : A quick property to reverse the order of the axes.
        - [`numpy.ndarray.transpose`][] : Numpy equivalent function.

        Examples
        --------
        We can change the order of the dimensions of any [`sparse.COO`][] array with this
        function.

        >>> x = np.add.outer(np.arange(5), np.arange(5)[::-1])
        >>> x  # doctest: +NORMALIZE_WHITESPACE
        array([[4, 3, 2, 1, 0],
               [5, 4, 3, 2, 1],
               [6, 5, 4, 3, 2],
               [7, 6, 5, 4, 3],
               [8, 7, 6, 5, 4]])
        >>> s = COO.from_numpy(x)
        >>> s.transpose((1, 0)).todense()  # doctest: +NORMALIZE_WHITESPACE
        array([[4, 5, 6, 7, 8],
               [3, 4, 5, 6, 7],
               [2, 3, 4, 5, 6],
               [1, 2, 3, 4, 5],
               [0, 1, 2, 3, 4]])

        Note that by default, this reverses the order of the axes rather than switching
        the last and second-to-last axes as required by some linear algebra operations.

        >>> x = np.random.rand(2, 3, 4)
        >>> s = COO.from_numpy(x)
        >>> s.transpose().shape
        (4, 3, 2)
        """
        if axes is None:
            axes = list(reversed(range(self.ndim)))

        # Normalize all axes indices to positive values
        axes = normalize_axis(axes, self.ndim)

        if len(np.unique(axes)) < len(axes):
            raise ValueError("repeated axis in transpose")

        if not len(axes) == self.ndim:
            raise ValueError("axes don't match array")

        axes = tuple(axes)

        if axes == tuple(range(self.ndim)):
            return self

        if self._cache is not None:
            for ax, value in self._cache["transpose"]:
                if ax == axes:
                    return value

        shape = tuple(self.shape[ax] for ax in axes)
        result = COO(
            self.coords[axes, :],
            self.data,
            shape,
            has_duplicates=False,
            cache=self._cache is not None,
            fill_value=self.fill_value,
        )

        if self._cache is not None:
            self._cache["transpose"].append((axes, result))
        return result

    @property
    def T(self):
        """
        Returns a new array which has the order of the axes reversed.

        Returns
        -------
        COO
            The new array with the axes in the desired order.

        See Also
        --------
        - [`sparse.COO.transpose`][] :
            A method where you can specify the order of the axes.
        - [`numpy.ndarray.T`][] :
            Numpy equivalent property.

        Examples
        --------
        We can change the order of the dimensions of any [`sparse.COO`][] array with this
        function.

        >>> x = np.add.outer(np.arange(5), np.arange(5)[::-1])
        >>> x  # doctest: +NORMALIZE_WHITESPACE
        array([[4, 3, 2, 1, 0],
               [5, 4, 3, 2, 1],
               [6, 5, 4, 3, 2],
               [7, 6, 5, 4, 3],
               [8, 7, 6, 5, 4]])
        >>> s = COO.from_numpy(x)
        >>> s.T.todense()  # doctest: +NORMALIZE_WHITESPACE
        array([[4, 5, 6, 7, 8],
               [3, 4, 5, 6, 7],
               [2, 3, 4, 5, 6],
               [1, 2, 3, 4, 5],
               [0, 1, 2, 3, 4]])

        Note that by default, this reverses the order of the axes rather than switching
        the last and second-to-last axes as required by some linear algebra operations.

        >>> x = np.random.rand(2, 3, 4)
        >>> s = COO.from_numpy(x)
        >>> s.T.shape
        (4, 3, 2)
        """
        return self.transpose(tuple(range(self.ndim))[::-1])

    @property
    def mT(self):
        """
        Transpose of a matrix (or a stack of matrices).
        If an array instance has fewer than two dimensions, an error should be raised.

        Returns
        -------
        COO
            array whose last two dimensions (axes) are permuted in reverse order relative to
            original array (i.e., for an array instance having shape (..., M, N), the returned
            array must have shape (..., N, M)). The returned array must have the same data
            type as the original array.

        See Also
        --------
        - [`sparse.COO.transpose`][] :
            A method where you can specify the order of the axes.
        - [`numpy.ndarray.mT`][] :
            Numpy equivalent property.

