_torch_docs.py

"""Adds docstrings to functions defined in the torch._C"""

import re

import torch._C
from torch._C import _add_docstr as add_docstr


def parse_kwargs(desc):
    """Maps a description of args to a dictionary of {argname: description}.
    Input:
        ('    weight (Tensor): a weight tensor\n' +
         '        Some optional description')
    Output: {
        'weight': \
        'weight (Tensor): a weight tensor\n        Some optional description'
    }
    """
    # Split on exactly 4 spaces after a newline
    regx = re.compile(r"\n\s{4}(?!\s)")
    kwargs = [section.strip() for section in regx.split(desc)]
    kwargs = [section for section in kwargs if len(section) > 0]
    return {desc.split(' ')[0]: desc for desc in kwargs}


def merge_dicts(*dicts):
    return {x: d[x] for d in dicts for x in d}


common_args = parse_kwargs("""
    input (Tensor): the input tensor.
    generator (:class:`torch.Generator`, optional): a pseudorandom number generator for sampling
    out (Tensor, optional): the output tensor.
    memory_format (:class:`torch.memory_format`, optional): the desired memory format of
        returned tensor. Default: ``torch.preserve_format``.
""")

reduceops_common_args = merge_dicts(common_args, parse_kwargs("""
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        If specified, the input tensor is casted to :attr:`dtype` before the operation
        is performed. This is useful for preventing data type overflows. Default: None.
    keepdim (bool): whether the output tensor has :attr:`dim` retained or not.
"""))

multi_dim_common = merge_dicts(reduceops_common_args, parse_kwargs("""
    dim (int or tuple of ints): the dimension or dimensions to reduce.
"""), {'keepdim_details': """
If :attr:`keepdim` is ``True``, the output tensor is of the same size
as :attr:`input` except in the dimension(s) :attr:`dim` where it is of size 1.
Otherwise, :attr:`dim` is squeezed (see :func:`torch.squeeze`), resulting in the
output tensor having 1 (or ``len(dim)``) fewer dimension(s).
"""})

single_dim_common = merge_dicts(reduceops_common_args, parse_kwargs("""
    dim (int): the dimension to reduce.
"""), {'keepdim_details': """If :attr:`keepdim` is ``True``, the output tensor is of the same size
as :attr:`input` except in the dimension :attr:`dim` where it is of size 1.
Otherwise, :attr:`dim` is squeezed (see :func:`torch.squeeze`), resulting in
the output tensor having 1 fewer dimension than :attr:`input`."""})

factory_common_args = merge_dicts(common_args, parse_kwargs("""
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        Default: if ``None``, uses a global default (see :func:`torch.set_default_tensor_type`).
    layout (:class:`torch.layout`, optional): the desired layout of returned Tensor.
        Default: ``torch.strided``.
    device (:class:`torch.device`, optional): the desired device of returned tensor.
        Default: if ``None``, uses the current device for the default tensor type
        (see :func:`torch.set_default_tensor_type`). :attr:`device` will be the CPU
        for CPU tensor types and the current CUDA device for CUDA tensor types.
    requires_grad (bool, optional): If autograd should record operations on the
        returned tensor. Default: ``False``.
    pin_memory (bool, optional): If set, returned tensor would be allocated in
        the pinned memory. Works only for CPU tensors. Default: ``False``.
    memory_format (:class:`torch.memory_format`, optional): the desired memory format of
        returned Tensor. Default: ``torch.contiguous_format``.
"""))

factory_like_common_args = parse_kwargs("""
    input (Tensor): the size of :attr:`input` will determine size of the output tensor.
    layout (:class:`torch.layout`, optional): the desired layout of returned tensor.
        Default: if ``None``, defaults to the layout of :attr:`input`.
    dtype (:class:`torch.dtype`, optional): the desired data type of returned Tensor.
        Default: if ``None``, defaults to the dtype of :attr:`input`.
    device (:class:`torch.device`, optional): the desired device of returned tensor.
        Default: if ``None``, defaults to the device of :attr:`input`.
    requires_grad (bool, optional): If autograd should record operations on the
        returned tensor. Default: ``False``.
    pin_memory (bool, optional): If set, returned tensor would be allocated in
        the pinned memory. Works only for CPU tensors. Default: ``False``.
    memory_format (:class:`torch.memory_format`, optional): the desired memory format of
        returned Tensor. Default: ``torch.preserve_format``.
""")

factory_data_common_args = parse_kwargs("""
    data (array_like): Initial data for the tensor. Can be a list, tuple,
        NumPy ``ndarray``, scalar, and other types.
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        Default: if ``None``, infers data type from :attr:`data`.
    device (:class:`torch.device`, optional): the desired device of returned tensor.
        Default: if ``None``, uses the current device for the default tensor type
        (see :func:`torch.set_default_tensor_type`). :attr:`device` will be the CPU
        for CPU tensor types and the current CUDA device for CUDA tensor types.
    requires_grad (bool, optional): If autograd should record operations on the
        returned tensor. Default: ``False``.
    pin_memory (bool, optional): If set, returned tensor would be allocated in
        the pinned memory. Works only for CPU tensors. Default: ``False``.
""")

tf32_notes = {
    "tf32_note": """This operator supports :ref:`TensorFloat32<tf32_on_ampere>`."""
}


reproducibility_notes = {
    "forward_reproducibility_note": """This operation may behave nondeterministically when given tensors on \
a CUDA device. See :doc:`/notes/randomness` for more information.""",
    "backward_reproducibility_note": """This operation may produce nondeterministic gradients when given tensors on \
a CUDA device. See :doc:`/notes/randomness` for more information.""",
    "cudnn_reproducibility_note": """In some circumstances when given tensors on a CUDA device \
and using CuDNN, this operator may select a nondeterministic algorithm to increase performance. If this is \
undesirable, you can try to make the operation deterministic (potentially at \
a performance cost) by setting ``torch.backends.cudnn.deterministic = True``. \
See :doc:`/notes/randomness` for more information."""
}

add_docstr(torch.abs, r"""
abs(input, *, out=None) -> Tensor

Computes the absolute value of each element in :attr:`input`.

.. math::
    \text{out}_{i} = |\text{input}_{i}|
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.abs(torch.tensor([-1, -2, 3]))
    tensor([ 1,  2,  3])
""".format(**common_args))

add_docstr(torch.absolute,
           r"""
absolute(input, *, out=None) -> Tensor

Alias for :func:`torch.abs`
""".format(**common_args))

add_docstr(torch.acos, r"""
acos(input, *, out=None) -> Tensor

Computes the inverse cosine of each element in :attr:`input`.

.. math::
    \text{out}_{i} = \cos^{-1}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.3348, -0.5889,  0.2005, -0.1584])
    >>> torch.acos(a)
    tensor([ 1.2294,  2.2004,  1.3690,  1.7298])
""".format(**common_args))

add_docstr(torch.arccos, r"""
arccos(input, *, out=None) -> Tensor

Alias for :func:`torch.acos`.
""")

add_docstr(torch.acosh, r"""
acosh(input, *, out=None) -> Tensor

Returns a new tensor with the inverse hyperbolic cosine of the elements of :attr:`input`.

Note:
    The domain of the inverse hyperbolic cosine is `[1, inf)` and values outside this range
    will be mapped to ``NaN``, except for `+ INF` for which the output is mapped to `+ INF`.

.. math::
    \text{out}_{i} = \cosh^{-1}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword arguments:
    {out}

Example::

    >>> a = torch.randn(4).uniform_(1, 2)
    >>> a
    tensor([ 1.3192, 1.9915, 1.9674, 1.7151 ])
    >>> torch.acosh(a)
    tensor([ 0.7791, 1.3120, 1.2979, 1.1341 ])
""".format(**common_args))

add_docstr(torch.arccosh, r"""
arccosh(input, *, out=None) -> Tensor

Alias for :func:`torch.acosh`.
""".format(**common_args))

add_docstr(torch.add, r"""
add(input, other, *, out=None)

Adds the scalar :attr:`other` to each element of the input :attr:`input`
and returns a new resulting tensor.

.. math::
    \text{{out}} = \text{{input}} + \text{{other}}

If :attr:`input` is of type FloatTensor or DoubleTensor, :attr:`other` must be
a real number, otherwise it should be an integer.

Args:
    {input}
    value (Number): the number to be added to each element of :attr:`input`

Keyword arguments:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.0202,  1.0985,  1.3506, -0.6056])
    >>> torch.add(a, 20)
    tensor([ 20.0202,  21.0985,  21.3506,  19.3944])

.. function:: add(input, other, *, alpha=1, out=None)

Each element of the tensor :attr:`other` is multiplied by the scalar
:attr:`alpha` and added to each element of the tensor :attr:`input`.
The resulting tensor is returned.

The shapes of :attr:`input` and :attr:`other` must be
:ref:`broadcastable <broadcasting-semantics>`.

.. math::
    \text{{out}} = \text{{input}} + \text{{alpha}} \times \text{{other}}

If :attr:`other` is of type FloatTensor or DoubleTensor, :attr:`alpha` must be
a real number, otherwise it should be an integer.

Args:
    input (Tensor): the first input tensor
    other (Tensor): the second input tensor

Keyword args:
    alpha (Number): the scalar multiplier for :attr:`other`
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.9732, -0.3497,  0.6245,  0.4022])
    >>> b = torch.randn(4, 1)
    >>> b
    tensor([[ 0.3743],
            [-1.7724],
            [-0.5811],
            [-0.8017]])
    >>> torch.add(a, b, alpha=10)
    tensor([[  2.7695,   3.3930,   4.3672,   4.1450],
            [-18.6971, -18.0736, -17.0994, -17.3216],
            [ -6.7845,  -6.1610,  -5.1868,  -5.4090],
            [ -8.9902,  -8.3667,  -7.3925,  -7.6147]])
""".format(**common_args))

add_docstr(torch.addbmm,
           r"""
addbmm(input, batch1, batch2, *, beta=1, alpha=1, out=None) -> Tensor

Performs a batch matrix-matrix product of matrices stored
in :attr:`batch1` and :attr:`batch2`,
with a reduced add step (all matrix multiplications get accumulated
along the first dimension).
:attr:`input` is added to the final result.

:attr:`batch1` and :attr:`batch2` must be 3-D tensors each containing the
same number of matrices.

If :attr:`batch1` is a :math:`(b \times n \times m)` tensor, :attr:`batch2` is a
:math:`(b \times m \times p)` tensor, :attr:`input` must be
:ref:`broadcastable <broadcasting-semantics>` with a :math:`(n \times p)` tensor
and :attr:`out` will be a :math:`(n \times p)` tensor.

.. math::
    out = \beta\ \text{input} + \alpha\ (\sum_{i=0}^{b-1} \text{batch1}_i \mathbin{@} \text{batch2}_i)

If :attr:`beta` is 0, then :attr:`input` will be ignored, and `nan` and `inf` in
it will not be propagated.
""" + r"""
For inputs of type `FloatTensor` or `DoubleTensor`, arguments :attr:`beta` and :attr:`alpha`
must be real numbers, otherwise they should be integers.

{tf32_note}

Args:
    batch1 (Tensor): the first batch of matrices to be multiplied
    batch2 (Tensor): the second batch of matrices to be multiplied

Keyword args:
    beta (Number, optional): multiplier for :attr:`input` (:math:`\beta`)
    input (Tensor): matrix to be added
    alpha (Number, optional): multiplier for `batch1 @ batch2` (:math:`\alpha`)
    {out}

Example::

    >>> M = torch.randn(3, 5)
    >>> batch1 = torch.randn(10, 3, 4)
    >>> batch2 = torch.randn(10, 4, 5)
    >>> torch.addbmm(M, batch1, batch2)
    tensor([[  6.6311,   0.0503,   6.9768, -12.0362,  -2.1653],
            [ -4.8185,  -1.4255,  -6.6760,   8.9453,   2.5743],
            [ -3.8202,   4.3691,   1.0943,  -1.1109,   5.4730]])
""".format(**common_args, **tf32_notes))

add_docstr(torch.addcdiv, r"""
addcdiv(input, tensor1, tensor2, *, value=1, out=None) -> Tensor

Performs the element-wise division of :attr:`tensor1` by :attr:`tensor2`,
multiply the result by the scalar :attr:`value` and add it to :attr:`input`.

.. warning::
    Integer division with addcdiv is no longer supported, and in a future
    release addcdiv will perform a true division of tensor1 and tensor2.
    The historic addcdiv behavior can be implemented as
    (input + value * torch.trunc(tensor1 / tensor2)).to(input.dtype)
    for integer inputs and as (input + value * tensor1 / tensor2) for float inputs.
    The future addcdiv behavior is just the latter implementation:
    (input + value * tensor1 / tensor2), for all dtypes.

.. math::
    \text{out}_i = \text{input}_i + \text{value} \times \frac{\text{tensor1}_i}{\text{tensor2}_i}
""" + r"""

The shapes of :attr:`input`, :attr:`tensor1`, and :attr:`tensor2` must be
:ref:`broadcastable <broadcasting-semantics>`.

For inputs of type `FloatTensor` or `DoubleTensor`, :attr:`value` must be
a real number, otherwise an integer.

Args:
    input (Tensor): the tensor to be added
    tensor1 (Tensor): the numerator tensor
    tensor2 (Tensor): the denominator tensor

Keyword args:
    value (Number, optional): multiplier for :math:`\text{{tensor1}} / \text{{tensor2}}`
    {out}

Example::

    >>> t = torch.randn(1, 3)
    >>> t1 = torch.randn(3, 1)
    >>> t2 = torch.randn(1, 3)
    >>> torch.addcdiv(t, t1, t2, value=0.1)
    tensor([[-0.2312, -3.6496,  0.1312],
            [-1.0428,  3.4292, -0.1030],
            [-0.5369, -0.9829,  0.0430]])
""".format(**common_args))

add_docstr(torch.addcmul,
           r"""
addcmul(input, tensor1, tensor2, *, value=1, out=None) -> Tensor

Performs the element-wise multiplication of :attr:`tensor1`
by :attr:`tensor2`, multiply the result by the scalar :attr:`value`
and add it to :attr:`input`.

.. math::
    \text{out}_i = \text{input}_i + \text{value} \times \text{tensor1}_i \times \text{tensor2}_i
""" + r"""
The shapes of :attr:`tensor`, :attr:`tensor1`, and :attr:`tensor2` must be
:ref:`broadcastable <broadcasting-semantics>`.

For inputs of type `FloatTensor` or `DoubleTensor`, :attr:`value` must be
a real number, otherwise an integer.

Args:
    input (Tensor): the tensor to be added
    tensor1 (Tensor): the tensor to be multiplied
    tensor2 (Tensor): the tensor to be multiplied

Keyword args:
    value (Number, optional): multiplier for :math:`tensor1 .* tensor2`
    {out}

Example::

    >>> t = torch.randn(1, 3)
    >>> t1 = torch.randn(3, 1)
    >>> t2 = torch.randn(1, 3)
    >>> torch.addcmul(t, t1, t2, value=0.1)
    tensor([[-0.8635, -0.6391,  1.6174],
            [-0.7617, -0.5879,  1.7388],
            [-0.8353, -0.6249,  1.6511]])
""".format(**common_args))

add_docstr(torch.addmm,
           r"""
addmm(input, mat1, mat2, *, beta=1, alpha=1, out=None) -> Tensor

Performs a matrix multiplication of the matrices :attr:`mat1` and :attr:`mat2`.
The matrix :attr:`input` is added to the final result.

If :attr:`mat1` is a :math:`(n \times m)` tensor, :attr:`mat2` is a
:math:`(m \times p)` tensor, then :attr:`input` must be
:ref:`broadcastable <broadcasting-semantics>` with a :math:`(n \times p)` tensor
and :attr:`out` will be a :math:`(n \times p)` tensor.

:attr:`alpha` and :attr:`beta` are scaling factors on matrix-vector product between
:attr:`mat1` and :attr:`mat2` and the added matrix :attr:`input` respectively.

