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# Source code for torch.functional

    from typing import (
        List, Tuple, Optional, Union, Any, Sequence, TYPE_CHECKING
    )

    import torch
    from torch._C import _add_docstr
    import torch.nn.functional as F
    from ._lowrank import svd_lowrank, pca_lowrank
    from .overrides import (
        has_torch_function, has_torch_function_unary, has_torch_function_variadic,
        handle_torch_function)
    from ._jit_internal import boolean_dispatch
    from ._jit_internal import _overload as overload

    Tensor = torch.Tensor
    from torch import _VF

    __all__ = [
        'atleast_1d',
        'atleast_2d',
        'atleast_3d',
        'align_tensors',
        'broadcast_shapes',
        'broadcast_tensors',
        'cartesian_prod',
        'block_diag',
        'cdist',
        'chain_matmul',
        'einsum',
        'istft',
        'lu',
        'norm',
        'meshgrid',
        'pca_lowrank',
        'split',
        'stft',
        'svd_lowrank',
        'tensordot',
        'unique',
        'unique_consecutive',
    ]

    def broadcast_tensors(*tensors):
        r"""broadcast_tensors(*tensors) -> List of Tensors

        Broadcasts the given tensors according to :ref:`broadcasting-semantics`.

        Args:
            *tensors: any number of tensors of the same type

        .. warning::

            More than one element of a broadcasted tensor may refer to a single
            memory location. As a result, in-place operations (especially ones that
            are vectorized) may result in incorrect behavior. If you need to write
            to the tensors, please clone them first.

        Example::

            >>> x = torch.arange(3).view(1, 3)
            >>> y = torch.arange(2).view(2, 1)
            >>> a, b = torch.broadcast_tensors(x, y)
            >>> a.size()
            torch.Size([2, 3])
            >>> a
            tensor([[0, 1, 2],
                    [0, 1, 2]])
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(tensors):
            return handle_torch_function(broadcast_tensors, tensors, *tensors)
        return _VF.broadcast_tensors(tensors)  # type: ignore[attr-defined]

    def broadcast_shapes(*shapes):
        r"""broadcast_shapes(*shapes) -> Size

        Similar to :func:`broadcast_tensors` but for shapes.

        This is equivalent to
        ``torch.broadcast_tensors(*map(torch.empty, shapes))[0].shape``
        but avoids the need create to intermediate tensors. This is useful for
        broadcasting tensors of common batch shape but different rightmost shape,
        e.g. to broadcast mean vectors with covariance matrices.

        Example::

            >>> torch.broadcast_shapes((2,), (3, 1), (1, 1, 1))
            torch.Size([1, 3, 2])

        Args:
            \*shapes (torch.Size): Shapes of tensors.

        Returns:
            shape (torch.Size): A shape compatible with all input shapes.

        Raises:
            RuntimeError: If shapes are incompatible.
        """
        # This wrapper exists to support variadic args.
        # TODO Move this to C++ once the jit has better support for torch.Size.
        if not torch.jit.is_tracing():
            max_len = 0
            for shape in shapes:
                if isinstance(shape, int):
                    if max_len < 1:
                        max_len = 1
                elif isinstance(shape, (tuple, list)):
                    s = len(shape)
                    if max_len < s:
                        max_len = s
            result = [1] * max_len
            for shape in shapes:
                if isinstance(shape, int):
                    shape = (shape,)
                if isinstance(shape, (tuple, list)):
                    for i in range(-1, -1 - len(shape), -1):
                        if shape[i] < 0:
                            raise RuntimeError(f"Trying to create tensor with negative dimension ({shape[i]}): ({shape[i]})")
                        if shape[i] == 1 or shape[i] == result[i]:
                            continue
                        if result[i] != 1:
                            raise RuntimeError("Shape mismatch: objects cannot be broadcast to a single shape")
                        result[i] = shape[i]
                else:
                    raise RuntimeError("Input shapes should be of type ints, a tuple of ints, or a list of ints, got ", shape)
            return torch.Size(result)
        else:
            # with implementation above, torch.jit.trace hardcodes the sizes which makes subsequent replays fail
            with torch.no_grad():
                scalar = torch.zeros((), device="cpu")
                tensors = [scalar.expand(shape) for shape in shapes]
                tensors = broadcast_tensors(*tensors)
                return tensors[0].shape

    def split(
        tensor: Tensor, split_size_or_sections: Union[int, List[int]], dim: int = 0
    ) -> Tuple[Tensor, ...]:
        r"""Splits the tensor into chunks. Each chunk is a view of the original tensor.

        If :attr:`split_size_or_sections` is an integer type, then :attr:`tensor` will
        be split into equally sized chunks (if possible). Last chunk will be smaller if
        the tensor size along the given dimension :attr:`dim` is not divisible by
        :attr:`split_size`.

        If :attr:`split_size_or_sections` is a list, then :attr:`tensor` will be split
        into ``len(split_size_or_sections)`` chunks with sizes in :attr:`dim` according
        to :attr:`split_size_or_sections`.

        Args:
            tensor (Tensor): tensor to split.
            split_size_or_sections (int) or (list(int)): size of a single chunk or
                list of sizes for each chunk
            dim (int): dimension along which to split the tensor.

        Example::

            >>> a = torch.arange(10).reshape(5, 2)
            >>> a
            tensor([[0, 1],
                    [2, 3],
                    [4, 5],
                    [6, 7],
                    [8, 9]])
            >>> torch.split(a, 2)
            (tensor([[0, 1],
                     [2, 3]]),
             tensor([[4, 5],
                     [6, 7]]),
             tensor([[8, 9]]))
            >>> torch.split(a, [1, 4])
            (tensor([[0, 1]]),
             tensor([[2, 3],
                     [4, 5],
                     [6, 7],
                     [8, 9]]))
        """
        if has_torch_function_unary(tensor):
            return handle_torch_function(
                split, (tensor,), tensor, split_size_or_sections, dim=dim)
        # Overwriting reason:
        # This dispatches to two ATen functions depending on the type of
        # split_size_or_sections. The branching code is in _tensor.py, which we
        # call here.
        return tensor.split(split_size_or_sections, dim)

    def einsum(*args: Any) -> Tensor:
        r"""einsum(equation, *operands) -> Tensor

        Sums the product of the elements of the input :attr:`operands` along dimensions specified using a notation
        based on the Einstein summation convention.

        Einsum allows computing many common multi-dimensional linear algebraic array operations by representing them
        in a short-hand format based on the Einstein summation convention, given by :attr:`equation`. The details of
        this format are described below, but the general idea is to label every dimension of the input :attr:`operands`
        with some subscript and define which subscripts are part of the output. The output is then computed by summing
        the product of the elements of the :attr:`operands` along the dimensions whose subscripts are not part of the
        output. For example, matrix multiplication can be computed using einsum as `torch.einsum("ij,jk->ik", A, B)`.
        Here, j is the summation subscript and i and k the output subscripts (see section below for more details on why).

        Equation:

            The :attr:`equation` string specifies the subscripts (letters in `[a-zA-Z]`) for each dimension of
            the input :attr:`operands` in the same order as the dimensions, separating subscripts for each operand by a
            comma (','), e.g. `'ij,jk'` specify subscripts for two 2D operands. The dimensions labeled with the same subscript
            must be broadcastable, that is, their size must either match or be `1`. The exception is if a subscript is
            repeated for the same input operand, in which case the dimensions labeled with this subscript for this operand
            must match in size and the operand will be replaced by its diagonal along these dimensions. The subscripts that
            appear exactly once in the :attr:`equation` will be part of the output, sorted in increasing alphabetical order.
            The output is computed by multiplying the input :attr:`operands` element-wise, with their dimensions aligned based
            on the subscripts, and then summing out the dimensions whose subscripts are not part of the output.

            Optionally, the output subscripts can be explicitly defined by adding an arrow ('->') at the end of the equation
            followed by the subscripts for the output. For instance, the following equation computes the transpose of a
            matrix multiplication: 'ij,jk->ki'. The output subscripts must appear at least once for some input operand and
            at most once for the output.

            Ellipsis ('...') can be used in place of subscripts to broadcast the dimensions covered by the ellipsis.
            Each input operand may contain at most one ellipsis which will cover the dimensions not covered by subscripts,
            e.g. for an input operand with 5 dimensions, the ellipsis in the equation `'ab...c'` cover the third and fourth
            dimensions. The ellipsis does not need to cover the same number of dimensions across the :attr:`operands` but the
            'shape' of the ellipsis (the size of the dimensions covered by them) must broadcast together. If the output is not
            explicitly defined with the arrow ('->') notation, the ellipsis will come first in the output (left-most dimensions),
            before the subscript labels that appear exactly once for the input operands. e.g. the following equation implements
            batch matrix multiplication `'...ij,...jk'`.

            A few final notes: the equation may contain whitespaces between the different elements (subscripts, ellipsis,
            arrow and comma) but something like `'. . .'` is not valid. An empty string `''` is valid for scalar operands.

        .. note::

            ``torch.einsum`` handles ellipsis ('...') differently from NumPy in that it allows dimensions
            covered by the ellipsis to be summed over, that is, ellipsis are not required to be part of the output.

        .. note::

            This function uses opt_einsum (https://optimized-einsum.readthedocs.io/en/stable/) to speed up computation or to
            consume less memory by optimizing contraction order. This optimization occurs when there are at least three
            inputs, since the order does not matter otherwise. Note that finding _the_ optimal path is an NP-hard problem,
            thus, opt_einsum relies on different heuristics to achieve near-optimal results. If opt_einsum is not available,
            the default order is to contract from left to right.