        Examples
        --------
        >>> x = np.arange(8).reshape((2, 2, 2))
        >>> x  # doctest: +NORMALIZE_WHITESPACE
        array([[[0, 1],
                [2, 3]],
               [[4, 5],
                [6, 7]]])
        >>> s = COO.from_numpy(x)
        >>> s.mT.todense()  # doctest: +NORMALIZE_WHITESPACE
        array([[[0, 2],
                [1, 3]],
               [[4, 6],
                [5, 7]]])
        """
        if self.ndim < 2:
            raise ValueError("Cannot compute matrix transpose if `ndim < 2`.")

        axis = list(range(self.ndim))
        axis[-1], axis[-2] = axis[-2], axis[-1]

        return self.transpose(axis)

    def swapaxes(self, axis1, axis2):
        """Returns array that has axes axis1 and axis2 swapped.

        Parameters
        ----------
        axis1 : int
            first axis to swap
        axis2 : int
            second axis to swap

        Returns
        -------
        COO
            The new array with the axes axis1 and axis2 swapped.

        Examples
        --------
        >>> x = COO.from_numpy(np.ones((2, 3, 4)))
        >>> x.swapaxes(0, 2)
        <COO: shape=(4, 3, 2), dtype=float64, nnz=24, fill_value=0.0>
        """
        # Normalize all axis1, axis2 to positive values
        axis1, axis2 = normalize_axis((axis1, axis2), self.ndim)  # checks if axis1,2 are in range + raises ValueError
        axes = list(range(self.ndim))
        axes[axis1], axes[axis2] = axes[axis2], axes[axis1]
        return self.transpose(axes)

    def dot(self, other):
        """
        Performs the equivalent of `x.dot(y)` for [`sparse.COO`][].

        Parameters
        ----------
        other : Union[COO, numpy.ndarray, scipy.sparse.spmatrix]
            The second operand of the dot product operation.

        Returns
        -------
        {COO, numpy.ndarray}
            The result of the dot product. If the result turns out to be dense,
            then a dense array is returned, otherwise, a sparse array.

        Raises
        ------
        ValueError
            If all arguments don't have zero fill-values.

        See Also
        --------
        - [`sparse.dot`][] : Equivalent function for two arguments.
        - [`numpy.dot`][] : Numpy equivalent function.
        - [`scipy.sparse.coo_matrix.dot`][] : Scipy equivalent function.

        Examples
        --------
        >>> x = np.arange(4).reshape((2, 2))
        >>> s = COO.from_numpy(x)
        >>> s.dot(s)  # doctest: +SKIP
        array([[ 2,  3],
               [ 6, 11]], dtype=int64)
        """
        from .._common import dot

        return dot(self, other)

    def __matmul__(self, other):
        from .._common import matmul

        try:
            return matmul(self, other)
        except NotImplementedError:
            return NotImplemented

    def __rmatmul__(self, other):
        from .._common import matmul

        try:
            return matmul(other, self)
        except NotImplementedError:
            return NotImplemented

    def linear_loc(self):
        """
        The nonzero coordinates of a flattened version of this array. Note that
        the coordinates may be out of order.

        Returns
        -------
        numpy.ndarray
            The flattened coordinates.

        See Also
        --------
        [`numpy.flatnonzero`][] : Equivalent Numpy function.

        Examples
        --------
        >>> x = np.eye(5)
        >>> s = COO.from_numpy(x)
        >>> s.linear_loc()  # doctest: +NORMALIZE_WHITESPACE
        array([ 0,  6, 12, 18, 24])
        >>> np.array_equal(np.flatnonzero(x), s.linear_loc())
        True
        """
        from .common import linear_loc

        return linear_loc(self.coords, self.shape)

    def flatten(self, order="C"):
        """
        Returns a new [`sparse.COO`][] array that is a flattened version of this array.

        Returns
        -------
        COO
            The flattened output array.

        Notes
        -----
        The `order` parameter is provided just for compatibility with
        Numpy and isn't actually supported.

        Examples
        --------
        >>> s = COO.from_numpy(np.arange(10))
        >>> s2 = s.reshape((2, 5)).flatten()
        >>> s2.todense()
        array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
        """
        if order not in {"C", None}:
            raise NotImplementedError("The `order` parameter is notsupported.")

        return self.reshape(-1)

    def reshape(self, shape, order="C"):
        """
        Returns a new [`sparse.COO`][] array that is a reshaped version of this array.

        Parameters
        ----------
        shape : tuple[int]
            The desired shape of the output array.

        Returns
        -------
        COO
            The reshaped output array.

        See Also
        --------
        [`numpy.ndarray.reshape`][] : The equivalent Numpy function.

        Notes
        -----
        The `order` parameter is provided just for compatibility with
        Numpy and isn't actually supported.