.. math::
    \text{out} = \beta\ \text{input} + \alpha\ (\text{mat1}_i \mathbin{@} \text{mat2}_i)

If :attr:`beta` is 0, then :attr:`input` will be ignored, and `nan` and `inf` in
it will not be propagated.
""" + r"""
For inputs of type `FloatTensor` or `DoubleTensor`, arguments :attr:`beta` and
:attr:`alpha` must be real numbers, otherwise they should be integers.

{tf32_note}

Args:
    input (Tensor): matrix to be added
    mat1 (Tensor): the first matrix to be matrix multiplied
    mat2 (Tensor): the second matrix to be matrix multiplied

Keyword args:
    beta (Number, optional): multiplier for :attr:`input` (:math:`\beta`)
    alpha (Number, optional): multiplier for :math:`mat1 @ mat2` (:math:`\alpha`)
    {out}

Example::

    >>> M = torch.randn(2, 3)
    >>> mat1 = torch.randn(2, 3)
    >>> mat2 = torch.randn(3, 3)
    >>> torch.addmm(M, mat1, mat2)
    tensor([[-4.8716,  1.4671, -1.3746],
            [ 0.7573, -3.9555, -2.8681]])
""".format(**common_args, **tf32_notes))

add_docstr(torch.sspaddmm,
           r"""
sspaddmm(input, mat1, mat2, *, beta=1, alpha=1, out=None) -> Tensor

Matrix multiplies a sparse tensor :attr:`mat1` with a dense tensor
:attr:`mat2`, then adds the sparse tensor :attr:`input` to the result.

Note: This function is equivalent to :func:`torch.addmm`, except
:attr:`input` and :attr:`mat1` are sparse.

Args:
    input (Tensor): a sparse matrix to be added
    mat1 (Tensor): a sparse matrix to be matrix multiplied
    mat2 (Tensor): a dense matrix to be matrix multiplied

Keyword args:
    beta (Number, optional): multiplier for :attr:`mat` (:math:`\beta`)
    alpha (Number, optional): multiplier for :math:`mat1 @ mat2` (:math:`\alpha`)
    {out}
""".format(**common_args))

add_docstr(torch.smm,
           r"""
smm(input, mat) -> Tensor

Performs a matrix multiplication of the sparse matrix :attr:`input`
with the dense matrix :attr:`mat`.

Args:
    input (Tensor): a sparse matrix to be matrix multiplied
    mat (Tensor): a dense matrix to be matrix multiplied
""")

add_docstr(torch.addmv,
           r"""
addmv(input, mat, vec, *, beta=1, alpha=1, out=None) -> Tensor

Performs a matrix-vector product of the matrix :attr:`mat` and
the vector :attr:`vec`.
The vector :attr:`input` is added to the final result.

If :attr:`mat` is a :math:`(n \times m)` tensor, :attr:`vec` is a 1-D tensor of
size `m`, then :attr:`input` must be
:ref:`broadcastable <broadcasting-semantics>` with a 1-D tensor of size `n` and
:attr:`out` will be 1-D tensor of size `n`.

:attr:`alpha` and :attr:`beta` are scaling factors on matrix-vector product between
:attr:`mat` and :attr:`vec` and the added tensor :attr:`input` respectively.

.. math::
    \text{out} = \beta\ \text{input} + \alpha\ (\text{mat} \mathbin{@} \text{vec})

If :attr:`beta` is 0, then :attr:`input` will be ignored, and `nan` and `inf` in
it will not be propagated.
""" + r"""
For inputs of type `FloatTensor` or `DoubleTensor`, arguments :attr:`beta` and
:attr:`alpha` must be real numbers, otherwise they should be integers

Args:
    input (Tensor): vector to be added
    mat (Tensor): matrix to be matrix multiplied
    vec (Tensor): vector to be matrix multiplied

Keyword args:
    beta (Number, optional): multiplier for :attr:`input` (:math:`\beta`)
    alpha (Number, optional): multiplier for :math:`mat @ vec` (:math:`\alpha`)
    {out}

Example::

    >>> M = torch.randn(2)
    >>> mat = torch.randn(2, 3)
    >>> vec = torch.randn(3)
    >>> torch.addmv(M, mat, vec)
    tensor([-0.3768, -5.5565])
""".format(**common_args))

add_docstr(torch.addr,
           r"""
addr(input, vec1, vec2, *, beta=1, alpha=1, out=None) -> Tensor

Performs the outer-product of vectors :attr:`vec1` and :attr:`vec2`
and adds it to the matrix :attr:`input`.

Optional values :attr:`beta` and :attr:`alpha` are scaling factors on the
outer product between :attr:`vec1` and :attr:`vec2` and the added matrix
:attr:`input` respectively.

.. math::
    \text{out} = \beta\ \text{input} + \alpha\ (\text{vec1} \otimes \text{vec2})

If :attr:`beta` is 0, then :attr:`input` will be ignored, and `nan` and `inf` in
it will not be propagated.
""" + r"""
If :attr:`vec1` is a vector of size `n` and :attr:`vec2` is a vector
of size `m`, then :attr:`input` must be
:ref:`broadcastable <broadcasting-semantics>` with a matrix of size
:math:`(n \times m)` and :attr:`out` will be a matrix of size
:math:`(n \times m)`.

For inputs of type `FloatTensor` or `DoubleTensor`, arguments :attr:`beta` and
:attr:`alpha` must be real numbers, otherwise they should be integers

.. warning::
    This function is deprecated and may be removed in a future release.
    It can be implemented using :func:`torch.outer` as
    ``alpha * torch.outer(vec1, vec2) + beta * input`` when :attr:`beta` is not zero,
    and as ``alpha * torch.outer(vec1, vec2)`` when :attr:`beta` is zero.

Args:
    input (Tensor): matrix to be added
    vec1 (Tensor): the first vector of the outer product
    vec2 (Tensor): the second vector of the outer product

Keyword args:
    beta (Number, optional): multiplier for :attr:`input` (:math:`\beta`)
    alpha (Number, optional): multiplier for :math:`\text{{vec1}} \otimes \text{{vec2}}` (:math:`\alpha`)
    {out}

Example::

    >>> vec1 = torch.arange(1., 4.)
    >>> vec2 = torch.arange(1., 3.)
    >>> M = torch.zeros(3, 2)
    >>> torch.addr(M, vec1, vec2)
    tensor([[ 1.,  2.],
            [ 2.,  4.],
            [ 3.,  6.]])
""".format(**common_args))

add_docstr(torch.allclose,
           r"""
allclose(input, other, rtol=1e-05, atol=1e-08, equal_nan=False) -> bool

This function checks if all :attr:`input` and :attr:`other` satisfy the condition:

.. math::
    \lvert \text{input} - \text{other} \rvert \leq \texttt{atol} + \texttt{rtol} \times \lvert \text{other} \rvert
""" + r"""
elementwise, for all elements of :attr:`input` and :attr:`other`. The behaviour of this function is analogous to
`numpy.allclose <https://docs.scipy.org/doc/numpy/reference/generated/numpy.allclose.html>`_

Args:
    input (Tensor): first tensor to compare
    other (Tensor): second tensor to compare
    atol (float, optional): absolute tolerance. Default: 1e-08
    rtol (float, optional): relative tolerance. Default: 1e-05
    equal_nan (bool, optional): if ``True``, then two ``NaN`` s will be considered equal. Default: ``False``

Example::

    >>> torch.allclose(torch.tensor([10000., 1e-07]), torch.tensor([10000.1, 1e-08]))
    False
    >>> torch.allclose(torch.tensor([10000., 1e-08]), torch.tensor([10000.1, 1e-09]))
    True
    >>> torch.allclose(torch.tensor([1.0, float('nan')]), torch.tensor([1.0, float('nan')]))
    False
    >>> torch.allclose(torch.tensor([1.0, float('nan')]), torch.tensor([1.0, float('nan')]), equal_nan=True)
    True
""")

add_docstr(torch.angle,
           r"""
angle(input, *, out=None) -> Tensor

Computes the element-wise angle (in radians) of the given :attr:`input` tensor.

.. math::
    \text{out}_{i} = angle(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.angle(torch.tensor([-1 + 1j, -2 + 2j, 3 - 3j]))*180/3.14159
    tensor([ 135.,  135,  -45])
""".format(**common_args))

add_docstr(torch.as_strided,
           r"""
as_strided(input, size, stride, storage_offset=0) -> Tensor

Create a view of an existing `torch.Tensor` :attr:`input` with specified
:attr:`size`, :attr:`stride` and :attr:`storage_offset`.

.. warning::
    More than one element of a created 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.

    Many PyTorch functions, which return a view of a tensor, are internally
    implemented with this function. Those functions, like
    :meth:`torch.Tensor.expand`, are easier to read and are therefore more
    advisable to use.


Args:
    {input}
    size (tuple or ints): the shape of the output tensor
    stride (tuple or ints): the stride of the output tensor
    storage_offset (int, optional): the offset in the underlying storage of the output tensor

Example::

    >>> x = torch.randn(3, 3)
    >>> x
    tensor([[ 0.9039,  0.6291,  1.0795],
            [ 0.1586,  2.1939, -0.4900],
            [-0.1909, -0.7503,  1.9355]])
    >>> t = torch.as_strided(x, (2, 2), (1, 2))
    >>> t
    tensor([[0.9039, 1.0795],
            [0.6291, 0.1586]])
    >>> t = torch.as_strided(x, (2, 2), (1, 2), 1)
    tensor([[0.6291, 0.1586],
            [1.0795, 2.1939]])
""".format(**common_args))

add_docstr(torch.as_tensor,
           r"""
as_tensor(data, dtype=None, device=None) -> Tensor

Convert the data into a `torch.Tensor`. If the data is already a `Tensor` with the same `dtype` and `device`,
no copy will be performed, otherwise a new `Tensor` will be returned with computational graph retained if data
`Tensor` has ``requires_grad=True``. Similarly, if the data is an ``ndarray`` of the corresponding `dtype` and
the `device` is the cpu, no copy will be performed.

Args:
    {data}
    {dtype}
    {device}

Example::

    >>> a = numpy.array([1, 2, 3])
    >>> t = torch.as_tensor(a)
    >>> t
    tensor([ 1,  2,  3])
    >>> t[0] = -1
    >>> a
    array([-1,  2,  3])

    >>> a = numpy.array([1, 2, 3])
    >>> t = torch.as_tensor(a, device=torch.device('cuda'))
    >>> t
    tensor([ 1,  2,  3])
    >>> t[0] = -1
    >>> a
    array([1,  2,  3])
""".format(**factory_data_common_args))

add_docstr(torch.asin, r"""
asin(input, *, out=None) -> Tensor

Returns a new tensor with the arcsine  of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \sin^{-1}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.5962,  1.4985, -0.4396,  1.4525])
    >>> torch.asin(a)
    tensor([-0.6387,     nan, -0.4552,     nan])
""".format(**common_args))

add_docstr(torch.arcsin, r"""
arcsin(input, *, out=None) -> Tensor

Alias for :func:`torch.asin`.
""")

add_docstr(torch.asinh,
           r"""
asinh(input, *, out=None) -> Tensor

Returns a new tensor with the inverse hyperbolic sine of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \sinh^{-1}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword arguments:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.1606, -1.4267, -1.0899, -1.0250 ])
    >>> torch.asinh(a)
    tensor([ 0.1599, -1.1534, -0.9435, -0.8990 ])
""".format(**common_args))

add_docstr(torch.arcsinh, r"""
arcsinh(input, *, out=None) -> Tensor

Alias for :func:`torch.asinh`.
""")

add_docstr(torch.atan, r"""
atan(input, *, out=None) -> Tensor

Returns a new tensor with the arctangent  of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \tan^{-1}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.2341,  0.2539, -0.6256, -0.6448])
    >>> torch.atan(a)
    tensor([ 0.2299,  0.2487, -0.5591, -0.5727])
""".format(**common_args))

add_docstr(torch.arctan, r"""
arctan(input, *, out=None) -> Tensor

Alias for :func:`torch.atan`.
""")

add_docstr(torch.atan2,
           r"""
atan2(input, other, *, out=None) -> Tensor

Element-wise arctangent of :math:`\text{{input}}_{{i}} / \text{{other}}_{{i}}`
with consideration of the quadrant. Returns a new tensor with the signed angles
in radians between vector :math:`(\text{{other}}_{{i}}, \text{{input}}_{{i}})`
and vector :math:`(1, 0)`. (Note that :math:`\text{{other}}_{{i}}`, the second
parameter, is the x-coordinate, while :math:`\text{{input}}_{{i}}`, the first
parameter, is the y-coordinate.)

The shapes of ``input`` and ``other`` must be
:ref:`broadcastable <broadcasting-semantics>`.

Args:
    input (Tensor): the first input tensor
    other (Tensor): the second input tensor

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.9041,  0.0196, -0.3108, -2.4423])
    >>> torch.atan2(a, torch.randn(4))
    tensor([ 0.9833,  0.0811, -1.9743, -1.4151])
""".format(**common_args))

add_docstr(torch.atanh, r"""
atanh(input, *, out=None) -> Tensor

Returns a new tensor with the inverse hyperbolic tangent of the elements of :attr:`input`.

Note:
    The domain of the inverse hyperbolic tangent is `(-1, 1)` and values outside this range
    will be mapped to ``NaN``, except for the values `1` and `-1` for which the output is
    mapped to `+/-INF` respectively.

.. math::
    \text{out}_{i} = \tanh^{-1}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword arguments:
    {out}

Example::

    >>> a = torch.randn(4).uniform_(-1, 1)
    >>> a
    tensor([ -0.9385, 0.2968, -0.8591, -0.1871 ])
    >>> torch.atanh(a)
    tensor([ -1.7253, 0.3060, -1.2899, -0.1893 ])
""".format(**common_args))

add_docstr(torch.arctanh, r"""
arctanh(input, *, out=None) -> Tensor

Alias for :func:`torch.atanh`.
""")

add_docstr(torch.baddbmm,
           r"""
baddbmm(input, batch1, batch2, *, beta=1, alpha=1, out=None) -> Tensor

Performs a batch matrix-matrix product of matrices in :attr:`batch1`
and :attr:`batch2`.
:attr:`input` is added to the final result.

:attr:`batch1` and :attr:`batch2` must be 3-D tensors each containing the same
number of matrices.

If :attr:`batch1` is a :math:`(b \times n \times m)` tensor, :attr:`batch2` is a
:math:`(b \times m \times p)` tensor, then :attr:`input` must be
:ref:`broadcastable <broadcasting-semantics>` with a
:math:`(b \times n \times p)` tensor and :attr:`out` will be a
:math:`(b \times n \times p)` tensor. Both :attr:`alpha` and :attr:`beta` mean the
same as the scaling factors used in :meth:`torch.addbmm`.

.. math::
    \text{out}_i = \beta\ \text{input}_i + \alpha\ (\text{batch1}_i \mathbin{@} \text{batch2}_i)

If :attr:`beta` is 0, then :attr:`input` will be ignored, and `nan` and `inf` in
it will not be propagated.
""" + r"""
For inputs of type `FloatTensor` or `DoubleTensor`, arguments :attr:`beta` and
:attr:`alpha` must be real numbers, otherwise they should be integers.

{tf32_note}

Args:
    input (Tensor): the tensor to be added
    batch1 (Tensor): the first batch of matrices to be multiplied
    batch2 (Tensor): the second batch of matrices to be multiplied

Keyword args:
    beta (Number, optional): multiplier for :attr:`input` (:math:`\beta`)
    alpha (Number, optional): multiplier for :math:`\text{{batch1}} \mathbin{{@}} \text{{batch2}}` (:math:`\alpha`)
    {out}

Example::

    >>> M = torch.randn(10, 3, 5)
    >>> batch1 = torch.randn(10, 3, 4)
    >>> batch2 = torch.randn(10, 4, 5)
    >>> torch.baddbmm(M, batch1, batch2).size()
    torch.Size([10, 3, 5])
""".format(**common_args, **tf32_notes))

add_docstr(torch.bernoulli,
           r"""
bernoulli(input, *, generator=None, out=None) -> Tensor

Draws binary random numbers (0 or 1) from a Bernoulli distribution.