            To bypass this default behavior, add the following line to disable the usage of opt_einsum and skip path
            calculation: `torch.backends.opt_einsum.enabled = False`

            To specify which strategy you'd like for opt_einsum to compute the contraction path, add the following line:
            `torch.backends.opt_einsum.strategy = 'auto'`. The default strategy is 'auto', and we also support 'greedy' and
            'optimal'. Disclaimer that the runtime of 'optimal' is factorial in the number of inputs! See more details in
            the opt_einsum documentation (https://optimized-einsum.readthedocs.io/en/stable/path_finding.html).

        .. note::

            As of PyTorch 1.10 :func:`torch.einsum` also supports the sublist format (see examples below). In this format,
            subscripts for each operand are specified by sublists, list of integers in the range [0, 52). These sublists
            follow their operands, and an extra sublist can appear at the end of the input to specify the output's
            subscripts., e.g. `torch.einsum(op1, sublist1, op2, sublist2, ..., [subslist_out])`. Python's `Ellipsis` object
            may be provided in a sublist to enable broadcasting as described in the Equation section above.

        Args:
            equation (str): The subscripts for the Einstein summation.
            operands (List[Tensor]): The tensors to compute the Einstein summation of.

        Examples::

            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> # trace
            >>> torch.einsum('ii', torch.randn(4, 4))
            tensor(-1.2104)

            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> # diagonal
            >>> torch.einsum('ii->i', torch.randn(4, 4))
            tensor([-0.1034,  0.7952, -0.2433,  0.4545])

            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> # outer product
            >>> x = torch.randn(5)
            >>> y = torch.randn(4)
            >>> torch.einsum('i,j->ij', x, y)
            tensor([[ 0.1156, -0.2897, -0.3918,  0.4963],
                    [-0.3744,  0.9381,  1.2685, -1.6070],
                    [ 0.7208, -1.8058, -2.4419,  3.0936],
                    [ 0.1713, -0.4291, -0.5802,  0.7350],
                    [ 0.5704, -1.4290, -1.9323,  2.4480]])

            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> # batch matrix multiplication
            >>> As = torch.randn(3, 2, 5)
            >>> Bs = torch.randn(3, 5, 4)
            >>> torch.einsum('bij,bjk->bik', As, Bs)
            tensor([[[-1.0564, -1.5904,  3.2023,  3.1271],
                    [-1.6706, -0.8097, -0.8025, -2.1183]],

                    [[ 4.2239,  0.3107, -0.5756, -0.2354],
                    [-1.4558, -0.3460,  1.5087, -0.8530]],

                    [[ 2.8153,  1.8787, -4.3839, -1.2112],
                    [ 0.3728, -2.1131,  0.0921,  0.8305]]])

            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> # with sublist format and ellipsis
            >>> torch.einsum(As, [..., 0, 1], Bs, [..., 1, 2], [..., 0, 2])
            tensor([[[-1.0564, -1.5904,  3.2023,  3.1271],
                    [-1.6706, -0.8097, -0.8025, -2.1183]],

                    [[ 4.2239,  0.3107, -0.5756, -0.2354],
                    [-1.4558, -0.3460,  1.5087, -0.8530]],

                    [[ 2.8153,  1.8787, -4.3839, -1.2112],
                    [ 0.3728, -2.1131,  0.0921,  0.8305]]])

            >>> # batch permute
            >>> A = torch.randn(2, 3, 4, 5)
            >>> torch.einsum('...ij->...ji', A).shape
            torch.Size([2, 3, 5, 4])

            >>> # equivalent to torch.nn.functional.bilinear
            >>> A = torch.randn(3, 5, 4)
            >>> l = torch.randn(2, 5)
            >>> r = torch.randn(2, 4)
            >>> torch.einsum('bn,anm,bm->ba', l, A, r)
            tensor([[-0.3430, -5.2405,  0.4494],
                    [ 0.3311,  5.5201, -3.0356]])
        """
        import torch.backends.opt_einsum as opt_einsum
        # This wrapper exists to support variadic args.
        if len(args) < 2:
            raise ValueError('einsum(): must specify the equation string and at least one operand, '
                             'or at least one operand and its subscripts list')

        equation = None
        operands = None

        if isinstance(args[0], torch.Tensor):
            # Convert the subscript list format which is an interleaving of operand and its subscripts
            # list with an optional output subscripts list at the end (see documentation for more details on this)
            # to the equation string format by creating the equation string from the subscripts list and grouping the
            # input operands into a tensorlist (List[Tensor]).
            def parse_subscript(n: int) -> str:
                if n == Ellipsis:
                    return '...'
                if n >= 0 and n < 26:
                    return chr(ord('A') + n)
                if n >= 26 and n < 52:
                    return chr(ord('a') + n - 26)
                raise ValueError('einsum(): subscript in subscript list is not within the valid range [0, 52)')

            # Parse subscripts for input operands
            equation = ','.join(''.join(parse_subscript(s) for s in l) for l in args[1::2])

            # Parse optional output subscripts (provided when the number of arguments is odd)
            if len(args) % 2 == 1:
                equation += '->' + ''.join(parse_subscript(s) for s in args[-1])
                operands = args[:-1:2]
            else:
                operands = args[::2]
        else:
            equation = args[0]
            operands = args[1:]

        if has_torch_function(operands):
            return handle_torch_function(einsum, operands, equation, *operands)

        if len(operands) == 1 and isinstance(operands[0], (list, tuple)):
            # the old interface of passing the operands as one list argument
            _operands = operands[0]
            # recurse incase operands contains value that has torch function
            # in the original implementation this line is omitted
            return einsum(equation, *_operands)

        if len(operands) <= 2 or not opt_einsum.enabled:
            # the path for contracting 0 or 1 time(s) is already optimized
            # or the user has disabled using opt_einsum
            return _VF.einsum(equation, operands)  # type: ignore[attr-defined]

        path = None
        if opt_einsum.is_available():
            _opt_einsum = opt_einsum.get_opt_einsum()
            tupled_path = _opt_einsum.contract_path(equation, *operands, optimize=opt_einsum.strategy)[0]
            # flatten path for dispatching to C++
            path = [item for pair in tupled_path for item in pair]
        return _VF.einsum(equation, operands, path=path)  # type: ignore[attr-defined]

    # This wrapper exists to support variadic args.
    if TYPE_CHECKING:
        # The JIT doesn't understand Union, so only add type annotation for mypy
        def meshgrid(*tensors: Union[Tensor, List[Tensor]],
                     indexing: Optional[str] = None) -> Tuple[Tensor, ...]:
            return _meshgrid(*tensors, indexing=indexing)
    else:
        def meshgrid(*tensors, indexing: Optional[str] = None) -> Tuple[Tensor, ...]:
            r"""Creates grids of coordinates specified by the 1D inputs in `attr`:tensors.

            This is helpful when you want to visualize data over some
            range of inputs. See below for a plotting example.

            Given :math:`N` 1D tensors :math:`T_0 \ldots T_{N-1}` as
            inputs with corresponding sizes :math:`S_0 \ldots S_{N-1}`,
            this creates :math:`N` N-dimensional tensors :math:`G_0 \ldots
            G_{N-1}`, each with shape :math:`(S_0, ..., S_{N-1})` where
            the output :math:`G_i` is constructed by expanding :math:`T_i`
            to the result shape.

            .. note::
                0D inputs are treated equivalently to 1D inputs of a
                single element.

            .. warning::
                `torch.meshgrid(*tensors)` currently has the same behavior
                as calling `numpy.meshgrid(*arrays, indexing='ij')`.

                In the future `torch.meshgrid` will transition to
                `indexing='xy'` as the default.

                https://github.com/pytorch/pytorch/issues/50276 tracks
                this issue with the goal of migrating to NumPy's behavior.

            .. seealso::

                :func:`torch.cartesian_prod` has the same effect but it
                collects the data in a tensor of vectors.

            Args:
                tensors (list of Tensor): list of scalars or 1 dimensional tensors. Scalars will be
                    treated as tensors of size :math:`(1,)` automatically

                indexing: (str, optional): the indexing mode, either "xy"
                    or "ij", defaults to "ij". See warning for future changes.

                    If "xy" is selected, the first dimension corresponds
                    to the cardinality of the second input and the second
                    dimension corresponds to the cardinality of the first
                    input.

                    If "ij" is selected, the dimensions are in the same
                    order as the cardinality of the inputs.

            Returns:
                seq (sequence of Tensors): If the input has :math:`N`
                tensors of size :math:`S_0 \ldots S_{N-1}``, then the
                output will also have :math:`N` tensors, where each tensor
                is of shape :math:`(S_0, ..., S_{N-1})`.