        Examples
        --------
        >>> s = COO.from_numpy(np.arange(25))
        >>> s2 = s.reshape((5, 5))
        >>> s2.todense()  # doctest: +NORMALIZE_WHITESPACE
        array([[ 0,  1,  2,  3,  4],
               [ 5,  6,  7,  8,  9],
               [10, 11, 12, 13, 14],
               [15, 16, 17, 18, 19],
               [20, 21, 22, 23, 24]])
        """
        shape = tuple(shape) if isinstance(shape, Iterable) else (shape,)

        if order not in {"C", None}:
            raise NotImplementedError("The `order` parameter is not supported")

        if self.shape == shape:
            return self
        if any(d == -1 for d in shape):
            extra = int(self.size / np.prod([d for d in shape if d != -1]))
            shape = tuple([d if d != -1 else extra for d in shape])

        if self.size != reduce(operator.mul, shape, 1):
            raise ValueError(f"cannot reshape array of size {self.size} into shape {shape}")

        if self._cache is not None:
            for sh, value in self._cache["reshape"]:
                if sh == shape:
                    return value

        # TODO: this self.size enforces a 2**64 limit to array size
        linear_loc = self.linear_loc()

        idx_dtype = self.coords.dtype
        if shape != () and not can_store(idx_dtype, max(shape)):
            idx_dtype = np.min_scalar_type(max(shape))
        coords = np.empty((len(shape), self.nnz), dtype=idx_dtype)
        strides = 1
        for i, d in enumerate(shape[::-1]):
            coords[-(i + 1), :] = (linear_loc // strides) % d
            strides *= d

        result = COO(
            coords,
            self.data,
            shape,
            has_duplicates=False,
            sorted=True,
            cache=self._cache is not None,
            fill_value=self.fill_value,
        )

        if self._cache is not None:
            self._cache["reshape"].append((shape, result))
        return result

    def squeeze(self, axis=None):
        """
        Removes singleton dimensions (axes) from ``x``.
        Parameters
        ----------
        axis : Union[None, int, Tuple[int, ...]]
            The axis (or axes) to squeeze. If a specified axis has a size greater than one,
            a `ValueError` is raised. ``axis=None`` removes all singleton dimensions.
            Default: ``None``.
        Returns
        -------
        COO
            The output array without ``axis`` dimensions.
        Examples
        --------
        >>> s = COO.from_numpy(np.eye(2)).reshape((2, 1, 2, 1))
        >>> s.squeeze().shape
        (2, 2)
        >>> s.squeeze(axis=1).shape
        (2, 2, 1)
        """
        squeezable_dims = tuple([d for d in range(self.ndim) if self.shape[d] == 1])

        if axis is None:
            axis = squeezable_dims
        if isinstance(axis, int):
            axis = (axis,)
        elif isinstance(axis, Iterable):
            axis = tuple(axis)
        else:
            raise ValueError(f"Invalid axis parameter: `{axis}`.")

        for d in axis:
            if d not in squeezable_dims:
                raise ValueError(f"Specified axis `{d}` has a size greater than one: {self.shape[d]}")

        retained_dims = [d for d in range(self.ndim) if d not in axis]

        coords = self.coords[retained_dims, :]
        shape = tuple([s for idx, s in enumerate(self.shape) if idx in retained_dims])

        return COO(
            coords,
            self.data,
            shape,
            has_duplicates=False,
            sorted=True,
            cache=self._cache is not None,
            fill_value=self.fill_value,
        )

    def to_scipy_sparse(self, /, *, accept_fv=None):
        """
        Converts this [`sparse.COO`][] object into a [`scipy.sparse.coo_matrix`][].

        Parameters
        ----------
        accept_fv : scalar or list of scalar, optional
            The list of accepted fill-values. The default accepts only zero.

        Returns
        -------
        scipy.sparse.coo_matrix
            The converted Scipy sparse matrix.

        Raises
        ------
        ValueError
            If the array is not two-dimensional.
        ValueError
            If all the array doesn't zero fill-values.

        See Also
        --------
        - [`sparse.COO.tocsr`][] : Convert to a [`scipy.sparse.csr_matrix`][].
        - [`sparse.COO.tocsc`][] : Convert to a [`scipy.sparse.csc_matrix`][].
        """
        import scipy.sparse

        check_fill_value(self, accept_fv=accept_fv)

        if self.ndim != 2:
            raise ValueError("Can only convert a 2-dimensional array to a Scipy sparse matrix.")

        result = scipy.sparse.coo_matrix((self.data, (self.coords[0], self.coords[1])), shape=self.shape)
        result.has_canonical_format = True
        return result

    def _tocsr(self):
        import scipy.sparse

        if self.ndim != 2:
            raise ValueError("This array must be two-dimensional for this conversion to work.")
        row, col = self.coords

        # Pass 3: count nonzeros in each row
        indptr = np.zeros(self.shape[0] + 1, dtype=np.int64)
        np.cumsum(np.bincount(row, minlength=self.shape[0]), out=indptr[1:])

        return scipy.sparse.csr_matrix((self.data, col, indptr), shape=self.shape)

    def tocsr(self):
        """
        Converts this array to a [`scipy.sparse.csr_matrix`][].