The :attr:`input` tensor should be a tensor containing probabilities
to be used for drawing the binary random number.
Hence, all values in :attr:`input` have to be in the range:
:math:`0 \leq \text{input}_i \leq 1`.

The :math:`\text{i}^{th}` element of the output tensor will draw a
value :math:`1` according to the :math:`\text{i}^{th}` probability value given
in :attr:`input`.

.. math::
    \text{out}_{i} \sim \mathrm{Bernoulli}(p = \text{input}_{i})
""" + r"""
The returned :attr:`out` tensor only has values 0 or 1 and is of the same
shape as :attr:`input`.

:attr:`out` can have integral ``dtype``, but :attr:`input` must have floating
point ``dtype``.

Args:
    input (Tensor): the input tensor of probability values for the Bernoulli distribution

Keyword args:
    {generator}
    {out}

Example::

    >>> a = torch.empty(3, 3).uniform_(0, 1)  # generate a uniform random matrix with range [0, 1]
    >>> a
    tensor([[ 0.1737,  0.0950,  0.3609],
            [ 0.7148,  0.0289,  0.2676],
            [ 0.9456,  0.8937,  0.7202]])
    >>> torch.bernoulli(a)
    tensor([[ 1.,  0.,  0.],
            [ 0.,  0.,  0.],
            [ 1.,  1.,  1.]])

    >>> a = torch.ones(3, 3) # probability of drawing "1" is 1
    >>> torch.bernoulli(a)
    tensor([[ 1.,  1.,  1.],
            [ 1.,  1.,  1.],
            [ 1.,  1.,  1.]])
    >>> a = torch.zeros(3, 3) # probability of drawing "1" is 0
    >>> torch.bernoulli(a)
    tensor([[ 0.,  0.,  0.],
            [ 0.,  0.,  0.],
            [ 0.,  0.,  0.]])
""".format(**common_args))

add_docstr(torch.bincount,
           r"""
bincount(input, weights=None, minlength=0) -> Tensor

Count the frequency of each value in an array of non-negative ints.

The number of bins (size 1) is one larger than the largest value in
:attr:`input` unless :attr:`input` is empty, in which case the result is a
tensor of size 0. If :attr:`minlength` is specified, the number of bins is at least
:attr:`minlength` and if :attr:`input` is empty, then the result is tensor of size
:attr:`minlength` filled with zeros. If ``n`` is the value at position ``i``,
``out[n] += weights[i]`` if :attr:`weights` is specified else
``out[n] += 1``.

Note:
    {backward_reproducibility_note}

Arguments:
    input (Tensor): 1-d int tensor
    weights (Tensor): optional, weight for each value in the input tensor.
        Should be of same size as input tensor.
    minlength (int): optional, minimum number of bins. Should be non-negative.

Returns:
    output (Tensor): a tensor of shape ``Size([max(input) + 1])`` if
    :attr:`input` is non-empty, else ``Size(0)``

Example::

    >>> input = torch.randint(0, 8, (5,), dtype=torch.int64)
    >>> weights = torch.linspace(0, 1, steps=5)
    >>> input, weights
    (tensor([4, 3, 6, 3, 4]),
     tensor([ 0.0000,  0.2500,  0.5000,  0.7500,  1.0000])

    >>> torch.bincount(input)
    tensor([0, 0, 0, 2, 2, 0, 1])

    >>> input.bincount(weights)
    tensor([0.0000, 0.0000, 0.0000, 1.0000, 1.0000, 0.0000, 0.5000])
""".format(**reproducibility_notes))

add_docstr(torch.bitwise_not,
           r"""
bitwise_not(input, *, out=None) -> Tensor

Computes the bitwise NOT of the given input tensor. The input tensor must be of
integral or Boolean types. For bool tensors, it computes the logical NOT.

Args:
    {input}

Keyword args:
    {out}

Example:

    >>> torch.bitwise_not(torch.tensor([-1, -2, 3], dtype=torch.int8))
    tensor([ 0,  1, -4], dtype=torch.int8)
""".format(**common_args))

# TODO: see https://github.com/pytorch/pytorch/issues/43667
add_docstr(torch.bmm,
           r"""
bmm(input, mat2, *, deterministic=False, out=None) -> Tensor

Performs a batch matrix-matrix product of matrices stored in :attr:`input`
and :attr:`mat2`.

:attr:`input` and :attr:`mat2` must be 3-D tensors each containing
the same number of matrices.

If :attr:`input` is a :math:`(b \times n \times m)` tensor, :attr:`mat2` is a
:math:`(b \times m \times p)` tensor, :attr:`out` will be a
:math:`(b \times n \times p)` tensor.

.. math::
    \text{out}_i = \text{input}_i \mathbin{@} \text{mat2}_i
""" + r"""
{tf32_note}

.. note:: This function does not :ref:`broadcast <broadcasting-semantics>`.
          For broadcasting matrix products, see :func:`torch.matmul`.

Args:
    input (Tensor): the first batch of matrices to be multiplied
    mat2 (Tensor): the second batch of matrices to be multiplied

Keyword Args:
    deterministic (bool, optional): flag to choose between a faster non-deterministic
                                    calculation, or a slower deterministic calculation.
                                    This argument is only available for sparse-dense CUDA bmm.
                                    Default: ``False``
    {out}

Example::

    >>> input = torch.randn(10, 3, 4)
    >>> mat2 = torch.randn(10, 4, 5)
    >>> res = torch.bmm(input, mat2)
    >>> res.size()
    torch.Size([10, 3, 5])
""".format(**common_args, **tf32_notes))

add_docstr(torch.bitwise_and,
           r"""
bitwise_and(input, other, *, out=None) -> Tensor

Computes the bitwise AND of :attr:`input` and :attr:`other`. The input tensor must be of
integral or Boolean types. For bool tensors, it computes the logical AND.

Args:
    input: the first input tensor
    other: the second input tensor

Keyword args:
    {out}

Example:

    >>> torch.bitwise_and(torch.tensor([-1, -2, 3], dtype=torch.int8), torch.tensor([1, 0, 3], dtype=torch.int8))
    tensor([1, 0,  3], dtype=torch.int8)
    >>> torch.bitwise_and(torch.tensor([True, True, False]), torch.tensor([False, True, False]))
    tensor([ False, True, False])
""".format(**common_args))

add_docstr(torch.bitwise_or,
           r"""
bitwise_or(input, other, *, out=None) -> Tensor

Computes the bitwise OR of :attr:`input` and :attr:`other`. The input tensor must be of
integral or Boolean types. For bool tensors, it computes the logical OR.

Args:
    input: the first input tensor
    other: the second input tensor

Keyword args:
    {out}

Example:

    >>> torch.bitwise_or(torch.tensor([-1, -2, 3], dtype=torch.int8), torch.tensor([1, 0, 3], dtype=torch.int8))
    tensor([-1, -2,  3], dtype=torch.int8)
    >>> torch.bitwise_or(torch.tensor([True, True, False]), torch.tensor([False, True, False]))
    tensor([ True, True, False])
""".format(**common_args))

add_docstr(torch.bitwise_xor,
           r"""
bitwise_xor(input, other, *, out=None) -> Tensor

Computes the bitwise XOR of :attr:`input` and :attr:`other`. The input tensor must be of
integral or Boolean types. For bool tensors, it computes the logical XOR.

Args:
    input: the first input tensor
    other: the second input tensor

Keyword args:
    {out}

Example:

    >>> torch.bitwise_xor(torch.tensor([-1, -2, 3], dtype=torch.int8), torch.tensor([1, 0, 3], dtype=torch.int8))
    tensor([-2, -2,  0], dtype=torch.int8)
    >>> torch.bitwise_xor(torch.tensor([True, True, False]), torch.tensor([False, True, False]))
    tensor([ True, False, False])
""".format(**common_args))

add_docstr(torch.stack,
           r"""
stack(tensors, dim=0, *, out=None) -> Tensor

Concatenates a sequence of tensors along a new dimension.

All tensors need to be of the same size.

Arguments:
    tensors (sequence of Tensors): sequence of tensors to concatenate
    dim (int): dimension to insert. Has to be between 0 and the number
        of dimensions of concatenated tensors (inclusive)

Keyword args:
    {out}
""".format(**common_args))

add_docstr(torch.hstack,
           r"""
hstack(tensors, *, out=None) -> Tensor

Stack tensors in sequence horizontally (column wise).

This is equivalent to concatenation along the first axis for 1-D tensors, and along the second axis for all other tensors.

Args:
    tensors (sequence of Tensors): sequence of tensors to concatenate

Keyword args:
    {out}

Example::

    >>> a = torch.tensor([1, 2, 3])
    >>> b = torch.tensor([4, 5, 6])
    >>> torch.hstack((a,b))
    tensor([1, 2, 3, 4, 5, 6])
    >>> a = torch.tensor([[1],[2],[3]])
    >>> b = torch.tensor([[4],[5],[6]])
    >>> torch.hstack((a,b))
    tensor([[1, 4],
            [2, 5],
            [3, 6]])

""".format(**common_args))

add_docstr(torch.vstack,
           r"""
vstack(tensors, *, out=None) -> Tensor

Stack tensors in sequence vertically (row wise).

This is equivalent to concatenation along the first axis after all 1-D tensors have been reshaped by :func:`torch.atleast_2d`.

Args:
    tensors (sequence of Tensors): sequence of tensors to concatenate

Keyword args:
    {out}

Example::

    >>> a = torch.tensor([1, 2, 3])
    >>> b = torch.tensor([4, 5, 6])
    >>> torch.vstack((a,b))
    tensor([[1, 2, 3],
            [4, 5, 6]])
    >>> a = torch.tensor([[1],[2],[3]])
    >>> b = torch.tensor([[4],[5],[6]])
    >>> torch.vstack((a,b))
    tensor([[1],
            [2],
            [3],
            [4],
            [5],
            [6]])


""".format(**common_args))

add_docstr(torch.dstack,
           r"""
dstack(tensors, *, out=None) -> Tensor

Stack tensors in sequence depthwise (along third axis).

This is equivalent to concatenation along the third axis after 1-D and 2-D tensors have been reshaped by :func:`torch.atleast_3d`.

Args:
    tensors (sequence of Tensors): sequence of tensors to concatenate

Keyword args:
    {out}

Example::
    >>> a = torch.tensor([1, 2, 3])
    >>> b = torch.tensor([4, 5, 6])
    >>> torch.dstack((a,b))
    tensor([[[1, 4],
             [2, 5],
             [3, 6]]])
    >>> a = torch.tensor([[1],[2],[3]])
    >>> b = torch.tensor([[4],[5],[6]])
    >>> torch.dstack((a,b))
    tensor([[[1, 4]],
            [[2, 5]],
            [[3, 6]]])


""".format(**common_args))

add_docstr(torch.tensor_split,
           r"""
tensor_split(input, indices_or_sections, dim=0) -> List of Tensors

Splits a tensor into multiple sub-tensors, all of which are views of :attr:`input`,
along dimension :attr:`dim` according to the indices or number of sections specified
by :attr:`indices_or_sections`. This function is based on NumPy's
:func:`numpy.array_split`.

Args:
    input (Tensor): the tensor to split
    indices_or_sections (int or (list(int))):
        If :attr:`indices_or_sections` is an integer ``n``, :attr:`input` is split
        into ``n`` sections along dimension :attr:`dim`. If :attr:`input` is divisible
        by ``n`` along dimension :attr:`dim`, each section will be of equal size,
        :code:`input.size(dim) / n`. If :attr:`input` is not divisible by ``n``, the
        sizes of the first :code:`int(input.size(dim) % n)` sections will have size
        :code:`int(input.size(dim) / n) + 1`, and the rest will have size
        :code:`int(input.size(dim) / n)`.

        If :attr:`indices_or_sections` is a list of ints, :attr:`input` is split along
        dimension :attr:`dim` at each of the indices in the list. For instance,
        :code:`indices_or_sections=[2, 3]` and :code:`dim=0` would result in the tensors
        :code:`input[:2]`, :code:`input[2:3]`, and :code:`input[3:]`.

    dim (int, optional): dimension along which to split the tensor. Default: ``0``

Example::
    >>> x = torch.arange(8)
    >>> torch.tensor_split(x, 3)
    (tensor([0, 1, 2]), tensor([3, 4, 5]), tensor([6, 7]))

    >>> x = torch.arange(7)
    >>> torch.tensor_split(x, 3)
    (tensor([0, 1, 2]), tensor([3, 4]), tensor([5, 6]))
    >>> torch.tensor_split(x, (1, 6))
    (tensor([0]), tensor([1, 2, 3, 4, 5]), tensor([6]))

    >>> x = torch.arange(14).reshape(2, 7)
    >>> x
    tensor([[ 0,  1,  2,  3,  4,  5,  6],
            [ 7,  8,  9, 10, 11, 12, 13]])
    >>> torch.tensor_split(x, 3, dim=1)
    (tensor([[0, 1, 2],
            [7, 8, 9]]),
     tensor([[ 3,  4],
            [10, 11]]),
     tensor([[ 5,  6],
            [12, 13]]))
    >>> torch.tensor_split(x, (1, 6), dim=1)
    (tensor([[0],
            [7]]),
     tensor([[ 1,  2,  3,  4,  5],
            [ 8,  9, 10, 11, 12]]),
     tensor([[ 6],
            [13]]))
""")

add_docstr(torch.chunk,
           r"""
chunk(input, chunks, dim=0) -> List of Tensors

Splits a tensor into a specific number of chunks. Each chunk is a view of
the input tensor.

Last chunk will be smaller if the tensor size along the given dimension
:attr:`dim` is not divisible by :attr:`chunks`.

Arguments:
    input (Tensor): the tensor to split
    chunks (int): number of chunks to return
    dim (int): dimension along which to split the tensor
""")

add_docstr(torch.unsafe_chunk,
           r"""
unsafe_chunk(input, chunks, dim=0) -> List of Tensors

Works like :func:`torch.chunk` but without enforcing the autograd restrictions
on inplace modification of the outputs.

.. warning::
    This function is safe to use as long as only the input, or only the outputs
    are modified inplace after calling this function. It is user's
    responsibility to ensure that is the case. If both the input and one or more
    of the outputs are modified inplace, gradients computed by autograd will be
    silently incorrect.
""")

add_docstr(torch.unsafe_split,
           r"""
unsafe_split(tensor, split_size_or_sections, dim=0) -> List of Tensors

Works like :func:`torch.split` but without enforcing the autograd restrictions
on inplace modification of the outputs.

.. warning::
    This function is safe to use as long as only the input, or only the outputs
    are modified inplace after calling this function. It is user's
    responsibility to ensure that is the case. If both the input and one or more
    of the outputs are modified inplace, gradients computed by autograd will be
    silently incorrect.
""")

add_docstr(torch.can_cast,
           r"""
can_cast(from, to) -> bool

Determines if a type conversion is allowed under PyTorch casting rules
described in the type promotion :ref:`documentation <type-promotion-doc>`.

Args:
    from (dtype): The original :class:`torch.dtype`.
    to (dtype): The target :class:`torch.dtype`.

Example::

    >>> torch.can_cast(torch.double, torch.float)
    True
    >>> torch.can_cast(torch.float, torch.int)
    False
""")

add_docstr(torch.cat,
           r"""
cat(tensors, dim=0, *, out=None) -> Tensor

Concatenates the given sequence of :attr:`seq` tensors in the given dimension.
All tensors must either have the same shape (except in the concatenating
dimension) or be empty.

:func:`torch.cat` can be seen as an inverse operation for :func:`torch.split`
and :func:`torch.chunk`.

:func:`torch.cat` can be best understood via examples.