            Example::

                >>> x = torch.tensor([1, 2, 3])
                >>> y = torch.tensor([4, 5, 6])

                Observe the element-wise pairings across the grid, (1, 4),
                (1, 5), ..., (3, 6). This is the same thing as the
                cartesian product.
                >>> grid_x, grid_y = torch.meshgrid(x, y, indexing='ij')
                >>> grid_x
                tensor([[1, 1, 1],
                        [2, 2, 2],
                        [3, 3, 3]])
                >>> grid_y
                tensor([[4, 5, 6],
                        [4, 5, 6],
                        [4, 5, 6]])

                This correspondence can be seen when these grids are
                stacked properly.
                >>> torch.equal(torch.cat(tuple(torch.dstack([grid_x, grid_y]))),
                ...             torch.cartesian_prod(x, y))
                True

                `torch.meshgrid` is commonly used to produce a grid for
                plotting.
                >>> # xdoctest: +REQUIRES(module:matplotlib)
                >>> # xdoctest: +REQUIRES(env:DOCTEST_SHOW)
                >>> import matplotlib.pyplot as plt
                >>> xs = torch.linspace(-5, 5, steps=100)
                >>> ys = torch.linspace(-5, 5, steps=100)
                >>> x, y = torch.meshgrid(xs, ys, indexing='xy')
                >>> z = torch.sin(torch.sqrt(x * x + y * y))
                >>> ax = plt.axes(projection='3d')
                >>> ax.plot_surface(x.numpy(), y.numpy(), z.numpy())
                >>> plt.show()

            .. image:: ../_static/img/meshgrid.png
                :width: 512

            """
            return _meshgrid(*tensors, indexing=indexing)

    def _meshgrid(*tensors, indexing: Optional[str]):
        if has_torch_function(tensors):
            return handle_torch_function(meshgrid, tensors, *tensors, indexing=indexing)
        if len(tensors) == 1 and isinstance(tensors[0], (list, tuple)):
            # the old interface of passing the operands as one list argument
            tensors = tensors[0]  # type: ignore[assignment]

        # Continue allowing call of old method that takes no indexing
        # kwarg for forward compatibility reasons.
        #
        # Remove this two weeks after landing.
        kwargs = {} if indexing is None else {'indexing': indexing}
        return _VF.meshgrid(tensors, **kwargs)  # type: ignore[attr-defined]

    def stft(input: Tensor, n_fft: int, hop_length: Optional[int] = None,
             win_length: Optional[int] = None, window: Optional[Tensor] = None,
             center: bool = True, pad_mode: str = 'reflect', normalized: bool = False,
             onesided: Optional[bool] = None,
             return_complex: Optional[bool] = None) -> Tensor:
        r"""Short-time Fourier transform (STFT).

        .. warning::
            From version 1.8.0, :attr:`return_complex` must always be given
            explicitly for real inputs and `return_complex=False` has been
            deprecated. Strongly prefer `return_complex=True` as in a future
            pytorch release, this function will only return complex tensors.

            Note that :func:`torch.view_as_real` can be used to recover a real
            tensor with an extra last dimension for real and imaginary components.

        The STFT computes the Fourier transform of short overlapping windows of the
        input. This giving frequency components of the signal as they change over
        time. The interface of this function is modeled after (but *not* a drop-in
        replacement for) librosa_ stft function.

        .. _librosa: https://librosa.org/doc/latest/generated/librosa.stft.html

        Ignoring the optional batch dimension, this method computes the following
        expression:

        .. math::
            X[\omega, m] = \sum_{k = 0}^{\text{win\_length-1}}%
                                \text{window}[k]\ \text{input}[m \times \text{hop\_length} + k]\ %
                                \exp\left(- j \frac{2 \pi \cdot \omega k}{\text{n\_fft}}\right),

        where :math:`m` is the index of the sliding window, and :math:`\omega` is
        the frequency :math:`0 \leq \omega < \text{n\_fft}` for ``onesided=False``,
        or :math:`0 \leq \omega < \lfloor \text{n\_fft} / 2 \rfloor + 1` for ``onesided=True``.

        * :attr:`input` must be either a 1-D time sequence or a 2-D batch of time
          sequences.

        * If :attr:`hop_length` is ``None`` (default), it is treated as equal to
          ``floor(n_fft / 4)``.

        * If :attr:`win_length` is ``None`` (default), it is treated as equal to
          :attr:`n_fft`.

        * :attr:`window` can be a 1-D tensor of size :attr:`win_length`, e.g., from
          :meth:`torch.hann_window`. If :attr:`window` is ``None`` (default), it is
          treated as if having :math:`1` everywhere in the window. If
          :math:`\text{win\_length} < \text{n\_fft}`, :attr:`window` will be padded on
          both sides to length :attr:`n_fft` before being applied.

        * If :attr:`center` is ``True`` (default), :attr:`input` will be padded on
          both sides so that the :math:`t`-th frame is centered at time
          :math:`t \times \text{hop\_length}`. Otherwise, the :math:`t`-th frame
          begins at time  :math:`t \times \text{hop\_length}`.

        * :attr:`pad_mode` determines the padding method used on :attr:`input` when
          :attr:`center` is ``True``. See :meth:`torch.nn.functional.pad` for
          all available options. Default is ``"reflect"``.

        * If :attr:`onesided` is ``True`` (default for real input), only values for
          :math:`\omega` in :math:`\left[0, 1, 2, \dots, \left\lfloor
          \frac{\text{n\_fft}}{2} \right\rfloor + 1\right]` are returned because
          the real-to-complex Fourier transform satisfies the conjugate symmetry,
          i.e., :math:`X[m, \omega] = X[m, \text{n\_fft} - \omega]^*`.
          Note if the input or window tensors are complex, then :attr:`onesided`
          output is not possible.

        * If :attr:`normalized` is ``True`` (default is ``False``), the function
          returns the normalized STFT results, i.e., multiplied by :math:`(\text{frame\_length})^{-0.5}`.

        * If :attr:`return_complex` is ``True`` (default if input is complex), the
          return is a ``input.dim() + 1`` dimensional complex tensor. If ``False``,
          the output is a ``input.dim() + 2`` dimensional real tensor where the last
          dimension represents the real and imaginary components.

        Returns either a complex tensor of size :math:`(* \times N \times T)` if
        :attr:`return_complex` is true, or a real tensor of size :math:`(* \times N
        \times T \times 2)`. Where :math:`*` is the optional batch size of
        :attr:`input`, :math:`N` is the number of frequencies where STFT is applied
        and :math:`T` is the total number of frames used.

        .. warning::
          This function changed signature at version 0.4.1. Calling with the
          previous signature may cause error or return incorrect result.

        Args:
            input (Tensor): the input tensor of shape `(B?, L)` where `B?` is an optional
                batch dimension
            n_fft (int): size of Fourier transform
            hop_length (int, optional): the distance between neighboring sliding window
                frames. Default: ``None`` (treated as equal to ``floor(n_fft / 4)``)
            win_length (int, optional): the size of window frame and STFT filter.
                Default: ``None``  (treated as equal to :attr:`n_fft`)
            window (Tensor, optional): the optional window function.
                Shape must be 1d and `<= n_fft`
                Default: ``None`` (treated as window of all :math:`1` s)
            center (bool, optional): whether to pad :attr:`input` on both sides so
                that the :math:`t`-th frame is centered at time :math:`t \times \text{hop\_length}`.
                Default: ``True``
            pad_mode (str, optional): controls the padding method used when
                :attr:`center` is ``True``. Default: ``"reflect"``
            normalized (bool, optional): controls whether to return the normalized STFT results
                 Default: ``False``
            onesided (bool, optional): controls whether to return half of results to
                avoid redundancy for real inputs.
                Default: ``True`` for real :attr:`input` and :attr:`window`, ``False`` otherwise.
            return_complex (bool, optional): whether to return a complex tensor, or
                a real tensor with an extra last dimension for the real and
                imaginary components.

                .. versionchanged:: 2.0
                   ``return_complex`` is now a required argument for real inputs,
                   as the default is being transitioned to ``True``.

                .. deprecated:: 2.0
                   ``return_complex=False`` is deprecated, instead use ``return_complex=True``
                   Note that calling :func:`torch.view_as_real` on the output will
                   recover the deprecated output format.

        Returns:
            Tensor: A tensor containing the STFT result with shape `(B?, N, T, C?)` where
               - `B?` is an optional batch dimnsion from the input
               - `N` is the number of frequency samples, `(n_fft // 2) + 1` for
                 `onesided=True`, or otherwise `n_fft`.
               - `T` is the number of frames, `1 + L // hop_length`
                 for `center=True`, or `1 + (L - n_fft) // hop_length` otherwise.
               - `C?` is an optional length-2 dimension of real and imaginary
                 components, present when `return_complex=False`.

        """
        if has_torch_function_unary(input):
            return handle_torch_function(
                stft, (input,), input, n_fft, hop_length=hop_length, win_length=win_length,
                window=window, center=center, pad_mode=pad_mode, normalized=normalized,
                onesided=onesided, return_complex=return_complex)
        # NOTE: Do not edit. This code will be removed once the forward-compatibility
        #       period is over for PR #73432
        if center:
            signal_dim = input.dim()
            extended_shape = [1] * (3 - signal_dim) + list(input.size())
            pad = int(n_fft // 2)
            input = F.pad(input.view(extended_shape), [pad, pad], pad_mode)
            input = input.view(input.shape[-signal_dim:])
        return _VF.stft(input, n_fft, hop_length, win_length, window,  # type: ignore[attr-defined]
                        normalized, onesided, return_complex)

    istft = _add_docstr(
        torch.istft,
        "istft(input, n_fft, hop_length=None, win_length=None, window=None, center=True, "
        "normalized=False, onesided=None, length=None, return_complex=False) -> Tensor:\n"
        r"""
    Inverse short time Fourier Transform. This is expected to be the inverse of :func:`~torch.stft`.

    It has the same parameters (+ additional optional parameter of :attr:`length`) and it should return the
    least squares estimation of the original signal. The algorithm will check using the NOLA condition (
    nonzero overlap).

    Important consideration in the parameters :attr:`window` and :attr:`center` so that the envelop
    created by the summation of all the windows is never zero at certain point in time. Specifically,
    :math:`\sum_{t=-\infty}^{\infty} |w|^2[n-t\times hop\_length] \cancel{=} 0`.

    Since :func:`~torch.stft` discards elements at the end of the signal if they do not fit in a frame,
    ``istft`` may return a shorter signal than the original signal (can occur if :attr:`center` is False
    since the signal isn't padded). If `length` is given in the arguments and is longer than expected,
    ``istft`` will pad zeros to the end of the returned signal.

    If :attr:`center` is ``True``, then there will be padding e.g. ``'constant'``, ``'reflect'``, etc.
    Left padding can be trimmed off exactly because they can be calculated but right padding cannot be
    calculated without additional information.