        Returns
        -------
        scipy.sparse.csr_matrix
            The result of the conversion.

        Raises
        ------
        ValueError
            If the array is not two-dimensional.
        ValueError
            If all the array doesn't have zero fill-values.

        See Also
        --------
        - [`sparse.COO.tocsc`][] : Convert to a [`scipy.sparse.csc_matrix`][].
        - [`sparse.COO.to_scipy_sparse`][] : Convert to a [`scipy.sparse.coo_matrix`][].
        - [`scipy.sparse.coo_matrix.tocsr`][] : Equivalent Scipy function.
        """
        check_zero_fill_value(self)

        if self._cache is not None:
            try:
                return self._csr
            except AttributeError:
                pass
            try:
                self._csr = self._csc.tocsr()
                return self._csr
            except AttributeError:
                pass

            self._csr = csr = self._tocsr()
        else:
            csr = self._tocsr()
        return csr

    def tocsc(self):
        """
        Converts this array to a [`scipy.sparse.csc_matrix`][].

        Returns
        -------
        scipy.sparse.csc_matrix
            The result of the conversion.

        Raises
        ------
        ValueError
            If the array is not two-dimensional.
        ValueError
            If the array doesn't have zero fill-values.

        See Also
        --------
        - [`sparse.COO.tocsr`][] : Convert to a [`scipy.sparse.csr_matrix`][].
        - [`sparse.COO.to_scipy_sparse`][] : Convert to a [`scipy.sparse.coo_matrix`][].
        - [`scipy.sparse.coo_matrix.tocsc`][] : Equivalent Scipy function.
        """
        check_zero_fill_value(self)

        if self._cache is not None:
            try:
                return self._csc
            except AttributeError:
                pass
            try:
                self._csc = self._csr.tocsc()
                return self._csc
            except AttributeError:
                pass

            self._csc = csc = self.tocsr().tocsc()
        else:
            csc = self.tocsr().tocsc()

        return csc

    def _sort_indices(self):
        """
        Sorts the :obj:`COO.coords` attribute. Also sorts the data in
        :obj:`COO.data` to match.

        Examples
        --------
        >>> coords = np.array([[1, 2, 0]], dtype=np.uint8)
        >>> data = np.array([4, 1, 3], dtype=np.uint8)
        >>> s = COO(coords, data, shape=(3,))
        >>> s._sort_indices()
        >>> s.coords  # doctest: +NORMALIZE_WHITESPACE
        array([[0, 1, 2]], dtype=uint8)
        >>> s.data  # doctest: +NORMALIZE_WHITESPACE
        array([3, 4, 1], dtype=uint8)
        """
        linear = self.linear_loc()

        if (np.diff(linear) >= 0).all():  # already sorted
            return

        order = np.argsort(linear, kind="mergesort")
        self.coords = self.coords[:, order]
        self.data = self.data[order]

    def _sum_duplicates(self):
        """
        Sums data corresponding to duplicates in :obj:`COO.coords`.

        See Also
        --------
        scipy.sparse.coo_matrix.sum_duplicates : Equivalent Scipy function.

        Examples
        --------
        >>> coords = np.array([[0, 1, 1, 2]], dtype=np.uint8)
        >>> data = np.array([6, 5, 2, 2], dtype=np.uint8)
        >>> s = COO(coords, data, shape=(3,))
        >>> s._sum_duplicates()
        >>> s.coords  # doctest: +NORMALIZE_WHITESPACE
        array([[0, 1, 2]], dtype=uint8)
        >>> s.data  # doctest: +NORMALIZE_WHITESPACE
        array([6, 7, 2], dtype=uint8)
        """
        # Inspired by scipy/sparse/coo.py::sum_duplicates
        # See https://github.com/scipy/scipy/blob/main/LICENSE.txt
        linear = self.linear_loc()
        unique_mask = np.diff(linear) != 0

        if unique_mask.sum() == len(unique_mask):  # already unique
            return

        unique_mask = np.append(True, unique_mask)

        coords = self.coords[:, unique_mask]
        (unique_inds,) = np.nonzero(unique_mask)
        data = np.add.reduceat(self.data, unique_inds, dtype=self.data.dtype)

        self.data = data
        self.coords = coords

    def _prune(self):
        """
        Prunes data so that if any fill-values are present, they are removed
        from both coordinates and data.