Args:
    tensors (sequence of Tensors): any python sequence of tensors of the same type.
        Non-empty tensors provided must have the same shape, except in the
        cat dimension.
    dim (int, optional): the dimension over which the tensors are concatenated

Keyword args:
    {out}

Example::

    >>> x = torch.randn(2, 3)
    >>> x
    tensor([[ 0.6580, -1.0969, -0.4614],
            [-0.1034, -0.5790,  0.1497]])
    >>> torch.cat((x, x, x), 0)
    tensor([[ 0.6580, -1.0969, -0.4614],
            [-0.1034, -0.5790,  0.1497],
            [ 0.6580, -1.0969, -0.4614],
            [-0.1034, -0.5790,  0.1497],
            [ 0.6580, -1.0969, -0.4614],
            [-0.1034, -0.5790,  0.1497]])
    >>> torch.cat((x, x, x), 1)
    tensor([[ 0.6580, -1.0969, -0.4614,  0.6580, -1.0969, -0.4614,  0.6580,
             -1.0969, -0.4614],
            [-0.1034, -0.5790,  0.1497, -0.1034, -0.5790,  0.1497, -0.1034,
             -0.5790,  0.1497]])
""".format(**common_args))

add_docstr(torch.ceil,
           r"""
ceil(input, *, out=None) -> Tensor

Returns a new tensor with the ceil of the elements of :attr:`input`,
the smallest integer greater than or equal to each element.

.. math::
    \text{out}_{i} = \left\lceil \text{input}_{i} \right\rceil = \left\lfloor \text{input}_{i} \right\rfloor + 1
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.6341, -1.4208, -1.0900,  0.5826])
    >>> torch.ceil(a)
    tensor([-0., -1., -1.,  1.])
""".format(**common_args))

add_docstr(torch.real,
           r"""
real(input) -> Tensor

Returns a new tensor containing real values of the :attr:`self` tensor.
The returned tensor and :attr:`self` share the same underlying storage.

.. warning::
    :func:`real` is only supported for tensors with complex dtypes.

Args:
    {input}

Example::
    >>> x=torch.randn(4, dtype=torch.cfloat)
    >>> x
    tensor([(0.3100+0.3553j), (-0.5445-0.7896j), (-1.6492-0.0633j), (-0.0638-0.8119j)])
    >>> x.real
    tensor([ 0.3100, -0.5445, -1.6492, -0.0638])

""".format(**common_args))

add_docstr(torch.imag,
           r"""
imag(input) -> Tensor

Returns a new tensor containing imaginary values of the :attr:`self` tensor.
The returned tensor and :attr:`self` share the same underlying storage.

.. warning::
    :func:`imag` is only supported for tensors with complex dtypes.

Args:
    {input}

Example::
    >>> x=torch.randn(4, dtype=torch.cfloat)
    >>> x
    tensor([(0.3100+0.3553j), (-0.5445-0.7896j), (-1.6492-0.0633j), (-0.0638-0.8119j)])
    >>> x.imag
    tensor([ 0.3553, -0.7896, -0.0633, -0.8119])

""".format(**common_args))

add_docstr(torch.view_as_real,
           r"""
view_as_real(input) -> Tensor

Returns a view of :attr:`input` as a real tensor. For an input complex tensor of
:attr:`size` :math:`m1, m2, \dots, mi`, this function returns a new
real tensor of size :math:`m1, m2, \dots, mi, 2`, where the last dimension of size 2
represents the real and imaginary components of complex numbers.

.. warning::
    :func:`view_as_real` is only supported for tensors with ``complex dtypes``.

Args:
    {input}

Example::
    >>> x=torch.randn(4, dtype=torch.cfloat)
    >>> x
    tensor([(0.4737-0.3839j), (-0.2098-0.6699j), (0.3470-0.9451j), (-0.5174-1.3136j)])
    >>> torch.view_as_real(x)
    tensor([[ 0.4737, -0.3839],
            [-0.2098, -0.6699],
            [ 0.3470, -0.9451],
            [-0.5174, -1.3136]])
""".format(**common_args))

add_docstr(torch.view_as_complex,
           r"""
view_as_complex(input) -> Tensor

Returns a view of :attr:`input` as a complex tensor. For an input complex
tensor of :attr:`size` :math:`m1, m2, \dots, mi, 2`, this function returns a
new complex tensor of :attr:`size` :math:`m1, m2, \dots, mi` where the last
dimension of the input tensor is expected to represent the real and imaginary
components of complex numbers.

.. warning::
    :func:`view_as_complex` is only supported for tensors with
    :class:`torch.dtype` ``torch.float64`` and ``torch.float32``.  The input is
    expected to have the last dimension of :attr:`size` 2. In addition, the
    tensor must have a `stride` of 1 for its last dimension. The strides of all
    other dimensions must be even numbers.

Args:
    {input}

Example::
    >>> x=torch.randn(4, 2)
    >>> x
    tensor([[ 1.6116, -0.5772],
            [-1.4606, -0.9120],
            [ 0.0786, -1.7497],
            [-0.6561, -1.6623]])
    >>> torch.view_as_complex(x)
    tensor([(1.6116-0.5772j), (-1.4606-0.9120j), (0.0786-1.7497j), (-0.6561-1.6623j)])
""".format(**common_args))

add_docstr(torch.reciprocal,
           r"""
reciprocal(input, *, out=None) -> Tensor

Returns a new tensor with the reciprocal of the elements of :attr:`input`

.. math::
    \text{out}_{i} = \frac{1}{\text{input}_{i}}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.4595, -2.1219, -1.4314,  0.7298])
    >>> torch.reciprocal(a)
    tensor([-2.1763, -0.4713, -0.6986,  1.3702])
""".format(**common_args))

add_docstr(torch.cholesky, r"""
cholesky(input, upper=False, *, out=None) -> Tensor

Computes the Cholesky decomposition of a symmetric positive-definite
matrix :math:`A` or for batches of symmetric positive-definite matrices.

If :attr:`upper` is ``True``, the returned matrix ``U`` is upper-triangular, and
the decomposition has the form:

.. math::

  A = U^TU

If :attr:`upper` is ``False``, the returned matrix ``L`` is lower-triangular, and
the decomposition has the form:

.. math::

    A = LL^T

If :attr:`upper` is ``True``, and :math:`A` is a batch of symmetric positive-definite
matrices, then the returned tensor will be composed of upper-triangular Cholesky factors
of each of the individual matrices. Similarly, when :attr:`upper` is ``False``, the returned
tensor will be composed of lower-triangular Cholesky factors of each of the individual
matrices.

Args:
    input (Tensor): the input tensor :math:`A` of size :math:`(*, n, n)` where `*` is zero or more
                batch dimensions consisting of symmetric positive-definite matrices.
    upper (bool, optional): flag that indicates whether to return a
                            upper or lower triangular matrix. Default: ``False``

Keyword args:
    out (Tensor, optional): the output matrix

Example::

    >>> a = torch.randn(3, 3)
    >>> a = torch.mm(a, a.t()) # make symmetric positive-definite
    >>> l = torch.cholesky(a)
    >>> a
    tensor([[ 2.4112, -0.7486,  1.4551],
            [-0.7486,  1.3544,  0.1294],
            [ 1.4551,  0.1294,  1.6724]])
    >>> l
    tensor([[ 1.5528,  0.0000,  0.0000],
            [-0.4821,  1.0592,  0.0000],
            [ 0.9371,  0.5487,  0.7023]])
    >>> torch.mm(l, l.t())
    tensor([[ 2.4112, -0.7486,  1.4551],
            [-0.7486,  1.3544,  0.1294],
            [ 1.4551,  0.1294,  1.6724]])
    >>> a = torch.randn(3, 2, 2)
    >>> a = torch.matmul(a, a.transpose(-1, -2)) + 1e-03 # make symmetric positive-definite
    >>> l = torch.cholesky(a)
    >>> z = torch.matmul(l, l.transpose(-1, -2))
    >>> torch.max(torch.abs(z - a)) # Max non-zero
    tensor(2.3842e-07)
""")

add_docstr(torch.cholesky_solve, r"""
cholesky_solve(input, input2, upper=False, *, out=None) -> Tensor

Solves a linear system of equations with a positive semidefinite
matrix to be inverted given its Cholesky factor matrix :math:`u`.

If :attr:`upper` is ``False``, :math:`u` is and lower triangular and `c` is
returned such that:

.. math::
    c = (u u^T)^{{-1}} b

If :attr:`upper` is ``True`` or not provided, :math:`u` is upper triangular
and `c` is returned such that:

.. math::
    c = (u^T u)^{{-1}} b

`torch.cholesky_solve(b, u)` can take in 2D inputs `b, u` or inputs that are
batches of 2D matrices. If the inputs are batches, then returns
batched outputs `c`

Args:
    input (Tensor): input matrix :math:`b` of size :math:`(*, m, k)`,
                where :math:`*` is zero or more batch dimensions
    input2 (Tensor): input matrix :math:`u` of size :math:`(*, m, m)`,
                where :math:`*` is zero of more batch dimensions composed of
                upper or lower triangular Cholesky factor
    upper (bool, optional): whether to consider the Cholesky factor as a
                            lower or upper triangular matrix. Default: ``False``.

Keyword args:
    out (Tensor, optional): the output tensor for `c`

Example::

    >>> a = torch.randn(3, 3)
    >>> a = torch.mm(a, a.t()) # make symmetric positive definite
    >>> u = torch.cholesky(a)
    >>> a
    tensor([[ 0.7747, -1.9549,  1.3086],
            [-1.9549,  6.7546, -5.4114],
            [ 1.3086, -5.4114,  4.8733]])
    >>> b = torch.randn(3, 2)
    >>> b
    tensor([[-0.6355,  0.9891],
            [ 0.1974,  1.4706],
            [-0.4115, -0.6225]])
    >>> torch.cholesky_solve(b, u)
    tensor([[ -8.1625,  19.6097],
            [ -5.8398,  14.2387],
            [ -4.3771,  10.4173]])
    >>> torch.mm(a.inverse(), b)
    tensor([[ -8.1626,  19.6097],
            [ -5.8398,  14.2387],
            [ -4.3771,  10.4173]])
""")

add_docstr(torch.cholesky_inverse, r"""
cholesky_inverse(input, upper=False, *, out=None) -> Tensor

Computes the inverse of a symmetric positive-definite matrix :math:`A` using its
Cholesky factor :math:`u`: returns matrix ``inv``. The inverse is computed using
LAPACK routines ``dpotri`` and ``spotri`` (and the corresponding MAGMA routines).

If :attr:`upper` is ``False``, :math:`u` is lower triangular
such that the returned tensor is

.. math::
    inv = (uu^{{T}})^{{-1}}

If :attr:`upper` is ``True`` or not provided, :math:`u` is upper
triangular such that the returned tensor is

.. math::
    inv = (u^T u)^{{-1}}

Args:
    input (Tensor): the input 2-D tensor :math:`u`, a upper or lower triangular
           Cholesky factor
    upper (bool, optional): whether to return a lower (default) or upper triangular matrix

Keyword args:
    out (Tensor, optional): the output tensor for `inv`

Example::

    >>> a = torch.randn(3, 3)
    >>> a = torch.mm(a, a.t()) + 1e-05 * torch.eye(3) # make symmetric positive definite
    >>> u = torch.cholesky(a)
    >>> a
    tensor([[  0.9935,  -0.6353,   1.5806],
            [ -0.6353,   0.8769,  -1.7183],
            [  1.5806,  -1.7183,  10.6618]])
    >>> torch.cholesky_inverse(u)
    tensor([[ 1.9314,  1.2251, -0.0889],
            [ 1.2251,  2.4439,  0.2122],
            [-0.0889,  0.2122,  0.1412]])
    >>> a.inverse()
    tensor([[ 1.9314,  1.2251, -0.0889],
            [ 1.2251,  2.4439,  0.2122],
            [-0.0889,  0.2122,  0.1412]])
""")

add_docstr(torch.clone, r"""
clone(input, *, memory_format=torch.preserve_format) -> Tensor

Returns a copy of :attr:`input`.

.. note::

    This function is differentiable, so gradients will flow back from the
    result of this operation to :attr:`input`. To create a tensor without an
    autograd relationship to :attr:`input` see :meth:`~Tensor.detach`.

Args:
    {input}

Keyword args:
    {memory_format}
""".format(**common_args))

add_docstr(torch.clamp, r"""
clamp(input, min, max, *, out=None) -> Tensor

Clamp all elements in :attr:`input` into the range `[` :attr:`min`, :attr:`max` `]`.
Let min_value and max_value be :attr:`min` and :attr:`max`, respectively, this returns:

.. math::
    y_i = \min(\max(x_i, \text{min\_value}), \text{max\_value})
""" + r"""

Args:
    {input}
    min (Number): lower-bound of the range to be clamped to
    max (Number): upper-bound of the range to be clamped to

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-1.7120,  0.1734, -0.0478, -0.0922])
    >>> torch.clamp(a, min=-0.5, max=0.5)
    tensor([-0.5000,  0.1734, -0.0478, -0.0922])

.. function:: clamp(input, *, min, out=None) -> Tensor

Clamps all elements in :attr:`input` to be larger or equal :attr:`min`.

Args:
    {input}

Keyword args:
    min (Number): minimal value of each element in the output
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.0299, -2.3184,  2.1593, -0.8883])
    >>> torch.clamp(a, min=0.5)
    tensor([ 0.5000,  0.5000,  2.1593,  0.5000])

.. function:: clamp(input, *, max, out=None) -> Tensor

Clamps all elements in :attr:`input` to be smaller or equal :attr:`max`.

Args:
    {input}

Keyword args:
    max (Number): maximal value of each element in the output
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.7753, -0.4702, -0.4599,  1.1899])
    >>> torch.clamp(a, max=0.5)
    tensor([ 0.5000, -0.4702, -0.4599,  0.5000])
""".format(**common_args))

add_docstr(torch.clip, r"""
clip(input, min, max, *, out=None) -> Tensor

Alias for :func:`torch.clamp`.
""".format(**common_args))

add_docstr(torch.column_stack,
           r"""
column_stack(tensors, *, out=None) -> Tensor

Creates a new tensor by horizontally stacking the tensors in :attr:`tensors`.

Equivalent to ``torch.hstack(tensors)``, except each zero or one dimensional tensor ``t``
in :attr:`tensors` is first reshaped into a ``(t.numel(), 1)`` column before being stacked horizontally.

Args:
    tensors (sequence of Tensors): sequence of tensors to concatenate

Keyword args:
    {out}

Example::

    >>> a = torch.tensor([1, 2, 3])
    >>> b = torch.tensor([4, 5, 6])
    >>> torch.column_stack((a, b))
    tensor([[1, 4],
        [2, 5],
        [3, 6]])
    >>> a = torch.arange(5)
    >>> b = torch.arange(10).reshape(5, 2)
    >>> torch.column_stack((a, b, b))
    tensor([[0, 0, 1, 0, 1],
            [1, 2, 3, 2, 3],
            [2, 4, 5, 4, 5],
            [3, 6, 7, 6, 7],
            [4, 8, 9, 8, 9]])

""".format(**common_args))

add_docstr(torch.complex,
           r"""
complex(real, imag, *, out=None) -> Tensor

Constructs a complex tensor with its real part equal to :attr:`real` and its
imaginary part equal to :attr:`imag`.

Args:
    real (Tensor): The real part of the complex tensor. Must be float or double.
    imag (Tensor): The imaginary part of the complex tensor. Must be same dtype
        as :attr:`real`.

Keyword args:
    out (Tensor): If the inputs are ``torch.float32``, must be
        ``torch.complex64``. If the inputs are ``torch.float64``, must be
        ``torch.complex128``.

Example::
    >>> real = torch.tensor([1, 2], dtype=torch.float32)
    >>> imag = torch.tensor([3, 4], dtype=torch.float32)
    >>> z = torch.complex(real, imag)
    >>> z
    tensor([(1.+3.j), (2.+4.j)])
    >>> z.dtype
    torch.complex64

""")

add_docstr(torch.polar,
           r"""
polar(abs, angle, *, out=None) -> Tensor

Constructs a complex tensor whose elements are Cartesian coordinates
corresponding to the polar coordinates with absolute value :attr:`abs` and angle
:attr:`angle`.

.. math::
    \text{out} = \text{abs} \cdot \cos(\text{angle}) + \text{abs} \cdot \sin(\text{angle}) \cdot j
""" + r"""
Args:
    abs (Tensor): The absolute value the complex tensor. Must be float or
        double.
    angle (Tensor): The angle of the complex tensor. Must be same dtype as
        :attr:`abs`.