    Example: Suppose the last window is:
    ``[17, 18, 0, 0, 0]`` vs ``[18, 0, 0, 0, 0]``

    The :attr:`n_fft`, :attr:`hop_length`, :attr:`win_length` are all the same which prevents the calculation
    of right padding. These additional values could be zeros or a reflection of the signal so providing
    :attr:`length` could be useful. If :attr:`length` is ``None`` then padding will be aggressively removed
    (some loss of signal).

    [1] D. W. Griffin and J. S. Lim, "Signal estimation from modified short-time Fourier transform,"
    IEEE Trans. ASSP, vol.32, no.2, pp.236-243, Apr. 1984.

    Args:
        input (Tensor): The input tensor. Expected to be in the format of :func:`~torch.stft`,
            output. That is a complex tensor of shape `(B?, N, T)` where

            - `B?` is an optional batch dimension
            - `N` is the number of frequency samples, `(n_fft // 2) + 1`
              for onesided input, or otherwise `n_fft`.
            - `T` is the number of frames, `1 + length // hop_length` for centered stft,
              or `1 + (length - n_fft) // hop_length` otherwise.

            .. versionchanged:: 2.0
                Real datatype inputs are no longer supported. Input must now have a
                complex datatype, as returned by ``stft(..., return_complex=True)``.
        n_fft (int): Size of Fourier transform
        hop_length (Optional[int]): The distance between neighboring sliding window frames.
            (Default: ``n_fft // 4``)
        win_length (Optional[int]): The size of window frame and STFT filter. (Default: ``n_fft``)
        window (Optional[torch.Tensor]): The optional window function.
            Shape must be 1d and `<= n_fft`
            (Default: ``torch.ones(win_length)``)
        center (bool): Whether :attr:`input` was padded on both sides so that the :math:`t`-th frame is
            centered at time :math:`t \times \text{hop\_length}`.
            (Default: ``True``)
        normalized (bool): Whether the STFT was normalized. (Default: ``False``)
        onesided (Optional[bool]): Whether the STFT was onesided.
            (Default: ``True`` if `n_fft != fft_size` in the input size)
        length (Optional[int]): The amount to trim the signal by (i.e. the
            original signal length). Defaults to `(T - 1) * hop_length` for
            centered stft, or `n_fft + (T - 1) * hop_length` otherwise, where `T`
            is the number of input frames.
        return_complex (Optional[bool]):
            Whether the output should be complex, or if the input should be
            assumed to derive from a real signal and window.
            Note that this is incompatible with ``onesided=True``.
            (Default: ``False``)

    Returns:
        Tensor: Least squares estimation of the original signal of shape `(B?, length)` where
            `B?` is an optional batch dimension from the input tensor.
    """)

    if TYPE_CHECKING:
        # These _impl functions return a variable number of tensors as output with
        # __torch_function__; tuple unpacking is done already rather than being
        # done by the caller of the _impl function
        _unique_impl_out = Any
    else:
        _unique_impl_out = Tuple[Tensor, Tensor, Tensor]

    def _unique_impl(input: Tensor, sorted: bool = True,
                     return_inverse: bool = False, return_counts: bool = False,
                     dim: Optional[int] = None) -> _unique_impl_out:
        r"""unique(input, sorted=True, return_inverse=False, return_counts=False, dim=None) -> Tuple[Tensor, Tensor, Tensor]

        Returns the unique elements of the input tensor.

        .. note:: This function is different from :func:`torch.unique_consecutive` in the sense that
            this function also eliminates non-consecutive duplicate values.

        .. note:: Currently in the CUDA implementation and the CPU implementation when dim is specified,
            `torch.unique` always sort the tensor at the beginning regardless of the `sort` argument.
            Sorting could be slow, so if your input tensor is already sorted, it is recommended to use
            :func:`torch.unique_consecutive` which avoids the sorting.

        Args:
            input (Tensor): the input tensor
            sorted (bool): Whether to sort the unique elements in ascending order
                before returning as output.
            return_inverse (bool): Whether to also return the indices for where
                elements in the original input ended up in the returned unique list.
            return_counts (bool): Whether to also return the counts for each unique
                element.
            dim (int, optional): the dimension to operate upon. If ``None``, the
                unique of the flattened input is returned. Otherwise, each of the
                tensors indexed by the given dimension is treated as one of the
                elements to apply the unique operation upon. See examples for more
                details. Default: ``None``

        Returns:
            (Tensor, Tensor (optional), Tensor (optional)): A tensor or a tuple of tensors containing

                - **output** (*Tensor*): the output list of unique scalar elements.
                - **inverse_indices** (*Tensor*): (optional) if
                  :attr:`return_inverse` is True, there will be an additional
                  returned tensor (same shape as input) representing the indices
                  for where elements in the original input map to in the output;
                  otherwise, this function will only return a single tensor.
                - **counts** (*Tensor*): (optional) if
                  :attr:`return_counts` is True, there will be an additional
                  returned tensor (same shape as output or output.size(dim),
                  if dim was specified) representing the number of occurrences
                  for each unique value or tensor.

        Example::

            >>> output = torch.unique(torch.tensor([1, 3, 2, 3], dtype=torch.long))
            >>> output
            tensor([1, 2, 3])

            >>> output, inverse_indices = torch.unique(
            ...     torch.tensor([1, 3, 2, 3], dtype=torch.long), sorted=True, return_inverse=True)
            >>> output
            tensor([1, 2, 3])
            >>> inverse_indices
            tensor([0, 2, 1, 2])

            >>> output, inverse_indices = torch.unique(
            ...     torch.tensor([[1, 3], [2, 3]], dtype=torch.long), sorted=True, return_inverse=True)
            >>> output
            tensor([1, 2, 3])
            >>> inverse_indices
            tensor([[0, 2],
                    [1, 2]])

            >>> a = torch.tensor([
            ...     [
            ...         [1, 1, 0, 0],
            ...         [1, 1, 0, 0],
            ...         [0, 0, 1, 1],
            ...     ],
            ...     [
            ...         [0, 0, 1, 1],
            ...         [0, 0, 1, 1],
            ...         [1, 1, 1, 1],
            ...     ],
            ...     [
            ...         [1, 1, 0, 0],
            ...         [1, 1, 0, 0],
            ...         [0, 0, 1, 1],
            ...     ],
            ... ])

            >>> # If we call `torch.unique(a, dim=0)`, each of the tensors `a[idx, :, :]`
            >>> # will be compared. We can see that `a[0, :, :]` and `a[2, :, :]` match
            >>> # each other, so one of them will be removed.
            >>> (a[0, :, :] == a[2, :, :]).all()
            tensor(True)
            >>> a_unique_dim0 = torch.unique(a, dim=0)
            >>> a_unique_dim0
            tensor([[[0, 0, 1, 1],
                     [0, 0, 1, 1],
                     [1, 1, 1, 1]],
                    [[1, 1, 0, 0],
                     [1, 1, 0, 0],
                     [0, 0, 1, 1]]])

            >>> # Notice which sub-tensors from `a` match with the sub-tensors from
            >>> # `a_unique_dim0`:
            >>> (a_unique_dim0[0, :, :] == a[1, :, :]).all()
            tensor(True)
            >>> (a_unique_dim0[1, :, :] == a[0, :, :]).all()
            tensor(True)

            >>> # For `torch.unique(a, dim=1)`, each of the tensors `a[:, idx, :]` are
            >>> # compared. `a[:, 0, :]` and `a[:, 1, :]` match each other, so one of
            >>> # them will be removed.
            >>> (a[:, 0, :] == a[:, 1, :]).all()
            tensor(True)
            >>> torch.unique(a, dim=1)
            tensor([[[0, 0, 1, 1],
                     [1, 1, 0, 0]],
                    [[1, 1, 1, 1],
                     [0, 0, 1, 1]],
                    [[0, 0, 1, 1],
                     [1, 1, 0, 0]]])

            >>> # For `torch.unique(a, dim=2)`, the tensors `a[:, :, idx]` are compared.
            >>> # `a[:, :, 0]` and `a[:, :, 1]` match each other. Also, `a[:, :, 2]` and
            >>> # `a[:, :, 3]` match each other as well. So in this case, two of the
            >>> # sub-tensors will be removed.
            >>> (a[:, :, 0] == a[:, :, 1]).all()
            tensor(True)
            >>> (a[:, :, 2] == a[:, :, 3]).all()
            tensor(True)
            >>> torch.unique(a, dim=2)
            tensor([[[0, 1],
                     [0, 1],
                     [1, 0]],
                    [[1, 0],
                     [1, 0],
                     [1, 1]],
                    [[0, 1],
                     [0, 1],
                     [1, 0]]])
        """
        if has_torch_function_unary(input):
            return handle_torch_function(
                unique, (input,), input, sorted=sorted, return_inverse=return_inverse,
                return_counts=return_counts, dim=dim)

        if dim is not None:
            output, inverse_indices, counts = _VF.unique_dim(
                input,
                dim,
                sorted=sorted,
                return_inverse=return_inverse,
                return_counts=return_counts,
            )
        else:
            output, inverse_indices, counts = torch._unique2(
                input,
                sorted=sorted,
                return_inverse=return_inverse,
                return_counts=return_counts,
            )
        return output, inverse_indices, counts

    def _unique_consecutive_impl(input: Tensor, return_inverse: bool = False,
                                 return_counts: bool = False,
                                 dim: Optional[int] = None) -> _unique_impl_out:
        r"""Eliminates all but the first element from every consecutive group of equivalent elements.

        .. note:: This function is different from :func:`torch.unique` in the sense that this function
            only eliminates consecutive duplicate values. This semantics is similar to `std::unique`
            in C++.