        Examples
        --------
        >>> coords = np.array([[0, 1, 2, 3]])
        >>> data = np.array([1, 0, 1, 2])
        >>> s = COO(coords, data, shape=(4,))
        >>> s._prune()
        >>> s.nnz
        3
        """
        mask = ~equivalent(self.data, self.fill_value)
        self.coords = self.coords[:, mask]
        self.data = self.data[mask]

    def broadcast_to(self, shape):
        """
        Performs the equivalent of [`sparse.COO`][]. Note that
        this function returns a new array instead of a view.

        Parameters
        ----------
        shape : tuple[int]
            The shape to broadcast the data to.

        Returns
        -------
        COO
            The broadcasted sparse array.

        Raises
        ------
        ValueError
            If the operand cannot be broadcast to the given shape.

        See Also
        --------
        [`numpy.broadcast_to`][] : NumPy equivalent function
        """
        return broadcast_to(self, shape)

    def maybe_densify(self, max_size=1000, min_density=0.25):
        """
        Converts this [`sparse.COO`][] array to a [`numpy.ndarray`][] if not too
        costly.

        Parameters
        ----------
        max_size : int
            Maximum number of elements in output
        min_density : float
            Minimum density of output

        Returns
        -------
        numpy.ndarray
            The dense array.

        Raises
        ------
        ValueError
            If the returned array would be too large.

        Examples
        --------
        Convert a small sparse array to a dense array.

        >>> s = COO.from_numpy(np.random.rand(2, 3, 4))
        >>> x = s.maybe_densify()
        >>> np.allclose(x, s.todense())
        True

        You can also specify the minimum allowed density or the maximum number
        of output elements. If both conditions are unmet, this method will throw
        an error.

        >>> x = np.zeros((5, 5), dtype=np.uint8)
        >>> x[2, 2] = 1
        >>> s = COO.from_numpy(x)
        >>> s.maybe_densify(max_size=5, min_density=0.25)
        Traceback (most recent call last):
            ...
        ValueError: Operation would require converting large sparse array to dense
        """
        if self.size > max_size and self.density < min_density:
            raise ValueError("Operation would require converting large sparse array to dense")

        return self.todense()

    def nonzero(self):
        """
        Get the indices where this array is nonzero.

        Returns
        -------
        idx : tuple[`numpy.ndarray`]
            The indices where this array is nonzero.

        See Also
        --------
        [`numpy.ndarray.nonzero`][] : NumPy equivalent function

        Raises
        ------
        ValueError
            If the array doesn't have zero fill-values.

        Examples
        --------
        >>> s = COO.from_numpy(np.eye(5))
        >>> s.nonzero()
        (array([0, 1, 2, 3, 4]), array([0, 1, 2, 3, 4]))
        """
        check_zero_fill_value(self)
        if self.ndim == 0:
            raise ValueError("`nonzero` is undefined for `self.ndim == 0`.")
        return tuple(self.coords)

    def asformat(self, format, **kwargs):
        """
        Convert this sparse array to a given format.

        Parameters
        ----------
        format : str
            A format string.

        Returns
        -------
        out : SparseArray
            The converted array.

        Raises
        ------
        NotImplementedError
            If the format isn't supported.
        """
        from .._utils import convert_format

        format = convert_format(format)

        if format == "gcxs":
            from .._compressed import GCXS

            return GCXS.from_coo(self, **kwargs)

        if len(kwargs) != 0:
            raise TypeError(f"Invalid keyword arguments provided: {kwargs}")

        if format == "coo":
            return self

        if format == "dok":
            from .._dok import DOK

            return DOK.from_coo(self, **kwargs)

        return self.asformat("gcxs", **kwargs).asformat(format, **kwargs)

    def isinf(self):
        """
        Tests each element ``x_i`` of the array to determine if equal to positive or negative infinity.
        """
        new_fill_value = bool(np.isinf(self.fill_value))
        new_data = np.isinf(self.data)

        return COO(
            self.coords,
            new_data,
            shape=self.shape,
            fill_value=new_fill_value,
            prune=True,
        )

    def isnan(self):
        """
        Tests each element ``x_i`` of the array to determine whether the element is ``NaN``.
        """
        new_fill_value = bool(np.isnan(self.fill_value))
        new_data = np.isnan(self.data)

        return COO(
            self.coords,
            new_data,
            shape=self.shape,
            fill_value=new_fill_value,
            prune=True,
        )
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