Keyword args:
    out (Tensor): If the inputs are ``torch.float32``, must be
        ``torch.complex64``. If the inputs are ``torch.float64``, must be
        ``torch.complex128``.

Example::
    >>> import numpy as np
    >>> abs = torch.tensor([1, 2], dtype=torch.float64)
    >>> angle = torch.tensor([np.pi / 2, 5 * np.pi / 4], dtype=torch.float64)
    >>> z = torch.polar(abs, angle)
    >>> z
    tensor([(0.0000+1.0000j), (-1.4142-1.4142j)], dtype=torch.complex128)
""")

add_docstr(torch.conj,
           r"""
conj(input, *, out=None) -> Tensor

Computes the element-wise conjugate of the given :attr:`input` tensor. If :attr`input` has a non-complex dtype,
this function just returns :attr:`input`.

.. warning:: In the future, :func:`torch.conj` may return a non-writeable view for an :attr:`input` of
             non-complex dtype. It's recommended that programs not modify the tensor returned by :func:`torch.conj`
             when :attr:`input` is of non-complex dtype to be compatible with this change.

.. math::
    \text{out}_{i} = conj(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.conj(torch.tensor([-1 + 1j, -2 + 2j, 3 - 3j]))
    tensor([-1 - 1j, -2 - 2j, 3 + 3j])
""".format(**common_args))

add_docstr(torch.cos,
           r"""
cos(input, *, out=None) -> Tensor

Returns a new tensor with the cosine  of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \cos(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 1.4309,  1.2706, -0.8562,  0.9796])
    >>> torch.cos(a)
    tensor([ 0.1395,  0.2957,  0.6553,  0.5574])
""".format(**common_args))

add_docstr(torch.cosh,
           r"""
cosh(input, *, out=None) -> Tensor

Returns a new tensor with the hyperbolic cosine  of the elements of
:attr:`input`.

.. math::
    \text{out}_{i} = \cosh(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.1632,  1.1835, -0.6979, -0.7325])
    >>> torch.cosh(a)
    tensor([ 1.0133,  1.7860,  1.2536,  1.2805])
""".format(**common_args))

add_docstr(torch.cross,
           r"""
cross(input, other, dim=None, *, out=None) -> Tensor


Returns the cross product of vectors in dimension :attr:`dim` of :attr:`input`
and :attr:`other`.

:attr:`input` and :attr:`other` must have the same size, and the size of their
:attr:`dim` dimension should be 3.

If :attr:`dim` is not given, it defaults to the first dimension found with the
size 3. Note that this might be unexpected.

Args:
    {input}
    other (Tensor): the second input tensor
    dim  (int, optional): the dimension to take the cross-product in.

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4, 3)
    >>> a
    tensor([[-0.3956,  1.1455,  1.6895],
            [-0.5849,  1.3672,  0.3599],
            [-1.1626,  0.7180, -0.0521],
            [-0.1339,  0.9902, -2.0225]])
    >>> b = torch.randn(4, 3)
    >>> b
    tensor([[-0.0257, -1.4725, -1.2251],
            [-1.1479, -0.7005, -1.9757],
            [-1.3904,  0.3726, -1.1836],
            [-0.9688, -0.7153,  0.2159]])
    >>> torch.cross(a, b, dim=1)
    tensor([[ 1.0844, -0.5281,  0.6120],
            [-2.4490, -1.5687,  1.9792],
            [-0.8304, -1.3037,  0.5650],
            [-1.2329,  1.9883,  1.0551]])
    >>> torch.cross(a, b)
    tensor([[ 1.0844, -0.5281,  0.6120],
            [-2.4490, -1.5687,  1.9792],
            [-0.8304, -1.3037,  0.5650],
            [-1.2329,  1.9883,  1.0551]])
""".format(**common_args))

add_docstr(torch.logcumsumexp,
           r"""
logcumsumexp(input, dim, *, out=None) -> Tensor
Returns the logarithm of the cumulative summation of the exponentiation of
elements of :attr:`input` in the dimension :attr:`dim`.

For summation index :math:`j` given by `dim` and other indices :math:`i`, the result is

    .. math::
        \text{{logcumsumexp}}(x)_{{ij}} = \log \sum\limits_{{j=0}}^{{i}} \exp(x_{{ij}})

Args:
    {input}
    dim  (int): the dimension to do the operation over

Keyword args:
    {out}

Example::
    >>> a = torch.randn(10)
    >>> torch.logcumsumexp(a, dim=0)
    tensor([-0.42296738, -0.04462666,  0.86278635,  0.94622083,  1.05277811,
             1.39202815,  1.83525007,  1.84492621,  2.06084887,  2.06844475]))
""".format(**reduceops_common_args))

add_docstr(torch.cummax,
           r"""
cummax(input, dim, *, out=None) -> (Tensor, LongTensor)
Returns a namedtuple ``(values, indices)`` where ``values`` is the cumulative maximum of
elements of :attr:`input` in the dimension :attr:`dim`. And ``indices`` is the index
location of each maximum value found in the dimension :attr:`dim`.

.. math::
    y_i = max(x_1, x_2, x_3, \dots, x_i)

Args:
    {input}
    dim  (int): the dimension to do the operation over

Keyword args:
    out (tuple, optional): the result tuple of two output tensors (values, indices)

Example::

    >>> a = torch.randn(10)
    >>> a
    tensor([-0.3449, -1.5447,  0.0685, -1.5104, -1.1706,  0.2259,  1.4696, -1.3284,
         1.9946, -0.8209])
    >>> torch.cummax(a, dim=0)
    torch.return_types.cummax(
        values=tensor([-0.3449, -0.3449,  0.0685,  0.0685,  0.0685,  0.2259,  1.4696,  1.4696,
         1.9946,  1.9946]),
        indices=tensor([0, 0, 2, 2, 2, 5, 6, 6, 8, 8]))
""".format(**reduceops_common_args))

add_docstr(torch.cummin,
           r"""
cummin(input, dim, *, out=None) -> (Tensor, LongTensor)
Returns a namedtuple ``(values, indices)`` where ``values`` is the cumulative minimum of
elements of :attr:`input` in the dimension :attr:`dim`. And ``indices`` is the index
location of each maximum value found in the dimension :attr:`dim`.

.. math::
    y_i = min(x_1, x_2, x_3, \dots, x_i)

Args:
    {input}
    dim  (int): the dimension to do the operation over

Keyword args:
    out (tuple, optional): the result tuple of two output tensors (values, indices)

Example::

    >>> a = torch.randn(10)
    >>> a
    tensor([-0.2284, -0.6628,  0.0975,  0.2680, -1.3298, -0.4220, -0.3885,  1.1762,
         0.9165,  1.6684])
    >>> torch.cummin(a, dim=0)
    torch.return_types.cummin(
        values=tensor([-0.2284, -0.6628, -0.6628, -0.6628, -1.3298, -1.3298, -1.3298, -1.3298,
        -1.3298, -1.3298]),
        indices=tensor([0, 1, 1, 1, 4, 4, 4, 4, 4, 4]))
""".format(**reduceops_common_args))

add_docstr(torch.cumprod,
           r"""
cumprod(input, dim, *, dtype=None, out=None) -> Tensor

Returns the cumulative product of elements of :attr:`input` in the dimension
:attr:`dim`.

For example, if :attr:`input` is a vector of size N, the result will also be
a vector of size N, with elements.

.. math::
    y_i = x_1 \times x_2\times x_3\times \dots \times x_i

Args:
    {input}
    dim  (int): the dimension to do the operation over

Keyword args:
    {dtype}
    {out}

Example::

    >>> a = torch.randn(10)
    >>> a
    tensor([ 0.6001,  0.2069, -0.1919,  0.9792,  0.6727,  1.0062,  0.4126,
            -0.2129, -0.4206,  0.1968])
    >>> torch.cumprod(a, dim=0)
    tensor([ 0.6001,  0.1241, -0.0238, -0.0233, -0.0157, -0.0158, -0.0065,
             0.0014, -0.0006, -0.0001])

    >>> a[5] = 0.0
    >>> torch.cumprod(a, dim=0)
    tensor([ 0.6001,  0.1241, -0.0238, -0.0233, -0.0157, -0.0000, -0.0000,
             0.0000, -0.0000, -0.0000])
""".format(**reduceops_common_args))

add_docstr(torch.cumsum,
           r"""
cumsum(input, dim, *, dtype=None, out=None) -> Tensor

Returns the cumulative sum of elements of :attr:`input` in the dimension
:attr:`dim`.

For example, if :attr:`input` is a vector of size N, the result will also be
a vector of size N, with elements.

.. math::
    y_i = x_1 + x_2 + x_3 + \dots + x_i

Args:
    {input}
    dim  (int): the dimension to do the operation over

Keyword args:
    {dtype}
    {out}

Example::

    >>> a = torch.randn(10)
    >>> a
    tensor([-0.8286, -0.4890,  0.5155,  0.8443,  0.1865, -0.1752, -2.0595,
             0.1850, -1.1571, -0.4243])
    >>> torch.cumsum(a, dim=0)
    tensor([-0.8286, -1.3175, -0.8020,  0.0423,  0.2289,  0.0537, -2.0058,
            -1.8209, -2.9780, -3.4022])
""".format(**reduceops_common_args))

add_docstr(torch.count_nonzero,
           r"""
count_nonzero(input, dim=None) -> Tensor

Counts the number of non-zero values in the tensor :attr:`input` along the given :attr:`dim`.
If no dim is specified then all non-zeros in the tensor are counted.

Args:
    {input}
    dim (int or tuple of ints, optional): Dim or tuple of dims along which to count non-zeros.

Example::

    >>> x = torch.zeros(3,3)
    >>> x[torch.randn(3,3) > 0.5] = 1
    >>> x
    tensor([[0., 1., 1.],
            [0., 0., 0.],
            [0., 0., 1.]])
    >>> torch.count_nonzero(x)
    tensor(3)
    >>> torch.count_nonzero(x, dim=0)
    tensor([0, 1, 2])
""".format(**reduceops_common_args))

add_docstr(torch.dequantize,
           r"""
dequantize(tensor) -> Tensor

Returns an fp32 Tensor by dequantizing a quantized Tensor

Args:
    tensor (Tensor): A quantized Tensor

.. function:: dequantize(tensors) -> sequence of Tensors

Given a list of quantized Tensors, dequantize them and return a list of fp32 Tensors

Args:
     tensors (sequence of Tensors): A list of quantized Tensors
""")

add_docstr(torch.diag,
           r"""
diag(input, diagonal=0, *, out=None) -> Tensor

- If :attr:`input` is a vector (1-D tensor), then returns a 2-D square tensor
  with the elements of :attr:`input` as the diagonal.
- If :attr:`input` is a matrix (2-D tensor), then returns a 1-D tensor with
  the diagonal elements of :attr:`input`.

The argument :attr:`diagonal` controls which diagonal to consider:

- If :attr:`diagonal` = 0, it is the main diagonal.
- If :attr:`diagonal` > 0, it is above the main diagonal.
- If :attr:`diagonal` < 0, it is below the main diagonal.

Args:
    {input}
    diagonal (int, optional): the diagonal to consider

Keyword args:
    {out}

.. seealso::

        :func:`torch.diagonal` always returns the diagonal of its input.

        :func:`torch.diagflat` always constructs a tensor with diagonal elements
        specified by the input.

Examples:

Get the square matrix where the input vector is the diagonal::

    >>> a = torch.randn(3)
    >>> a
    tensor([ 0.5950,-0.0872, 2.3298])
    >>> torch.diag(a)
    tensor([[ 0.5950, 0.0000, 0.0000],
            [ 0.0000,-0.0872, 0.0000],
            [ 0.0000, 0.0000, 2.3298]])
    >>> torch.diag(a, 1)
    tensor([[ 0.0000, 0.5950, 0.0000, 0.0000],
            [ 0.0000, 0.0000,-0.0872, 0.0000],
            [ 0.0000, 0.0000, 0.0000, 2.3298],
            [ 0.0000, 0.0000, 0.0000, 0.0000]])

Get the k-th diagonal of a given matrix::

    >>> a = torch.randn(3, 3)
    >>> a
    tensor([[-0.4264, 0.0255,-0.1064],
            [ 0.8795,-0.2429, 0.1374],
            [ 0.1029,-0.6482,-1.6300]])
    >>> torch.diag(a, 0)
    tensor([-0.4264,-0.2429,-1.6300])
    >>> torch.diag(a, 1)
    tensor([ 0.0255, 0.1374])
""".format(**common_args))

add_docstr(torch.diag_embed,
           r"""
diag_embed(input, offset=0, dim1=-2, dim2=-1) -> Tensor

Creates a tensor whose diagonals of certain 2D planes (specified by
:attr:`dim1` and :attr:`dim2`) are filled by :attr:`input`.
To facilitate creating batched diagonal matrices, the 2D planes formed by
the last two dimensions of the returned tensor are chosen by default.

The argument :attr:`offset` controls which diagonal to consider:

- If :attr:`offset` = 0, it is the main diagonal.
- If :attr:`offset` > 0, it is above the main diagonal.
- If :attr:`offset` < 0, it is below the main diagonal.

The size of the new matrix will be calculated to make the specified diagonal
of the size of the last input dimension.
Note that for :attr:`offset` other than :math:`0`, the order of :attr:`dim1`
and :attr:`dim2` matters. Exchanging them is equivalent to changing the
sign of :attr:`offset`.

Applying :meth:`torch.diagonal` to the output of this function with
the same arguments yields a matrix identical to input. However,
:meth:`torch.diagonal` has different default dimensions, so those
need to be explicitly specified.

Args:
    {input} Must be at least 1-dimensional.
    offset (int, optional): which diagonal to consider. Default: 0
        (main diagonal).
    dim1 (int, optional): first dimension with respect to which to
        take diagonal. Default: -2.
    dim2 (int, optional): second dimension with respect to which to
        take diagonal. Default: -1.

Example::

    >>> a = torch.randn(2, 3)
    >>> torch.diag_embed(a)
    tensor([[[ 1.5410,  0.0000,  0.0000],
             [ 0.0000, -0.2934,  0.0000],
             [ 0.0000,  0.0000, -2.1788]],

            [[ 0.5684,  0.0000,  0.0000],
             [ 0.0000, -1.0845,  0.0000],
             [ 0.0000,  0.0000, -1.3986]]])

    >>> torch.diag_embed(a, offset=1, dim1=0, dim2=2)
    tensor([[[ 0.0000,  1.5410,  0.0000,  0.0000],
             [ 0.0000,  0.5684,  0.0000,  0.0000]],

            [[ 0.0000,  0.0000, -0.2934,  0.0000],
             [ 0.0000,  0.0000, -1.0845,  0.0000]],

            [[ 0.0000,  0.0000,  0.0000, -2.1788],
             [ 0.0000,  0.0000,  0.0000, -1.3986]],

            [[ 0.0000,  0.0000,  0.0000,  0.0000],
             [ 0.0000,  0.0000,  0.0000,  0.0000]]])
""".format(**common_args))


add_docstr(torch.diagflat,
           r"""
diagflat(input, offset=0) -> Tensor

- If :attr:`input` is a vector (1-D tensor), then returns a 2-D square tensor
  with the elements of :attr:`input` as the diagonal.
- If :attr:`input` is a tensor with more than one dimension, then returns a
  2-D tensor with diagonal elements equal to a flattened :attr:`input`.

The argument :attr:`offset` controls which diagonal to consider:

- If :attr:`offset` = 0, it is the main diagonal.
- If :attr:`offset` > 0, it is above the main diagonal.
- If :attr:`offset` < 0, it is below the main diagonal.

Args:
    {input}
    offset (int, optional): the diagonal to consider. Default: 0 (main
        diagonal).