        Args:
            input (Tensor): the input tensor
            return_inverse (bool): Whether to also return the indices for where
                elements in the original input ended up in the returned unique list.
            return_counts (bool): Whether to also return the counts for each unique
                element.
            dim (int): the dimension to apply unique. If ``None``, the unique of the
                flattened input is returned. default: ``None``

        Returns:
            (Tensor, Tensor (optional), Tensor (optional)): A tensor or a tuple of tensors containing

                - **output** (*Tensor*): the output list of unique scalar elements.
                - **inverse_indices** (*Tensor*): (optional) if
                  :attr:`return_inverse` is True, there will be an additional
                  returned tensor (same shape as input) representing the indices
                  for where elements in the original input map to in the output;
                  otherwise, this function will only return a single tensor.
                - **counts** (*Tensor*): (optional) if
                  :attr:`return_counts` is True, there will be an additional
                  returned tensor (same shape as output or output.size(dim),
                  if dim was specified) representing the number of occurrences
                  for each unique value or tensor.

        Example::

            >>> x = torch.tensor([1, 1, 2, 2, 3, 1, 1, 2])
            >>> output = torch.unique_consecutive(x)
            >>> output
            tensor([1, 2, 3, 1, 2])

            >>> output, inverse_indices = torch.unique_consecutive(x, return_inverse=True)
            >>> output
            tensor([1, 2, 3, 1, 2])
            >>> inverse_indices
            tensor([0, 0, 1, 1, 2, 3, 3, 4])

            >>> output, counts = torch.unique_consecutive(x, return_counts=True)
            >>> output
            tensor([1, 2, 3, 1, 2])
            >>> counts
            tensor([2, 2, 1, 2, 1])
        """
        if has_torch_function_unary(input):
            return handle_torch_function(
                unique_consecutive, (input,), input, return_inverse=return_inverse,
                return_counts=return_counts, dim=dim)
        output, inverse_indices, counts = _VF.unique_consecutive(  # type: ignore[attr-defined]
            input, return_inverse=return_inverse, return_counts=return_counts, dim=dim)
        return output, inverse_indices, counts

    def _return_counts(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
        # type: (Tensor, bool, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

        if has_torch_function_unary(input):
            return _unique_impl(input, sorted, return_inverse, return_counts, dim)

        output, _, counts = _unique_impl(input, sorted, return_inverse, return_counts, dim)
        return output, counts

    def _return_output(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
        # type: (Tensor, bool, bool, bool, Optional[int]) -> Tensor

        if has_torch_function_unary(input):
            return _unique_impl(input, sorted, return_inverse, return_counts, dim)

        output, _, _ = _unique_impl(input, sorted, return_inverse, return_counts, dim)
        return output

    def _return_inverse(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
        # type: (Tensor, bool, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

        if has_torch_function_unary(input):
            return _unique_impl(input, sorted, return_inverse, return_counts, dim)

        output, inverse_indices, _ = _unique_impl(input, sorted, return_inverse, return_counts, dim)
        return output, inverse_indices

    _return_inverse_false = boolean_dispatch(
        arg_name='return_counts',
        arg_index=3,
        default=False,
        if_true=_return_counts,
        if_false=_return_output,
        module_name=__name__,
        func_name='unique')

    _return_inverse_true = boolean_dispatch(
        arg_name='return_counts',
        arg_index=3,
        default=False,
        if_true=_unique_impl,
        if_false=_return_inverse,
        module_name=__name__,
        func_name='unique')

    # The return type of unique depends on `return_inverse`, and `return_counts` so in order to
    # resolve the output type in TorchScript we need to statically know the value of both parameters

    unique = boolean_dispatch(
        arg_name='return_inverse',
        arg_index=2,
        default=False,
        if_true=_return_inverse_true,
        if_false=_return_inverse_false,
        module_name=__name__,
        func_name='unique')
    unique.__doc__ = _unique_impl.__doc__

    def _consecutive_return_counts(input, return_inverse=False, return_counts=False, dim=None):
        # type: (Tensor, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

        if has_torch_function_unary(input):
            return _unique_consecutive_impl(input, return_inverse, return_counts, dim)

        output, _, counts = _unique_consecutive_impl(input, return_inverse, return_counts, dim)
        return output, counts

    def _consecutive_return_output(input, return_inverse=False, return_counts=False, dim=None):
        # type: (Tensor, bool, bool, Optional[int]) -> Tensor

        if has_torch_function_unary(input):
            return _unique_consecutive_impl(input, return_inverse, return_counts, dim)

        output, _, _ = _unique_consecutive_impl(input, return_inverse, return_counts, dim)
        return output

    def _consecutive_return_inverse(input, return_inverse=False, return_counts=False, dim=None):
        # type: (Tensor, bool, bool, Optional[int]) -> Tuple[Tensor, Tensor]

        if has_torch_function_unary(input):
            return _unique_consecutive_impl(input, return_inverse, return_counts, dim)

        output, inverse_indices, _ = _unique_consecutive_impl(input, return_inverse, return_counts, dim)
        return output, inverse_indices

    _consecutive_return_inverse_false = boolean_dispatch(
        arg_name='return_counts',
        arg_index=1,
        default=False,
        if_true=_consecutive_return_counts,
        if_false=_consecutive_return_output,
        module_name=__name__,
        func_name='unique_consecutive')

    _consecutive_return_inverse_true = boolean_dispatch(
        arg_name='return_counts',
        arg_index=1,
        default=False,
        if_true=_unique_consecutive_impl,
        if_false=_consecutive_return_inverse,
        module_name=__name__,
        func_name='unique_consecutive')

    # The return type of unique depends on `return_inverse`, and `return_counts` so in order to
    # resolve the output type in TorchScript we need to statically know the value of both parameters

    unique_consecutive = boolean_dispatch(
        arg_name='return_inverse',
        arg_index=2,
        default=False,
        if_true=_consecutive_return_inverse_true,
        if_false=_consecutive_return_inverse_false,
        module_name=__name__,
        func_name='unique_consecutive')
    unique_consecutive.__doc__ = _unique_consecutive_impl.__doc__

    if TYPE_CHECKING:
        pass
        # There's no good way to use this type annotation without breaking JIT
        # overloads. So leave untyped for mypy for now.
    else:
        @overload
        def tensordot(a, b, dims: int = 2, out: Optional[torch.Tensor] = None):
            pass

        @overload  # noqa: F811
        def tensordot(a, b, dims: Tuple[List[int], List[int]], out: Optional[torch.Tensor] = None):  # noqa: F811
            pass

        @overload  # noqa: F811
        def tensordot(a, b, dims: List[List[int]], out: Optional[torch.Tensor] = None):  # noqa: F811
            pass

        @overload  # noqa: F811
        def tensordot(a, b, dims: torch.Tensor, out: Optional[torch.Tensor] = None):  # noqa: F811
            pass

    def tensordot(a, b, dims=2, out: Optional[torch.Tensor] = None):  # noqa: F811
        r"""Returns a contraction of a and b over multiple dimensions.

        :attr:`tensordot` implements a generalized matrix product.

        Args:
          a (Tensor): Left tensor to contract
          b (Tensor): Right tensor to contract
          dims (int or Tuple[List[int], List[int]] or List[List[int]] containing two lists or Tensor): number of dimensions to
             contract or explicit lists of dimensions for :attr:`a` and
             :attr:`b` respectively

        When called with a non-negative integer argument :attr:`dims` = :math:`d`, and
        the number of dimensions of :attr:`a` and :attr:`b` is :math:`m` and :math:`n`,
        respectively, :func:`~torch.tensordot` computes

        .. math::
            r_{i_0,...,i_{m-d}, i_d,...,i_n}
              = \sum_{k_0,...,k_{d-1}} a_{i_0,...,i_{m-d},k_0,...,k_{d-1}} \times b_{k_0,...,k_{d-1}, i_d,...,i_n}.

        When called with :attr:`dims` of the list form, the given dimensions will be contracted
        in place of the last :math:`d` of :attr:`a` and the first :math:`d` of :math:`b`. The sizes
        in these dimensions must match, but :func:`~torch.tensordot` will deal with broadcasted
        dimensions.

        Examples::

            >>> a = torch.arange(60.).reshape(3, 4, 5)
            >>> b = torch.arange(24.).reshape(4, 3, 2)
            >>> torch.tensordot(a, b, dims=([1, 0], [0, 1]))
            tensor([[4400., 4730.],
                    [4532., 4874.],
                    [4664., 5018.],
                    [4796., 5162.],
                    [4928., 5306.]])

            >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_CUDA)
            >>> a = torch.randn(3, 4, 5, device='cuda')
            >>> b = torch.randn(4, 5, 6, device='cuda')
            >>> c = torch.tensordot(a, b, dims=2).cpu()
            tensor([[ 8.3504, -2.5436,  6.2922,  2.7556, -1.0732,  3.2741],
                    [ 3.3161,  0.0704,  5.0187, -0.4079, -4.3126,  4.8744],
                    [ 0.8223,  3.9445,  3.2168, -0.2400,  3.4117,  1.7780]])

            >>> a = torch.randn(3, 5, 4, 6)
            >>> b = torch.randn(6, 4, 5, 3)
            >>> torch.tensordot(a, b, dims=([2, 1, 3], [1, 2, 0]))
            tensor([[  7.7193,  -2.4867, -10.3204],
                    [  1.5513, -14.4737,  -6.5113],
                    [ -0.2850,   4.2573,  -3.5997]])
        """
        if has_torch_function_variadic(a, b):
            return handle_torch_function(tensordot, (a, b), a, b, dims=dims, out=out)

        if not isinstance(dims, (tuple, list, torch.Tensor, int)):
            raise RuntimeError("tensordot expects dims to be int or "
                               + "Tuple[List[int], List[int]] or "
                               + "List[List[int]] containing two lists, but got "
                               + f"dims={dims}")

        dims_a: List[int] = []
        dims_b: List[int] = []

        if isinstance(dims, (tuple, list)):
            dims_a, dims_b = dims

        if isinstance(dims, torch.Tensor):
            num_elements = dims.numel()
            if num_elements > 1:
                assert dims.size()[0] == 2
                dims_a = torch.jit.annotate(List[int], dims[0].tolist())
                dims_b = torch.jit.annotate(List[int], dims[1].tolist())
            else:
                dims_val = int(dims.item())
                if dims_val < 0:
                    raise RuntimeError(f"tensordot expects dims >= 0, but got dims={dims}")
                dims_a = list(range(-dims_val, 0))
                dims_b = list(range(dims_val))

        if isinstance(dims, int):
            if dims < 0:
                raise RuntimeError(f"tensordot expects dims >= 0, but got dims={dims}")
            if dims > min(a.dim(), b.dim()):
                raise RuntimeError(f"tensordot expects dims < ndim_a or ndim_b, but got dims={dims}")
            dims_a = list(range(-dims, 0))
            dims_b = list(range(dims))

        if out is None:
            return _VF.tensordot(a, b, dims_a, dims_b)  # type: ignore[attr-defined]
        else:
            return _VF.tensordot(a, b, dims_a, dims_b, out=out)  # type: ignore[attr-defined]

    def cartesian_prod(*tensors: Tensor) -> Tensor:
        """Do cartesian product of the given sequence of tensors. The behavior is similar to
        python's `itertools.product`.

        Args:
            *tensors: any number of 1 dimensional tensors.

        Returns:
            Tensor: A tensor equivalent to converting all the input tensors into lists,
            do `itertools.product` on these lists, and finally convert the resulting list
            into tensor.

        Example::

            >>> import itertools
            >>> a = [1, 2, 3]
            >>> b = [4, 5]
            >>> list(itertools.product(a, b))
            [(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)]
            >>> tensor_a = torch.tensor(a)
            >>> tensor_b = torch.tensor(b)
            >>> torch.cartesian_prod(tensor_a, tensor_b)
            tensor([[1, 4],
                    [1, 5],
                    [2, 4],
                    [2, 5],
                    [3, 4],
                    [3, 5]])
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(tensors):
            return handle_torch_function(cartesian_prod, tensors, *tensors)
        return _VF.cartesian_prod(tensors)  # type: ignore[attr-defined]

    def block_diag(*tensors):
        """Create a block diagonal matrix from provided tensors.

        Args:
            *tensors: One or more tensors with 0, 1, or 2 dimensions.

        Returns:
            Tensor: A 2 dimensional tensor with all the input tensors arranged in
            order such that their upper left and lower right corners are
            diagonally adjacent. All other elements are set to 0.

        Example::

            >>> import torch
            >>> A = torch.tensor([[0, 1], [1, 0]])
            >>> B = torch.tensor([[3, 4, 5], [6, 7, 8]])
            >>> C = torch.tensor(7)
            >>> D = torch.tensor([1, 2, 3])
            >>> E = torch.tensor([[4], [5], [6]])
            >>> torch.block_diag(A, B, C, D, E)
            tensor([[0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
                    [1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                    [0, 0, 3, 4, 5, 0, 0, 0, 0, 0],
                    [0, 0, 6, 7, 8, 0, 0, 0, 0, 0],
                    [0, 0, 0, 0, 0, 7, 0, 0, 0, 0],
                    [0, 0, 0, 0, 0, 0, 1, 2, 3, 0],
                    [0, 0, 0, 0, 0, 0, 0, 0, 0, 4],
                    [0, 0, 0, 0, 0, 0, 0, 0, 0, 5],
                    [0, 0, 0, 0, 0, 0, 0, 0, 0, 6]])
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(tensors):
            return handle_torch_function(block_diag, tensors, *tensors)
        return torch._C._VariableFunctions.block_diag(tensors)  # type: ignore[attr-defined]

    def cdist(x1, x2, p=2., compute_mode='use_mm_for_euclid_dist_if_necessary'):
        # type: (Tensor, Tensor, float, str) -> (Tensor)
        r"""Computes batched the p-norm distance between each pair of the two collections of row vectors.

        Args:
            x1 (Tensor): input tensor of shape :math:`B \times P \times M`.
            x2 (Tensor): input tensor of shape :math:`B \times R \times M`.
            p: p value for the p-norm distance to calculate between each vector pair
                :math:`\in [0, \infty]`.
            compute_mode:
                'use_mm_for_euclid_dist_if_necessary' - will use matrix multiplication approach to calculate
                euclidean distance (p = 2) if P > 25 or R > 25
                'use_mm_for_euclid_dist' - will always use matrix multiplication approach to calculate
                euclidean distance (p = 2)
                'donot_use_mm_for_euclid_dist' - will never use matrix multiplication approach to calculate
                euclidean distance (p = 2)
                Default: use_mm_for_euclid_dist_if_necessary.

        If x1 has shape :math:`B \times P \times M` and x2 has shape :math:`B \times R \times M` then the
        output will have shape :math:`B \times P \times R`.

        This function is equivalent to `scipy.spatial.distance.cdist(input,'minkowski', p=p)`
        if :math:`p \in (0, \infty)`. When :math:`p = 0` it is equivalent to
        `scipy.spatial.distance.cdist(input, 'hamming') * M`. When :math:`p = \infty`, the closest
        scipy function is `scipy.spatial.distance.cdist(xn, lambda x, y: np.abs(x - y).max())`.

        Example:

            >>> a = torch.tensor([[0.9041,  0.0196], [-0.3108, -2.4423], [-0.4821,  1.059]])
            >>> a
            tensor([[ 0.9041,  0.0196],
                    [-0.3108, -2.4423],
                    [-0.4821,  1.0590]])
            >>> b = torch.tensor([[-2.1763, -0.4713], [-0.6986,  1.3702]])
            >>> b
            tensor([[-2.1763, -0.4713],
                    [-0.6986,  1.3702]])
            >>> torch.cdist(a, b, p=2)
            tensor([[3.1193, 2.0959],
                    [2.7138, 3.8322],
                    [2.2830, 0.3791]])
        """
        if has_torch_function_variadic(x1, x2):
            return handle_torch_function(
                cdist, (x1, x2), x1, x2, p=p, compute_mode=compute_mode)
        if compute_mode == 'use_mm_for_euclid_dist_if_necessary':
            return _VF.cdist(x1, x2, p, None)  # type: ignore[attr-defined]
        elif compute_mode == 'use_mm_for_euclid_dist':
            return _VF.cdist(x1, x2, p, 1)  # type: ignore[attr-defined]
        elif compute_mode == 'donot_use_mm_for_euclid_dist':
            return _VF.cdist(x1, x2, p, 2)  # type: ignore[attr-defined]
        else:
            raise ValueError(f"{compute_mode} is not a valid value for compute_mode")

    def atleast_1d(*tensors):
        r"""
        Returns a 1-dimensional view of each input tensor with zero dimensions.
        Input tensors with one or more dimensions are returned as-is.

        Args:
            input (Tensor or list of Tensors)

        Returns:
            output (Tensor or tuple of Tensors)

        Example::

            >>> x = torch.arange(2)
            >>> x
            tensor([0, 1])
            >>> torch.atleast_1d(x)
            tensor([0, 1])
            >>> x = torch.tensor(1.)
            >>> x
            tensor(1.)
            >>> torch.atleast_1d(x)
            tensor([1.])
            >>> x = torch.tensor(0.5)
            >>> y = torch.tensor(1.)
            >>> torch.atleast_1d((x, y))
            (tensor([0.5000]), tensor([1.]))
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(tensors):
            return handle_torch_function(atleast_1d, tensors, *tensors)
        if len(tensors) == 1:
            tensors = tensors[0]
        return _VF.atleast_1d(tensors)  # type: ignore[attr-defined]

    def atleast_2d(*tensors):
        r"""
        Returns a 2-dimensional view of each input tensor with zero dimensions.
        Input tensors with two or more dimensions are returned as-is.

        Args:
            input (Tensor or list of Tensors)

        Returns:
            output (Tensor or tuple of Tensors)

        Example::

            >>> x = torch.tensor(1.)
            >>> x
            tensor(1.)
            >>> torch.atleast_2d(x)
            tensor([[1.]])
            >>> x = torch.arange(4).view(2, 2)
            >>> x
            tensor([[0, 1],
                    [2, 3]])
            >>> torch.atleast_2d(x)
            tensor([[0, 1],
                    [2, 3]])
            >>> x = torch.tensor(0.5)
            >>> y = torch.tensor(1.)
            >>> torch.atleast_2d((x, y))
            (tensor([[0.5000]]), tensor([[1.]]))
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(tensors):
            return handle_torch_function(atleast_2d, tensors, *tensors)
        if len(tensors) == 1:
            tensors = tensors[0]
        return _VF.atleast_2d(tensors)  # type: ignore[attr-defined]

    def atleast_3d(*tensors):
        r"""
        Returns a 3-dimensional view of each input tensor with zero dimensions.
        Input tensors with three or more dimensions are returned as-is.