Examples::

    >>> a = torch.randn(3)
    >>> a
    tensor([-0.2956, -0.9068,  0.1695])
    >>> torch.diagflat(a)
    tensor([[-0.2956,  0.0000,  0.0000],
            [ 0.0000, -0.9068,  0.0000],
            [ 0.0000,  0.0000,  0.1695]])
    >>> torch.diagflat(a, 1)
    tensor([[ 0.0000, -0.2956,  0.0000,  0.0000],
            [ 0.0000,  0.0000, -0.9068,  0.0000],
            [ 0.0000,  0.0000,  0.0000,  0.1695],
            [ 0.0000,  0.0000,  0.0000,  0.0000]])

    >>> a = torch.randn(2, 2)
    >>> a
    tensor([[ 0.2094, -0.3018],
            [-0.1516,  1.9342]])
    >>> torch.diagflat(a)
    tensor([[ 0.2094,  0.0000,  0.0000,  0.0000],
            [ 0.0000, -0.3018,  0.0000,  0.0000],
            [ 0.0000,  0.0000, -0.1516,  0.0000],
            [ 0.0000,  0.0000,  0.0000,  1.9342]])
""".format(**common_args))

add_docstr(torch.diagonal,
           r"""
diagonal(input, offset=0, dim1=0, dim2=1) -> Tensor

Returns a partial view of :attr:`input` with the its diagonal elements
with respect to :attr:`dim1` and :attr:`dim2` appended as a dimension
at the end of the shape.

The argument :attr:`offset` controls which diagonal to consider:

- If :attr:`offset` = 0, it is the main diagonal.
- If :attr:`offset` > 0, it is above the main diagonal.
- If :attr:`offset` < 0, it is below the main diagonal.

Applying :meth:`torch.diag_embed` to the output of this function with
the same arguments yields a diagonal matrix with the diagonal entries
of the input. However, :meth:`torch.diag_embed` has different default
dimensions, so those need to be explicitly specified.

Args:
    {input} Must be at least 2-dimensional.
    offset (int, optional): which diagonal to consider. Default: 0
        (main diagonal).
    dim1 (int, optional): first dimension with respect to which to
        take diagonal. Default: 0.
    dim2 (int, optional): second dimension with respect to which to
        take diagonal. Default: 1.

.. note::  To take a batch diagonal, pass in dim1=-2, dim2=-1.

Examples::

    >>> a = torch.randn(3, 3)
    >>> a
    tensor([[-1.0854,  1.1431, -0.1752],
            [ 0.8536, -0.0905,  0.0360],
            [ 0.6927, -0.3735, -0.4945]])


    >>> torch.diagonal(a, 0)
    tensor([-1.0854, -0.0905, -0.4945])


    >>> torch.diagonal(a, 1)
    tensor([ 1.1431,  0.0360])


    >>> x = torch.randn(2, 5, 4, 2)
    >>> torch.diagonal(x, offset=-1, dim1=1, dim2=2)
    tensor([[[-1.2631,  0.3755, -1.5977, -1.8172],
             [-1.1065,  1.0401, -0.2235, -0.7938]],

            [[-1.7325, -0.3081,  0.6166,  0.2335],
             [ 1.0500,  0.7336, -0.3836, -1.1015]]])
""".format(**common_args))

add_docstr(torch.digamma,
           r"""
digamma(input, *, out=None) -> Tensor

Computes the logarithmic derivative of the gamma function on `input`.

.. math::
    \psi(x) = \frac{d}{dx} \ln\left(\Gamma\left(x\right)\right) = \frac{\Gamma'(x)}{\Gamma(x)}
""" + r"""
Args:
    input (Tensor): the tensor to compute the digamma function on

Keyword args:
    {out}

Example::

    >>> a = torch.tensor([1, 0.5])
    >>> torch.digamma(a)
    tensor([-0.5772, -1.9635])
""".format(**common_args))


add_docstr(torch.dist,
           r"""
dist(input, other, p=2) -> Tensor

Returns the p-norm of (:attr:`input` - :attr:`other`)

The shapes of :attr:`input` and :attr:`other` must be
:ref:`broadcastable <broadcasting-semantics>`.

Args:
    {input}
    other (Tensor): the Right-hand-side input tensor
    p (float, optional): the norm to be computed

Example::

    >>> x = torch.randn(4)
    >>> x
    tensor([-1.5393, -0.8675,  0.5916,  1.6321])
    >>> y = torch.randn(4)
    >>> y
    tensor([ 0.0967, -1.0511,  0.6295,  0.8360])
    >>> torch.dist(x, y, 3.5)
    tensor(1.6727)
    >>> torch.dist(x, y, 3)
    tensor(1.6973)
    >>> torch.dist(x, y, 0)
    tensor(inf)
    >>> torch.dist(x, y, 1)
    tensor(2.6537)
""".format(**common_args))

add_docstr(torch.div, r"""
div(input, other, *, out=None) -> Tensor

Divides each element of the input ``input`` by the corresponding element of
:attr:`other`.

.. math::
    \text{{out}}_i = \frac{{\text{{input}}_i}}{{\text{{other}}_i}}

.. note::
    Performs a "true" division like Python 3. See :func:`torch.floor_divide`
    for floor division.

Supports :ref:`broadcasting to a common shape <broadcasting-semantics>`,
:ref:`type promotion <type-promotion-doc>`, and integer, float, and complex inputs.
Always promotes integer types to the default scalar type.

Args:
    input (Tensor): the dividend
    other (Tensor or Number): the divisor

Keyword args:
    {out}

Examples::

    >>> a = torch.randn(5)
    >>> a
    tensor([ 0.3810,  1.2774, -0.2972, -0.3719,  0.4637])
    >>> torch.div(a, 0.5)
    tensor([ 0.7620,  2.5548, -0.5944, -0.7439,  0.9275])
    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[-0.3711, -1.9353, -0.4605, -0.2917],
            [ 0.1815, -1.0111,  0.9805, -1.5923],
            [ 0.1062,  1.4581,  0.7759, -1.2344],
            [-0.1830, -0.0313,  1.1908, -1.4757]])
    >>> b = torch.randn(4)
    >>> b
    tensor([ 0.8032,  0.2930, -0.8113, -0.2308])
    >>> torch.div(a, b)
    tensor([[-0.4620, -6.6051,  0.5676,  1.2637],
            [ 0.2260, -3.4507, -1.2086,  6.8988],
            [ 0.1322,  4.9764, -0.9564,  5.3480],
            [-0.2278, -0.1068, -1.4678,  6.3936]])
""".format(**common_args))

add_docstr(torch.divide, r"""
divide(input, other, *, out=None) -> Tensor

Alias for :func:`torch.div`.
""")

add_docstr(torch.dot,
           r"""
dot(input, tensor) -> Tensor

Computes the dot product (inner product) of two tensors.

.. note:: This function does not :ref:`broadcast <broadcasting-semantics>`.

Example::

    >>> torch.dot(torch.tensor([2, 3]), torch.tensor([2, 1]))
    tensor(7)
""")

add_docstr(torch.vdot,
           r"""
vdot(input, other, *, out=None) -> Tensor

Computes the dot product (inner product) of two tensors. The vdot(a, b) function
handles complex numbers differently than dot(a, b). If the first argument is complex,
the complex conjugate of the first argument is used for the calculation of the dot product.

.. note:: This function does not :ref:`broadcast <broadcasting-semantics>`.

Args:
    input (Tensor): first tensor in the dot product. Its conjugate is used if it's complex.
    other (Tensor): second tensor in the dot product.

Keyword args:
    {out}

Example::

    >>> torch.vdot(torch.tensor([2, 3]), torch.tensor([2, 1]))
    tensor(7)
    >>> a = torch.tensor((1 +2j, 3 - 1j))
    >>> b = torch.tensor((2 +1j, 4 - 0j))
    >>> torch.vdot(a, b)
    tensor([16.+1.j])
    >>> torch.vdot(b, a)
    tensor([16.-1.j])
""".format(**common_args))

add_docstr(torch.eig,
           r"""
eig(input, eigenvectors=False, *, out=None) -> (Tensor, Tensor)

Computes the eigenvalues and eigenvectors of a real square matrix.

.. note::
    Since eigenvalues and eigenvectors might be complex, backward pass is supported only
    for :func:`torch.symeig`

Args:
    input (Tensor): the square matrix of shape :math:`(n \times n)` for which the eigenvalues and eigenvectors
        will be computed
    eigenvectors (bool): ``True`` to compute both eigenvalues and eigenvectors;
        otherwise, only eigenvalues will be computed

Keyword args:
    out (tuple, optional): the output tensors

Returns:
    (Tensor, Tensor): A namedtuple (eigenvalues, eigenvectors) containing

        - **eigenvalues** (*Tensor*): Shape :math:`(n \times 2)`. Each row is an eigenvalue of ``input``,
          where the first element is the real part and the second element is the imaginary part.
          The eigenvalues are not necessarily ordered.
        - **eigenvectors** (*Tensor*): If ``eigenvectors=False``, it's an empty tensor.
          Otherwise, this tensor of shape :math:`(n \times n)` can be used to compute normalized (unit length)
          eigenvectors of corresponding eigenvalues as follows.
          If the corresponding `eigenvalues[j]` is a real number, column `eigenvectors[:, j]` is the eigenvector
          corresponding to `eigenvalues[j]`.
          If the corresponding `eigenvalues[j]` and `eigenvalues[j + 1]` form a complex conjugate pair, then the
          true eigenvectors can be computed as
          :math:`\text{true eigenvector}[j] = eigenvectors[:, j] + i \times eigenvectors[:, j + 1]`,
          :math:`\text{true eigenvector}[j + 1] = eigenvectors[:, j] - i \times eigenvectors[:, j + 1]`.
""")

add_docstr(torch.eq, r"""
eq(input, other, *, out=None) -> Tensor

Computes element-wise equality

The second argument can be a number or a tensor whose shape is
:ref:`broadcastable <broadcasting-semantics>` with the first argument.

Args:
    input (Tensor): the tensor to compare
    other (Tensor or float): the tensor or value to compare

Keyword args:
    {out}

Returns:
    A boolean tensor that is True where :attr:`input` is equal to :attr:`other` and False elsewhere

Example::

    >>> torch.eq(torch.tensor([[1, 2], [3, 4]]), torch.tensor([[1, 1], [4, 4]]))
    tensor([[ True, False],
            [False, True]])
""".format(**common_args))

add_docstr(torch.equal,
           r"""
equal(input, other) -> bool

``True`` if two tensors have the same size and elements, ``False`` otherwise.

Example::

    >>> torch.equal(torch.tensor([1, 2]), torch.tensor([1, 2]))
    True
""")

add_docstr(torch.erf,
           r"""
erf(input, *, out=None) -> Tensor

Computes the error function of each element. The error function is defined as follows:

.. math::
    \mathrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.erf(torch.tensor([0, -1., 10.]))
    tensor([ 0.0000, -0.8427,  1.0000])
""".format(**common_args))

add_docstr(torch.erfc,
           r"""
erfc(input, *, out=None) -> Tensor

Computes the complementary error function of each element of :attr:`input`.
The complementary error function is defined as follows:

.. math::
    \mathrm{erfc}(x) = 1 - \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.erfc(torch.tensor([0, -1., 10.]))
    tensor([ 1.0000, 1.8427,  0.0000])
""".format(**common_args))

add_docstr(torch.erfinv,
           r"""
erfinv(input, *, out=None) -> Tensor

Computes the inverse error function of each element of :attr:`input`.
The inverse error function is defined in the range :math:`(-1, 1)` as:

.. math::
    \mathrm{erfinv}(\mathrm{erf}(x)) = x
""" + r"""

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.erfinv(torch.tensor([0, 0.5, -1.]))
    tensor([ 0.0000,  0.4769,    -inf])
""".format(**common_args))

add_docstr(torch.exp,
           r"""
exp(input, *, out=None) -> Tensor

Returns a new tensor with the exponential of the elements
of the input tensor :attr:`input`.

.. math::
    y_{i} = e^{x_{i}}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.exp(torch.tensor([0, math.log(2.)]))
    tensor([ 1.,  2.])
""".format(**common_args))

add_docstr(torch.exp2,
           r"""
exp2(input, *, out=None) -> Tensor

Computes the base two exponential function of :attr:`input`.

.. math::
    y_{i} = 2^{x_{i}}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.exp2(torch.tensor([0, math.log2(2.), 3, 4]))
    tensor([ 1.,  2.,  8., 16.])
""".format(**common_args))

add_docstr(torch.expm1,
           r"""
expm1(input, *, out=None) -> Tensor

Returns a new tensor with the exponential of the elements minus 1
of :attr:`input`.

.. math::
    y_{i} = e^{x_{i}} - 1
""" + r"""

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.expm1(torch.tensor([0, math.log(2.)]))
    tensor([ 0.,  1.])
""".format(**common_args))

# TODO: see https://github.com/pytorch/pytorch/issues/43667
add_docstr(torch.eye,
           r"""
eye(n, m=None, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a 2-D tensor with ones on the diagonal and zeros elsewhere.

Args:
    n (int): the number of rows
    m (int, optional): the number of columns with default being :attr:`n`
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Returns:
    Tensor: A 2-D tensor with ones on the diagonal and zeros elsewhere

Example::

    >>> torch.eye(3)
    tensor([[ 1.,  0.,  0.],
            [ 0.,  1.,  0.],
            [ 0.,  0.,  1.]])
""".format(**factory_common_args))

add_docstr(torch.floor,
           r"""
floor(input, *, out=None) -> Tensor

Returns a new tensor with the floor of the elements of :attr:`input`,
the largest integer less than or equal to each element.

.. math::
    \text{out}_{i} = \left\lfloor \text{input}_{i} \right\rfloor
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.8166,  1.5308, -0.2530, -0.2091])
    >>> torch.floor(a)
    tensor([-1.,  1., -1., -1.])
""".format(**common_args))

add_docstr(torch.floor_divide, r"""
floor_divide(input, other, *, out=None) -> Tensor

.. warning::
    This function's name is a misnomer. It actually rounds the
    quotient towards zero instead of taking its floor. This behavior
    will be deprecated in a future PyTorch release.

Computes :attr:`input` divided by :attr:`other`, elementwise, and rounds each
quotient towards zero. Equivalently, it truncates the quotient(s):

.. math::
    \text{{out}}_i = \text{trunc} \left( \frac{{\text{{input}}_i}}{{\text{{other}}_i}} \right)

""" + r"""

Supports broadcasting to a common shape, type promotion, and integer and float inputs.

Args:
    input (Tensor or Number): the dividend
    other (Tensor or Number): the divisor

Keyword args:
    {out}

Example::

    >>> a = torch.tensor([4.0, 3.0])
    >>> b = torch.tensor([2.0, 2.0])
    >>> torch.floor_divide(a, b)
    tensor([2.0, 1.0])
    >>> torch.floor_divide(a, 1.4)
    tensor([2.0, 2.0])
""".format(**common_args))

add_docstr(torch.fmod,
           r"""
fmod(input, other, *, out=None) -> Tensor

Computes the element-wise remainder of division.

The dividend and divisor may contain both for integer and floating point
numbers. The remainder has the same sign as the dividend :attr:`input`.

When :attr:`other` is a tensor, the shapes of :attr:`input` and
:attr:`other` must be :ref:`broadcastable <broadcasting-semantics>`.

Args:
    input (Tensor): the dividend
    other (Tensor or float): the divisor, which may be either a number or a tensor of the same shape as the dividend

Keyword args:
    {out}

Example::

    >>> torch.fmod(torch.tensor([-3., -2, -1, 1, 2, 3]), 2)
    tensor([-1., -0., -1.,  1.,  0.,  1.])
    >>> torch.fmod(torch.tensor([1., 2, 3, 4, 5]), 1.5)
    tensor([ 1.0000,  0.5000,  0.0000,  1.0000,  0.5000])


""".format(**common_args))

add_docstr(torch.frac,
           r"""
frac(input, *, out=None) -> Tensor

Computes the fractional portion of each element in :attr:`input`.

.. math::
    \text{out}_{i} = \text{input}_{i} - \left\lfloor |\text{input}_{i}| \right\rfloor * \operatorname{sgn}(\text{input}_{i})

Example::

    >>> torch.frac(torch.tensor([1, 2.5, -3.2]))
    tensor([ 0.0000,  0.5000, -0.2000])
""")

add_docstr(torch.from_numpy,
           r"""
from_numpy(ndarray) -> Tensor

Creates a :class:`Tensor` from a :class:`numpy.ndarray`.