        Args:
            input (Tensor or list of Tensors)

        Returns:
            output (Tensor or tuple of Tensors)

        Example:

            >>> x = torch.tensor(0.5)
            >>> x
            tensor(0.5000)
            >>> torch.atleast_3d(x)
            tensor([[[0.5000]]])
            >>> y = torch.arange(4).view(2, 2)
            >>> y
            tensor([[0, 1],
                    [2, 3]])
            >>> torch.atleast_3d(y)
            tensor([[[0],
                     [1]],
                    <BLANKLINE>
                    [[2],
                     [3]]])
            >>> x = torch.tensor(1).view(1, 1, 1)
            >>> x
            tensor([[[1]]])
            >>> torch.atleast_3d(x)
            tensor([[[1]]])
            >>> x = torch.tensor(0.5)
            >>> y = torch.tensor(1.)
            >>> torch.atleast_3d((x, y))
            (tensor([[[0.5000]]]), tensor([[[1.]]]))
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(tensors):
            return handle_torch_function(atleast_3d, tensors, *tensors)
        if len(tensors) == 1:
            tensors = tensors[0]
        return _VF.atleast_3d(tensors)  # type: ignore[attr-defined]

    if TYPE_CHECKING:
        pass
        # There's no good way to use this type annotation; cannot rename norm() to
        # _norm_impl() in a way that doesn't break JIT overloads. So leave untyped
        # for mypy for now.
        #    def norm(input: Tensor,
        #             p: Optional[Union[str, Number]] = "fro",
        #             dim: Optional[Union[int, List[int]]] = None,
        #             keepdim: bool = False,
        #             out: Optional[Tensor] = None,
        #             dtype: _dtype = None) -> Tensor:
        #        return _norm_impl(input, p, dim, keepdim, out, dtype)
    else:
        # TODO: type dim as BroadcastingList when
        # https://github.com/pytorch/pytorch/issues/33782 is fixed
        @overload
        def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):
            # type: (Tensor, str, Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
            pass

        @overload  # noqa: F811
        def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
            # type: (Tensor, Optional[number], Optional[List[int]], bool, Optional[Tensor], Optional[int]) -> Tensor
            pass

        @overload  # noqa: F811
        def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
            # type: (Tensor, Optional[number], Optional[int], bool, Optional[Tensor], Optional[int]) -> Tensor
            pass

        @overload  # noqa: F811
        def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
            # type: (Tensor, str, Optional[int], bool, Optional[Tensor], Optional[int]) -> Tensor
            pass

    def norm(input, p: Optional[Union[float, str]] = "fro", dim=None, keepdim=False, out=None, dtype=None):  # noqa: F811
        r"""Returns the matrix norm or vector norm of a given tensor.

        .. warning::

            torch.norm is deprecated and may be removed in a future PyTorch release.
            Its documentation and behavior may be incorrect, and it is no longer
            actively maintained.

            Use :func:`torch.linalg.vector_norm` when computing vector norms and
            :func:`torch.linalg.matrix_norm` when computing matrix norms.
            For a function with a similar behavior as this one see :func:`torch.linalg.norm`.
            Note, however, the signature for these functions is slightly different than the
            signature for ``torch.norm``.

        Args:
            input (Tensor): The input tensor. Its data type must be either a floating
                point or complex type. For complex inputs, the norm is calculated using the
                absolute value of each element. If the input is complex and neither
                :attr:`dtype` nor :attr:`out` is specified, the result's data type will
                be the corresponding floating point type (e.g. float if :attr:`input` is
                complexfloat).

            p (int, float, inf, -inf, 'fro', 'nuc', optional): the order of norm. Default: ``'fro'``
                The following norms can be calculated:

                ======  ==============  ==========================
                ord     matrix norm     vector norm
                ======  ==============  ==========================
                'fro'   Frobenius norm  --
                'nuc'   nuclear norm    --
                Number  --              sum(abs(x)**ord)**(1./ord)
                ======  ==============  ==========================

                The vector norm can be calculated across any number of dimensions.
                The corresponding dimensions of :attr:`input` are flattened into
                one dimension, and the norm is calculated on the flattened
                dimension.

                Frobenius norm produces the same result as ``p=2`` in all cases
                except when :attr:`dim` is a list of three or more dims, in which
                case Frobenius norm throws an error.

                Nuclear norm can only be calculated across exactly two dimensions.

            dim (int, tuple of ints, list of ints, optional):
                Specifies which dimension or dimensions of :attr:`input` to
                calculate the norm across. If :attr:`dim` is ``None``, the norm will
                be calculated across all dimensions of :attr:`input`. If the norm
                type indicated by :attr:`p` does not support the specified number of
                dimensions, an error will occur.
            keepdim (bool, optional): whether the output tensors have :attr:`dim`
                retained or not. Ignored if :attr:`dim` = ``None`` and
                :attr:`out` = ``None``. Default: ``False``
            out (Tensor, optional): the output tensor. Ignored if
                :attr:`dim` = ``None`` and :attr:`out` = ``None``.
            dtype (:class:`torch.dtype`, optional): the desired data type of
                returned tensor. If specified, the input tensor is casted to
                :attr:`dtype` while performing the operation. Default: None.

        .. note::
            Even though ``p='fro'`` supports any number of dimensions, the true
            mathematical definition of Frobenius norm only applies to tensors with
            exactly two dimensions. :func:`torch.linalg.matrix_norm` with ``ord='fro'``
            aligns with the mathematical definition, since it can only be applied across
            exactly two dimensions.

        Example::

            >>> import torch
            >>> a = torch.arange(9, dtype= torch.float) - 4
            >>> b = a.reshape((3, 3))
            >>> torch.norm(a)
            tensor(7.7460)
            >>> torch.norm(b)
            tensor(7.7460)
            >>> torch.norm(a, float('inf'))
            tensor(4.)
            >>> torch.norm(b, float('inf'))
            tensor(4.)
            >>> c = torch.tensor([[ 1, 2, 3], [-1, 1, 4]] , dtype=torch.float)
            >>> torch.norm(c, dim=0)
            tensor([1.4142, 2.2361, 5.0000])
            >>> torch.norm(c, dim=1)
            tensor([3.7417, 4.2426])
            >>> torch.norm(c, p=1, dim=1)
            tensor([6., 6.])
            >>> d = torch.arange(8, dtype=torch.float).reshape(2, 2, 2)
            >>> torch.norm(d, dim=(1, 2))
            tensor([ 3.7417, 11.2250])
            >>> torch.norm(d[0, :, :]), torch.norm(d[1, :, :])
            (tensor(3.7417), tensor(11.2250))
        """

        if has_torch_function_unary(input):
            return handle_torch_function(
                norm, (input,), input, p=p, dim=dim, keepdim=keepdim, out=out, dtype=dtype)

        # NB. All the repeated code and weird python is to please TorchScript.
        #     For a more compact implementation see the relevant function in `_refs/__init__.py`

        # We don't do this for MPS or sparse tensors
        if input.layout == torch.strided and input.device.type in \
                ("cpu", "cuda", "meta", torch.utils.backend_registration._privateuse1_backend_name):
            if dim is not None:
                if isinstance(dim, int):
                    _dim = [dim]
                else:
                    _dim = dim
            else:
                _dim = None  # type: ignore[assignment]

            if isinstance(p, str):
                if p == "fro" and (dim is None or isinstance(dim, int) or len(dim) <= 2):
                    if out is None:
                        return torch.linalg.vector_norm(input, 2, _dim, keepdim, dtype=dtype)
                    else:
                        return torch.linalg.vector_norm(input, 2, _dim, keepdim, dtype=dtype, out=out)

                # Here we either call the nuclear norm, or we call matrix_norm with some arguments
                # that will throw an error
                if _dim is None:
                    _dim = list(range(input.ndim))
                if out is None:
                    return torch.linalg.matrix_norm(input, p, _dim, keepdim, dtype=dtype)
                else:
                    return torch.linalg.matrix_norm(input, p, _dim, keepdim, dtype=dtype, out=out)
            else:
                # NB. p should be Union[str, number], not Optional!
                _p = 2.0 if p is None else p
                if out is None:
                    return torch.linalg.vector_norm(input, _p, _dim, keepdim, dtype=dtype)
                else:
                    return torch.linalg.vector_norm(input, _p, _dim, keepdim, dtype=dtype, out=out)

        ndim = input.dim()

        # catch default case
        if dim is None and out is None and dtype is None and p is not None:
            if isinstance(p, str):
                if p == "fro":
                    return _VF.frobenius_norm(input, dim=(), keepdim=keepdim)
            if not isinstance(p, str):
                _dim = [i for i in range(ndim)]  # noqa: C416 TODO: rewrite as list(range(m))
                return _VF.norm(input, p, dim=_dim, keepdim=keepdim)  # type: ignore[attr-defined]

        # TODO: when https://github.com/pytorch/pytorch/issues/33782 is fixed
        # remove the overloads where dim is an int and replace with BraodcastingList1
        # and remove next four lines, replace _dim with dim
        if dim is not None:
            if isinstance(dim, int):
                _dim = [dim]
            else:
                _dim = dim
        else:
            _dim = None  # type: ignore[assignment]

        if isinstance(p, str):
            if p == "fro":
                if dtype is not None:
                    raise ValueError("dtype argument is not supported in frobenius norm")

                if _dim is None:
                    _dim = list(range(ndim))
                if out is None:
                    return _VF.frobenius_norm(input, _dim, keepdim=keepdim)
                else:
                    return _VF.frobenius_norm(input, _dim, keepdim=keepdim, out=out)
            elif p == "nuc":
                if dtype is not None:
                    raise ValueError("dtype argument is not supported in nuclear norm")
                if _dim is None:
                    if out is None:
                        return _VF.nuclear_norm(input, keepdim=keepdim)
                    else:
                        return _VF.nuclear_norm(input, keepdim=keepdim, out=out)
                else:
                    if out is None:
                        return _VF.nuclear_norm(input, _dim, keepdim=keepdim)
                    else:
                        return _VF.nuclear_norm(input, _dim, keepdim=keepdim, out=out)
            raise RuntimeError(f"only valid string values are 'fro' and 'nuc', found {p}")
        else:
            if _dim is None:
                _dim = list(range(ndim))

            if out is None:
                if dtype is None:
                    return _VF.norm(input, p, _dim, keepdim=keepdim)  # type: ignore[attr-defined]
                else:
                    return _VF.norm(input, p, _dim, keepdim=keepdim, dtype=dtype)  # type: ignore[attr-defined]
            else:
                if dtype is None:
                    return _VF.norm(input, p, _dim, keepdim=keepdim, out=out)  # type: ignore[attr-defined]
                else:
                    return _VF.norm(input, p, _dim, keepdim=keepdim, dtype=dtype, out=out)  # type: ignore[attr-defined]

    def chain_matmul(*matrices, out=None):
        r"""Returns the matrix product of the :math:`N` 2-D tensors. This product is efficiently computed
        using the matrix chain order algorithm which selects the order in which incurs the lowest cost in terms
        of arithmetic operations (`[CLRS]`_). Note that since this is a function to compute the product, :math:`N`
        needs to be greater than or equal to 2; if equal to 2 then a trivial matrix-matrix product is returned.
        If :math:`N` is 1, then this is a no-op - the original matrix is returned as is.