The returned tensor and :attr:`ndarray` share the same memory. Modifications to
the tensor will be reflected in the :attr:`ndarray` and vice versa. The returned
tensor is not resizable.

It currently accepts :attr:`ndarray` with dtypes of ``numpy.float64``,
``numpy.float32``, ``numpy.float16``, ``numpy.complex64``, ``numpy.complex128``,
``numpy.int64``, ``numpy.int32``, ``numpy.int16``, ``numpy.int8``, ``numpy.uint8``,
and ``numpy.bool``.

Example::

    >>> a = numpy.array([1, 2, 3])
    >>> t = torch.from_numpy(a)
    >>> t
    tensor([ 1,  2,  3])
    >>> t[0] = -1
    >>> a
    array([-1,  2,  3])
""")

add_docstr(torch.flatten,
           r"""
flatten(input, start_dim=0, end_dim=-1) -> Tensor

Flattens a contiguous range of dims in a tensor.

Args:
    {input}
    start_dim (int): the first dim to flatten
    end_dim (int): the last dim to flatten

Example::

    >>> t = torch.tensor([[[1, 2],
                           [3, 4]],
                          [[5, 6],
                           [7, 8]]])
    >>> torch.flatten(t)
    tensor([1, 2, 3, 4, 5, 6, 7, 8])
    >>> torch.flatten(t, start_dim=1)
    tensor([[1, 2, 3, 4],
            [5, 6, 7, 8]])
""".format(**common_args))

# TODO: see https://github.com/pytorch/pytorch/issues/43667
add_docstr(torch.gather,
           r"""
gather(input, dim, index, *, sparse_grad=False, out=None) -> Tensor

Gathers values along an axis specified by `dim`.

For a 3-D tensor the output is specified by::

    out[i][j][k] = input[index[i][j][k]][j][k]  # if dim == 0
    out[i][j][k] = input[i][index[i][j][k]][k]  # if dim == 1
    out[i][j][k] = input[i][j][index[i][j][k]]  # if dim == 2

If :attr:`input` is an n-dimensional tensor with size
:math:`(x_0, x_1..., x_{i-1}, x_i, x_{i+1}, ..., x_{n-1})`
and ``dim = i``, then :attr:`index` must be an :math:`n`-dimensional tensor with
size :math:`(x_0, x_1, ..., x_{i-1}, y, x_{i+1}, ..., x_{n-1})` where :math:`y \geq 1`
and :attr:`out` will have the same size as :attr:`index`.
""" + r"""
Args:
    input (Tensor): the source tensor
    dim (int): the axis along which to index
    index (LongTensor): the indices of elements to gather
    sparse_grad(bool,optional): If ``True``, gradient w.r.t. :attr:`input` will be a sparse tensor.
    out (Tensor, optional): the destination tensor

Example::

    >>> t = torch.tensor([[1,2],[3,4]])
    >>> torch.gather(t, 1, torch.tensor([[0,0],[1,0]]))
    tensor([[ 1,  1],
            [ 4,  3]])
""")


add_docstr(torch.gcd,
           r"""
gcd(input, other, *, out=None) -> Tensor

Computes the element-wise greatest common divisor (GCD) of :attr:`input` and :attr:`other`.

Both :attr:`input` and :attr:`other` must have integer types.

.. note::
    This defines :math:`gcd(0, 0) = 0`.

Args:
    {input}
    other (Tensor): the second input tensor

Keyword arguments:
    {out}

Example::

    >>> a = torch.tensor([5, 10, 15])
    >>> b = torch.tensor([3, 4, 5])
    >>> torch.gcd(a, b)
    tensor([1, 2, 5])
    >>> c = torch.tensor([3])
    >>> torch.gcd(a, c)
    tensor([1, 1, 3])
""".format(**common_args))

add_docstr(torch.ge, r"""
ge(input, other, *, out=None) -> Tensor

Computes :math:`\text{input} \geq \text{other}` element-wise.
""" + r"""

The second argument can be a number or a tensor whose shape is
:ref:`broadcastable <broadcasting-semantics>` with the first argument.

Args:
    input (Tensor): the tensor to compare
    other (Tensor or float): the tensor or value to compare

Keyword args:
    {out}

Returns:
    A boolean tensor that is True where :attr:`input` is greater than or equal to :attr:`other` and False elsewhere

Example::

    >>> torch.ge(torch.tensor([[1, 2], [3, 4]]), torch.tensor([[1, 1], [4, 4]]))
    tensor([[True, True], [False, True]])
""".format(**common_args))

add_docstr(torch.greater_equal, r"""
greater_equal(input, other, *, out=None) -> Tensor

Alias for :func:`torch.ge`.
""")

add_docstr(torch.geqrf,
           r"""
geqrf(input, *, out=None) -> (Tensor, Tensor)

This is a low-level function for calling LAPACK directly. This function
returns a namedtuple (a, tau) as defined in `LAPACK documentation for geqrf`_ .

You'll generally want to use :func:`torch.qr` instead.

Computes a QR decomposition of :attr:`input`, but without constructing
:math:`Q` and :math:`R` as explicit separate matrices.

Rather, this directly calls the underlying LAPACK function `?geqrf`
which produces a sequence of 'elementary reflectors'.

See `LAPACK documentation for geqrf`_ for further details.

Args:
    input (Tensor): the input matrix

Keyword args:
    out (tuple, optional): the output tuple of (Tensor, Tensor)

.. _LAPACK documentation for geqrf:
    https://software.intel.com/en-us/node/521004

""")

add_docstr(torch.outer, r"""
outer(input, vec2, *, out=None) -> Tensor

Outer product of :attr:`input` and :attr:`vec2`.
If :attr:`input` is a vector of size :math:`n` and :attr:`vec2` is a vector of
size :math:`m`, then :attr:`out` must be a matrix of size :math:`(n \times m)`.

.. note:: This function does not :ref:`broadcast <broadcasting-semantics>`.

Args:
    input (Tensor): 1-D input vector
    vec2 (Tensor): 1-D input vector

Keyword args:
    out (Tensor, optional): optional output matrix

Example::

    >>> v1 = torch.arange(1., 5.)
    >>> v2 = torch.arange(1., 4.)
    >>> torch.outer(v1, v2)
    tensor([[  1.,   2.,   3.],
            [  2.,   4.,   6.],
            [  3.,   6.,   9.],
            [  4.,   8.,  12.]])
""")

add_docstr(torch.ger,
           r"""
ger(input, vec2, *, out=None) -> Tensor

Alias of :func:`torch.outer`.

.. warning::
    This function is deprecated and will be removed in a future PyTorch release.
    Use :func:`torch.outer` instead.
""")

add_docstr(torch.solve,
           r"""
torch.solve(input, A, *, out=None) -> (Tensor, Tensor)

This function returns the solution to the system of linear
equations represented by :math:`AX = B` and the LU factorization of
A, in order as a namedtuple `solution, LU`.

`LU` contains `L` and `U` factors for LU factorization of `A`.

`torch.solve(B, A)` can take in 2D inputs `B, A` or inputs that are
batches of 2D matrices. If the inputs are batches, then returns
batched outputs `solution, LU`.

.. note::

    Irrespective of the original strides, the returned matrices
    `solution` and `LU` will be transposed, i.e. with strides like
    `B.contiguous().transpose(-1, -2).stride()` and
    `A.contiguous().transpose(-1, -2).stride()` respectively.

Args:
    input (Tensor): input matrix :math:`B` of size :math:`(*, m, k)` , where :math:`*`
                is zero or more batch dimensions.
    A (Tensor): input square matrix of size :math:`(*, m, m)`, where
                :math:`*` is zero or more batch dimensions.

Keyword args:
    out ((Tensor, Tensor), optional): optional output tuple.

Example::

    >>> A = torch.tensor([[6.80, -2.11,  5.66,  5.97,  8.23],
                          [-6.05, -3.30,  5.36, -4.44,  1.08],
                          [-0.45,  2.58, -2.70,  0.27,  9.04],
                          [8.32,  2.71,  4.35,  -7.17,  2.14],
                          [-9.67, -5.14, -7.26,  6.08, -6.87]]).t()
    >>> B = torch.tensor([[4.02,  6.19, -8.22, -7.57, -3.03],
                          [-1.56,  4.00, -8.67,  1.75,  2.86],
                          [9.81, -4.09, -4.57, -8.61,  8.99]]).t()
    >>> X, LU = torch.solve(B, A)
    >>> torch.dist(B, torch.mm(A, X))
    tensor(1.00000e-06 *
           7.0977)

    >>> # Batched solver example
    >>> A = torch.randn(2, 3, 1, 4, 4)
    >>> B = torch.randn(2, 3, 1, 4, 6)
    >>> X, LU = torch.solve(B, A)
    >>> torch.dist(B, A.matmul(X))
    tensor(1.00000e-06 *
       3.6386)

""")

add_docstr(torch.get_default_dtype,
           r"""
get_default_dtype() -> torch.dtype

Get the current default floating point :class:`torch.dtype`.

Example::

    >>> torch.get_default_dtype()  # initial default for floating point is torch.float32
    torch.float32
    >>> torch.set_default_dtype(torch.float64)
    >>> torch.get_default_dtype()  # default is now changed to torch.float64
    torch.float64
    >>> torch.set_default_tensor_type(torch.FloatTensor)  # setting tensor type also affects this
    >>> torch.get_default_dtype()  # changed to torch.float32, the dtype for torch.FloatTensor
    torch.float32

""")

add_docstr(torch.get_num_threads,
           r"""
get_num_threads() -> int

Returns the number of threads used for parallelizing CPU operations
""")

add_docstr(torch.get_num_interop_threads,
           r"""
get_num_interop_threads() -> int

Returns the number of threads used for inter-op parallelism on CPU
(e.g. in JIT interpreter)
""")

add_docstr(torch.gt, r"""
gt(input, other, *, out=None) -> Tensor

Computes :math:`\text{input} > \text{other}` element-wise.
""" + r"""

The second argument can be a number or a tensor whose shape is
:ref:`broadcastable <broadcasting-semantics>` with the first argument.

Args:
    input (Tensor): the tensor to compare
    other (Tensor or float): the tensor or value to compare

Keyword args:
    {out}

Returns:
    A boolean tensor that is True where :attr:`input` is greater than :attr:`other` and False elsewhere

Example::

    >>> torch.gt(torch.tensor([[1, 2], [3, 4]]), torch.tensor([[1, 1], [4, 4]]))
    tensor([[False, True], [False, False]])
""".format(**common_args))

add_docstr(torch.greater, r"""
greater(input, other, *, out=None) -> Tensor

Alias for :func:`torch.gt`.
""")

add_docstr(torch.histc,
           r"""
histc(input, bins=100, min=0, max=0, *, out=None) -> Tensor

Computes the histogram of a tensor.

The elements are sorted into equal width bins between :attr:`min` and
:attr:`max`. If :attr:`min` and :attr:`max` are both zero, the minimum and
maximum values of the data are used.

Elements lower than min and higher than max are ignored.

Args:
    {input}
    bins (int): number of histogram bins
    min (int): lower end of the range (inclusive)
    max (int): upper end of the range (inclusive)

Keyword args:
    {out}

Returns:
    Tensor: Histogram represented as a tensor

Example::

    >>> torch.histc(torch.tensor([1., 2, 1]), bins=4, min=0, max=3)
    tensor([ 0.,  2.,  1.,  0.])
""".format(**common_args))

add_docstr(torch.hypot,
           r"""
hypot(input, other, *, out=None) -> Tensor

Given the legs of a right triangle, return its hypotenuse.

.. math::
    \text{out}_{i} = \sqrt{\text{input}_{i}^{2} + \text{other}_{i}^{2}}

The shapes of ``input`` and ``other`` must be
:ref:`broadcastable <broadcasting-semantics>`.
""" + r"""
Args:
    input (Tensor): the first input tensor
    other (Tensor): the second input tensor

Keyword args:
    {out}

Example::

    >>> a = torch.hypot(torch.tensor([4.0]), torch.tensor([3.0, 4.0, 5.0]))
    tensor([5.0000, 5.6569, 6.4031])

""".format(**common_args))

add_docstr(torch.i0,
           r"""
i0(input, *, out=None) -> Tensor

Computes the zeroth order modified Bessel function of the first kind for each element of :attr:`input`.

.. math::
    \text{out}_{i} = I_0(\text{input}_{i}) = \sum_{k=0}^{\infty} \frac{(\text{input}_{i}^2/4)^k}{(k!)^2}

""" + r"""
Args:
    input (Tensor): the input tensor

Keyword args:
    {out}

Example::

    >>> torch.i0(torch.arange(5, dtype=torch.float32))
    tensor([ 1.0000,  1.2661,  2.2796,  4.8808, 11.3019])

""".format(**common_args))

add_docstr(torch.igamma,
           r"""
igamma(input, other, *, out=None) -> Tensor

Computes the regularized lower incomplete gamma function:

.. math::
    \text{out}_{i} = \frac{1}{\Gamma(\text{input}_i)} \int_0^{\text{other}_i} t^{\text{input}_i-1} e^{-t} dt

where both :math:`\text{input}_i` and :math:`\text{other}_i` are weakly positive
and at least one is strictly positive.
If both are zero or either is negative then :math:`\text{out}_i=\text{nan}`.
:math:`\Gamma(\cdot)` in the equation above is the gamma function,

.. math::
    \Gamma(\text{input}_i) = \int_0^\infty t^{(\text{input}_i-1)} e^{-t} dt.

See :func:`torch.lgamma` for a related function.

Supports :ref:`broadcasting to a common shape <broadcasting-semantics>`
and float inputs.

.. note::
    The backward pass with respect to :attr:`input` is not yet supported.
    Please open an issue on PyTorch's Github to request it.

""" + r"""
Args:
    input (Tensor): the first non-negative input tensor
    other (Tensor): the second non-negative input tensor

Keyword args:
    {out}

Example::

    >>> a = torch.igamma(torch.tensor([4.0]), torch.tensor([3.0, 4.0, 5.0]))
    tensor([0.3528, 0.5665, 0.7350])

""".format(**common_args))

add_docstr(torch.index_select,
           r"""
index_select(input, dim, index, *, out=None) -> Tensor

Returns a new tensor which indexes the :attr:`input` tensor along dimension
:attr:`dim` using the entries in :attr:`index` which is a `LongTensor`.

The returned tensor has the same number of dimensions as the original tensor
(:attr:`input`).  The :attr:`dim`\ th dimension has the same size as the length
of :attr:`index`; other dimensions have the same size as in the original tensor.

.. note:: The returned tensor does **not** use the same storage as the original
          tensor.  If :attr:`out` has a different shape than expected, we
          silently change it to the correct shape, reallocating the underlying
          storage if necessary.

Args:
    {input}
    dim (int): the dimension in which we index
    index (LongTensor): the 1-D tensor containing the indices to index

Keyword args:
    {out}

Example::

    >>> x = torch.randn(3, 4)
    >>> x
    tensor([[ 0.1427,  0.0231, -0.5414, -1.0009],
            [-0.4664,  0.2647, -0.1228, -1.1068],
            [-1.1734, -0.6571,  0.7230, -0.6004]])
    >>> indices = torch.tensor([0, 2])
    >>> torch.index_select(x, 0, indices)
    tensor([[ 0.1427,  0.0231, -0.5414, -1.0009],
            [-1.1734, -0.6571,  0.7230, -0.6004]])
    >>> torch.index_select(x, 1, indices)
    tensor([[ 0.1427, -0.5414],
            [-0.4664, -0.1228],
            [-1.1734,  0.7230]])
""".format(**common_args))

add_docstr(torch.inverse,
           r"""
inverse(input, *, out=None) -> Tensor

Takes the inverse of the square matrix :attr:`input`. :attr:`input` can be batches
of 2D square tensors, in which case this function would return a tensor composed of
individual inverses.