        .. warning::

            :func:`torch.chain_matmul` is deprecated and will be removed in a future PyTorch release.
            Use :func:`torch.linalg.multi_dot` instead, which accepts a list of two or more tensors
            rather than multiple arguments.

        Args:
            matrices (Tensors...): a sequence of 2 or more 2-D tensors whose product is to be determined.
            out (Tensor, optional): the output tensor. Ignored if :attr:`out` = ``None``.

        Returns:
            Tensor: if the :math:`i^{th}` tensor was of dimensions :math:`p_{i} \times p_{i + 1}`, then the product
            would be of dimensions :math:`p_{1} \times p_{N + 1}`.

        Example::

            >>> # xdoctest: +SKIP
            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> a = torch.randn(3, 4)
            >>> b = torch.randn(4, 5)
            >>> c = torch.randn(5, 6)
            >>> d = torch.randn(6, 7)
            >>> # will raise a deprecation warning
            >>> torch.chain_matmul(a, b, c, d)
            tensor([[ -2.3375,  -3.9790,  -4.1119,  -6.6577,   9.5609, -11.5095,  -3.2614],
                    [ 21.4038,   3.3378,  -8.4982,  -5.2457, -10.2561,  -2.4684,   2.7163],
                    [ -0.9647,  -5.8917,  -2.3213,  -5.2284,  12.8615, -12.2816,  -2.5095]])

        .. _`[CLRS]`: https://mitpress.mit.edu/books/introduction-algorithms-third-edition
        """
        # This wrapper exists to support variadic args.
        if has_torch_function(matrices):
            return handle_torch_function(chain_matmul, matrices, *matrices)

        if out is None:
            return _VF.chain_matmul(matrices)  # type: ignore[attr-defined]
        else:
            return _VF.chain_matmul(matrices, out=out)  # type: ignore[attr-defined]

    def _lu_impl(A, pivot=True, get_infos=False, out=None):
        # type: (Tensor, bool, bool, Any) -> Tuple[Tensor, Tensor, Tensor]
        r"""Computes the LU factorization of a matrix or batches of matrices
        :attr:`A`. Returns a tuple containing the LU factorization and
        pivots of :attr:`A`.  Pivoting is done if :attr:`pivot` is set to
        ``True``.

        .. warning::

            :func:`torch.lu` is deprecated in favor of :func:`torch.linalg.lu_factor`
            and :func:`torch.linalg.lu_factor_ex`. :func:`torch.lu` will be removed in a
            future PyTorch release.
            ``LU, pivots, info = torch.lu(A, compute_pivots)`` should be replaced with

            .. code:: python

                LU, pivots = torch.linalg.lu_factor(A, compute_pivots)

            ``LU, pivots, info = torch.lu(A, compute_pivots, get_infos=True)`` should be replaced with

            .. code:: python

                LU, pivots, info = torch.linalg.lu_factor_ex(A, compute_pivots)

        .. note::
            * The returned permutation matrix for every matrix in the batch is
              represented by a 1-indexed vector of size ``min(A.shape[-2], A.shape[-1])``.
              ``pivots[i] == j`` represents that in the ``i``-th step of the algorithm,
              the ``i``-th row was permuted with the ``j-1``-th row.
            * LU factorization with :attr:`pivot` = ``False`` is not available
              for CPU, and attempting to do so will throw an error. However,
              LU factorization with :attr:`pivot` = ``False`` is available for
              CUDA.
            * This function does not check if the factorization was successful
              or not if :attr:`get_infos` is ``True`` since the status of the
              factorization is present in the third element of the return tuple.
            * In the case of batches of square matrices with size less or equal
              to 32 on a CUDA device, the LU factorization is repeated for
              singular matrices due to the bug in the MAGMA library
              (see magma issue 13).
            * ``L``, ``U``, and ``P`` can be derived using :func:`torch.lu_unpack`.

        .. warning::
            The gradients of this function will only be finite when :attr:`A` is full rank.
            This is because the LU decomposition is just differentiable at full rank matrices.
            Furthermore, if :attr:`A` is close to not being full rank,
            the gradient will be numerically unstable as it depends on the computation of :math:`L^{-1}` and :math:`U^{-1}`.

        Args:
            A (Tensor): the tensor to factor of size :math:`(*, m, n)`
            pivot (bool, optional): controls whether pivoting is done. Default: ``True``
            get_infos (bool, optional): if set to ``True``, returns an info IntTensor.
                                        Default: ``False``
            out (tuple, optional): optional output tuple. If :attr:`get_infos` is ``True``,
                                   then the elements in the tuple are Tensor, IntTensor,
                                   and IntTensor. If :attr:`get_infos` is ``False``, then the
                                   elements in the tuple are Tensor, IntTensor. Default: ``None``

        Returns:
            (Tensor, IntTensor, IntTensor (optional)): A tuple of tensors containing

                - **factorization** (*Tensor*): the factorization of size :math:`(*, m, n)`

                - **pivots** (*IntTensor*): the pivots of size :math:`(*, \text{min}(m, n))`.
                  ``pivots`` stores all the intermediate transpositions of rows.
                  The final permutation ``perm`` could be reconstructed by
                  applying ``swap(perm[i], perm[pivots[i] - 1])`` for ``i = 0, ..., pivots.size(-1) - 1``,
                  where ``perm`` is initially the identity permutation of :math:`m` elements
                  (essentially this is what :func:`torch.lu_unpack` is doing).

                - **infos** (*IntTensor*, *optional*): if :attr:`get_infos` is ``True``, this is a tensor of
                  size :math:`(*)` where non-zero values indicate whether factorization for the matrix or
                  each minibatch has succeeded or failed

        Example::

            >>> # xdoctest: +REQUIRES(env:TORCH_DOCTEST_LAPACK)
            >>> # xdoctest: +IGNORE_WANT("non-deterministic")
            >>> A = torch.randn(2, 3, 3)
            >>> A_LU, pivots = torch.lu(A)
            >>> A_LU
            tensor([[[ 1.3506,  2.5558, -0.0816],
                     [ 0.1684,  1.1551,  0.1940],
                     [ 0.1193,  0.6189, -0.5497]],

                    [[ 0.4526,  1.2526, -0.3285],
                     [-0.7988,  0.7175, -0.9701],
                     [ 0.2634, -0.9255, -0.3459]]])
            >>> pivots
            tensor([[ 3,  3,  3],
                    [ 3,  3,  3]], dtype=torch.int32)
            >>> A_LU, pivots, info = torch.lu(A, get_infos=True)
            >>> if info.nonzero().size(0) == 0:
            ...     print('LU factorization succeeded for all samples!')
            LU factorization succeeded for all samples!
        """
        # If get_infos is True, then we don't need to check for errors and vice versa
        return torch._lu_with_info(A, pivot=pivot, check_errors=(not get_infos))

    if TYPE_CHECKING:
        _ListOrSeq = Sequence[Tensor]
    else:
        _ListOrSeq = List[Tensor]

    def _check_list_size(out_len: int, get_infos: bool, out: _ListOrSeq) -> None:
        get_infos_int = 1 if get_infos else 0
        if out_len - get_infos_int != 2:
            raise TypeError(f"expected tuple of {2 + int(get_infos)} elements but got {out_len}")
        if not isinstance(out, (tuple, list)):
            raise TypeError(f"argument 'out' must be tuple of Tensors, not {type(out).__name__}")

    def _lu_with_infos(A, pivot=True, get_infos=False, out=None):
        # type: (Tensor, bool, bool, Optional[Tuple[Tensor, Tensor, Tensor]]) -> Tuple[Tensor, Tensor, Tensor]
        if has_torch_function_unary(A):
            return handle_torch_function(
                lu, (A,), A, pivot=pivot, get_infos=get_infos, out=out)
        result = _lu_impl(A, pivot, get_infos, out)
        if out is not None:
            _check_list_size(len(out), get_infos, out)
            for i in range(len(out)):
                out[i].resize_as_(result[i]).copy_(result[i])
            return out
        else:
            return result  # A_LU, pivots, infos

    def _lu_no_infos(A, pivot=True, get_infos=False, out=None):
        # type: (Tensor, bool, bool, Optional[Tuple[Tensor, Tensor]]) -> Tuple[Tensor, Tensor]
        # need to check for torch_function here so that we exit if
        if has_torch_function_unary(A):
            return handle_torch_function(
                lu, (A,), A, pivot=pivot, get_infos=get_infos, out=out)
        result = _lu_impl(A, pivot, get_infos, out)
        if out is not None:
            _check_list_size(len(out), get_infos, out)
            for i in range(len(out)):
                out[i].resize_as_(result[i]).copy_(result[i])
            return out
        else:
            return result[0], result[1]  # A_LU, pivots

    # The return type of lu depends on `get_infos`, so in order to resolve the output type
    # of lu in TorchScript we need to statically know the value of `get_infos`
    lu = boolean_dispatch(
        arg_name='get_infos',
        arg_index=2,
        default=False,
        if_true=_lu_with_infos,
        if_false=_lu_no_infos,
        module_name=__name__,
        func_name='lu')
    lu.__doc__ = _lu_impl.__doc__

    def align_tensors(*tensors):
        raise RuntimeError('`align_tensors` not yet implemented.')

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