.. note::

    Irrespective of the original strides, the returned tensors will be
    transposed, i.e. with strides like `input.contiguous().transpose(-2, -1).stride()`

Args:
    input (Tensor): the input tensor of size :math:`(*, n, n)` where `*` is zero or more
                    batch dimensions

Keyword args:
    {out}

Example::

    >>> x = torch.rand(4, 4)
    >>> y = torch.inverse(x)
    >>> z = torch.mm(x, y)
    >>> z
    tensor([[ 1.0000, -0.0000, -0.0000,  0.0000],
            [ 0.0000,  1.0000,  0.0000,  0.0000],
            [ 0.0000,  0.0000,  1.0000,  0.0000],
            [ 0.0000, -0.0000, -0.0000,  1.0000]])
    >>> torch.max(torch.abs(z - torch.eye(4))) # Max non-zero
    tensor(1.1921e-07)
    >>> # Batched inverse example
    >>> x = torch.randn(2, 3, 4, 4)
    >>> y = torch.inverse(x)
    >>> z = torch.matmul(x, y)
    >>> torch.max(torch.abs(z - torch.eye(4).expand_as(x))) # Max non-zero
    tensor(1.9073e-06)
""".format(**common_args))

add_docstr(torch.isinf, r"""
isinf(input) -> Tensor

Tests if each element of :attr:`input` is infinite
(positive or negative infinity) or not.

.. note::
    Complex values are infinite when their real or imaginary part is
    infinite.

    Arguments:
        {input}

    Returns:
        A boolean tensor that is True where :attr:`input` is infinite and False elsewhere

    Example::

        >>> torch.isinf(torch.tensor([1, float('inf'), 2, float('-inf'), float('nan')]))
        tensor([False,  True,  False,  True,  False])
""")

add_docstr(torch.isposinf,
           r"""
isposinf(input, *, out=None) -> Tensor
Tests if each element of :attr:`input` is positive infinity or not.

Args:
  {input}

Keyword args:
  {out}

Example::
    >>> a = torch.tensor([-float('inf'), float('inf'), 1.2])
    >>> torch.isposinf(a)
    tensor([False,  True, False])
""".format(**common_args))

add_docstr(torch.isneginf,
           r"""
isneginf(input, *, out=None) -> Tensor
Tests if each element of :attr:`input` is negative infinity or not.

Args:
  {input}

Keyword args:
  {out}

Example::
    >>> a = torch.tensor([-float('inf'), float('inf'), 1.2])
    >>> torch.isneginf(a)
    tensor([ True, False, False])
""".format(**common_args))

add_docstr(torch.isclose, r"""
isclose(input, other, rtol=1e-05, atol=1e-08, equal_nan=False) -> Tensor

Returns a new tensor with boolean elements representing if each element of
:attr:`input` is "close" to the corresponding element of :attr:`other`.
Closeness is defined as:

.. math::
    \lvert \text{input} - \text{other} \rvert \leq \texttt{atol} + \texttt{rtol} \times \lvert \text{other} \rvert
""" + r"""

where :attr:`input` and :attr:`other` are finite. Where :attr:`input`
and/or :attr:`other` are nonfinite they are close if and only if
they are equal, with NaNs being considered equal to each other when
:attr:`equal_nan` is True.

Args:
    input (Tensor): first tensor to compare
    other (Tensor): second tensor to compare
    atol (float, optional): absolute tolerance. Default: 1e-08
    rtol (float, optional): relative tolerance. Default: 1e-05
    equal_nan (bool, optional): if ``True``, then two ``NaN`` s will be considered equal. Default: ``False``

Examples::

    >>> torch.isclose(torch.tensor((1., 2, 3)), torch.tensor((1 + 1e-10, 3, 4)))
    tensor([ True, False, False])
    >>> torch.isclose(torch.tensor((float('inf'), 4)), torch.tensor((float('inf'), 6)), rtol=.5)
    tensor([True, True])
""")

add_docstr(torch.isfinite, r"""
isfinite(input) -> Tensor

Returns a new tensor with boolean elements representing if each element is `finite` or not.

Real values are finite when they are not NaN, negative infinity, or infinity.
Complex values are finite when both their real and imaginary parts are finite.

    Arguments:
        {input}

    Returns:
        A boolean tensor that is True where :attr:`input` is finite and False elsewhere

    Example::

        >>> torch.isfinite(torch.tensor([1, float('inf'), 2, float('-inf'), float('nan')]))
        tensor([True,  False,  True,  False,  False])
""".format(**common_args))

add_docstr(torch.isnan, r"""
isnan(input) -> Tensor

Returns a new tensor with boolean elements representing if each element of :attr:`input`
is NaN or not. Complex values are considered NaN when either their real
and/or imaginary part is NaN.

Arguments:
    {input}

Returns:
    A boolean tensor that is True where :attr:`input` is NaN and False elsewhere

Example::

    >>> torch.isnan(torch.tensor([1, float('nan'), 2]))
    tensor([False, True, False])
""".format(**common_args))

add_docstr(torch.isreal, r"""
isreal(input) -> Tensor

Returns a new tensor with boolean elements representing if each element of :attr:`input` is real-valued or not.
All real-valued types are considered real. Complex values are considered real when their imaginary part is 0.

Arguments:
    {input}

Returns:
    A boolean tensor that is True where :attr:`input` is real and False elsewhere

Example::

    >>> torch.isreal(torch.tensor([1, 1+1j, 2+0j]))
    tensor([True, False, True])
""".format(**common_args))

add_docstr(torch.is_floating_point, r"""
is_floating_point(input) -> (bool)

Returns True if the data type of :attr:`input` is a floating point data type i.e.,
one of ``torch.float64``, ``torch.float32`` and ``torch.float16``.

Args:
    {input}
""".format(**common_args))

add_docstr(torch.is_complex, r"""
is_complex(input) -> (bool)

Returns True if the data type of :attr:`input` is a complex data type i.e.,
one of ``torch.complex64``, and ``torch.complex128``.

Args:
    {input}
""".format(**common_args))

add_docstr(torch.is_nonzero, r"""
is_nonzero(input) -> (bool)

Returns True if the :attr:`input` is a single element tensor which is not equal to zero
after type conversions.
i.e. not equal to ``torch.tensor([0.])`` or ``torch.tensor([0])`` or
``torch.tensor([False])``.
Throws a ``RuntimeError`` if ``torch.numel() != 1`` (even in case
of sparse tensors).

Args:
    {input}

Examples::

    >>> torch.is_nonzero(torch.tensor([0.]))
    False
    >>> torch.is_nonzero(torch.tensor([1.5]))
    True
    >>> torch.is_nonzero(torch.tensor([False]))
    False
    >>> torch.is_nonzero(torch.tensor([3]))
    True
    >>> torch.is_nonzero(torch.tensor([1, 3, 5]))
    Traceback (most recent call last):
    ...
    RuntimeError: bool value of Tensor with more than one value is ambiguous
    >>> torch.is_nonzero(torch.tensor([]))
    Traceback (most recent call last):
    ...
    RuntimeError: bool value of Tensor with no values is ambiguous
""".format(**common_args))

add_docstr(torch.kthvalue,
           r"""
kthvalue(input, k, dim=None, keepdim=False, *, out=None) -> (Tensor, LongTensor)

Returns a namedtuple ``(values, indices)`` where ``values`` is the :attr:`k` th
smallest element of each row of the :attr:`input` tensor in the given dimension
:attr:`dim`. And ``indices`` is the index location of each element found.

If :attr:`dim` is not given, the last dimension of the `input` is chosen.

If :attr:`keepdim` is ``True``, both the :attr:`values` and :attr:`indices` tensors
are the same size as :attr:`input`, except in the dimension :attr:`dim` where
they are of size 1. Otherwise, :attr:`dim` is squeezed
(see :func:`torch.squeeze`), resulting in both the :attr:`values` and
:attr:`indices` tensors having 1 fewer dimension than the :attr:`input` tensor.

Args:
    {input}
    k (int): k for the k-th smallest element
    dim (int, optional): the dimension to find the kth value along
    {keepdim}

Keyword args:
    out (tuple, optional): the output tuple of (Tensor, LongTensor)
                           can be optionally given to be used as output buffers

Example::

    >>> x = torch.arange(1., 6.)
    >>> x
    tensor([ 1.,  2.,  3.,  4.,  5.])
    >>> torch.kthvalue(x, 4)
    torch.return_types.kthvalue(values=tensor(4.), indices=tensor(3))

    >>> x=torch.arange(1.,7.).resize_(2,3)
    >>> x
    tensor([[ 1.,  2.,  3.],
            [ 4.,  5.,  6.]])
    >>> torch.kthvalue(x, 2, 0, True)
    torch.return_types.kthvalue(values=tensor([[4., 5., 6.]]), indices=tensor([[1, 1, 1]]))
""".format(**single_dim_common))

add_docstr(torch.lcm,
           r"""
lcm(input, other, *, out=None) -> Tensor

Computes the element-wise least common multiple (LCM) of :attr:`input` and :attr:`other`.

Both :attr:`input` and :attr:`other` must have integer types.

.. note::
    This defines :math:`lcm(0, 0) = 0` and :math:`lcm(0, a) = 0`.

Args:
    {input}
    other (Tensor): the second input tensor

Keyword arguments:
    {out}

Example::

    >>> a = torch.tensor([5, 10, 15])
    >>> b = torch.tensor([3, 4, 5])
    >>> torch.lcm(a, b)
    tensor([15, 20, 15])
    >>> c = torch.tensor([3])
    >>> torch.lcm(a, c)
    tensor([15, 30, 15])
""".format(**common_args))

add_docstr(torch.le, r"""
le(input, other, *, out=None) -> Tensor

Computes :math:`\text{input} \leq \text{other}` element-wise.
""" + r"""

The second argument can be a number or a tensor whose shape is
:ref:`broadcastable <broadcasting-semantics>` with the first argument.

Args:
    input (Tensor): the tensor to compare
    other (Tensor or Scalar): the tensor or value to compare

Keyword args:
    {out}

Returns:
    A boolean tensor that is True where :attr:`input` is less than or equal to
    :attr:`other` and False elsewhere

Example::

    >>> torch.le(torch.tensor([[1, 2], [3, 4]]), torch.tensor([[1, 1], [4, 4]]))
    tensor([[True, False], [True, True]])
""".format(**common_args))

add_docstr(torch.less_equal, r"""
less_equal(input, other, *, out=None) -> Tensor

Alias for :func:`torch.le`.
""")

add_docstr(torch.lerp,
           r"""
lerp(input, end, weight, *, out=None)

Does a linear interpolation of two tensors :attr:`start` (given by :attr:`input`) and :attr:`end` based
on a scalar or tensor :attr:`weight` and returns the resulting :attr:`out` tensor.

.. math::
    \text{out}_i = \text{start}_i + \text{weight}_i \times (\text{end}_i - \text{start}_i)
""" + r"""
The shapes of :attr:`start` and :attr:`end` must be
:ref:`broadcastable <broadcasting-semantics>`. If :attr:`weight` is a tensor, then
the shapes of :attr:`weight`, :attr:`start`, and :attr:`end` must be :ref:`broadcastable <broadcasting-semantics>`.

Args:
    input (Tensor): the tensor with the starting points
    end (Tensor): the tensor with the ending points
    weight (float or tensor): the weight for the interpolation formula

Keyword args:
    {out}

Example::

    >>> start = torch.arange(1., 5.)
    >>> end = torch.empty(4).fill_(10)
    >>> start
    tensor([ 1.,  2.,  3.,  4.])
    >>> end
    tensor([ 10.,  10.,  10.,  10.])
    >>> torch.lerp(start, end, 0.5)
    tensor([ 5.5000,  6.0000,  6.5000,  7.0000])
    >>> torch.lerp(start, end, torch.full_like(start, 0.5))
    tensor([ 5.5000,  6.0000,  6.5000,  7.0000])
""".format(**common_args))

add_docstr(torch.lgamma,
           r"""
lgamma(input, *, out=None) -> Tensor

Computes the logarithm of the gamma function on :attr:`input`.

.. math::
    \text{out}_{i} = \log \Gamma(\text{input}_{i})
""" + """
Args:
    {input}
    {out}

Example::

    >>> a = torch.arange(0.5, 2, 0.5)
    >>> torch.lgamma(a)
    tensor([ 0.5724,  0.0000, -0.1208])
""".format(**common_args))

# TODO: update kwargs formatting (see https://github.com/pytorch/pytorch/issues/43667)
add_docstr(torch.linspace, r"""
linspace(start, end, steps, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Creates a one-dimensional tensor of size :attr:`steps` whose values are evenly
spaced from :attr:`start` to :attr:`end`, inclusive. That is, the value are:

.. math::
    (\text{start},
    \text{start} + \frac{\text{end} - \text{start}}{\text{steps} - 1},
    \ldots,
    \text{start} + (\text{steps} - 2) * \frac{\text{end} - \text{start}}{\text{steps} - 1},
    \text{end})
""" + """

.. warning::
    Not providing a value for :attr:`steps` is deprecated. For backwards
    compatibility, not providing a value for :attr:`steps` will create a tensor
    with 100 elements. Note that this behavior is not reflected in the
    documented function signature and should not be relied on. In a future
    PyTorch release, failing to provide a value for :attr:`steps` will throw a
    runtime error.

Args:
    start (float): the starting value for the set of points
    end (float): the ending value for the set of points
    steps (int): size of the constructed tensor
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}


Example::

    >>> torch.linspace(3, 10, steps=5)
    tensor([  3.0000,   4.7500,   6.5000,   8.2500,  10.0000])
    >>> torch.linspace(-10, 10, steps=5)
    tensor([-10.,  -5.,   0.,   5.,  10.])
    >>> torch.linspace(start=-10, end=10, steps=5)
    tensor([-10.,  -5.,   0.,   5.,  10.])
    >>> torch.linspace(start=-10, end=10, steps=1)
    tensor([-10.])
""".format(**factory_common_args))

add_docstr(torch.log,
           r"""
log(input, *, out=None) -> Tensor

Returns a new tensor with the natural logarithm of the elements
of :attr:`input`.

.. math::
    y_{i} = \log_{e} (x_{i})
""" + r"""

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(5)
    >>> a
    tensor([-0.7168, -0.5471, -0.8933, -1.4428, -0.1190])
    >>> torch.log(a)
    tensor([ nan,  nan,  nan,  nan,  nan])
""".format(**common_args))

add_docstr(torch.log10,
           r"""
log10(input, *, out=None) -> Tensor

Returns a new tensor with the logarithm to the base 10 of the elements
of :attr:`input`.

.. math::
    y_{i} = \log_{10} (x_{i})
""" + r"""

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.rand(5)
    >>> a
    tensor([ 0.5224,  0.9354,  0.7257,  0.1301,  0.2251])


    >>> torch.log10(a)
    tensor([-0.2820, -0.0290, -0.1392, -0.8857, -0.6476])

""".format(**common_args))

add_docstr(torch.log1p,
           r"""
log1p(input, *, out=None) -> Tensor

Returns a new tensor with the natural logarithm of (1 + :attr:`input`).

.. math::
    y_i = \log_{e} (x_i + 1)
""" + r"""
.. note:: This function is more accurate than :func:`torch.log` for small
          values of :attr:`input`

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(5)
    >>> a
    tensor([-1.0090, -0.9923,  1.0249, -0.5372,  0.2492])
    >>> torch.log1p(a)
    tensor([    nan, -4.8653,  0.7055, -0.7705,  0.2225])
""".format(**common_args))

add_docstr(torch.log2,
           r"""
log2(input, *, out=None) -> Tensor

Returns a new tensor with the logarithm to the base 2 of the elements
of :attr:`input`.

.. math::
    y_{i} = \log_{2} (x_{i})
""" + r"""

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.rand(5)
    >>> a
    tensor([ 0.8419,  0.8003,  0.9971,  0.5287,  0.0490])


    >>> torch.log2(a)
    tensor([-0.2483, -0.3213, -0.0042, -0.9196, -4.3504])

""".format(**common_args))

add_docstr(torch.logaddexp,
           r"""
logaddexp(input, other, *, out=None) -> Tensor

Logarithm of the sum of exponentiations of the inputs.

Calculates pointwise :math:`\log\left(e^x + e^y\right)`. This function is useful
in statistics where the calculated probabilities of events may be so small as to
exceed the range of normal floating point numbers. In such cases the logarithm
of the calculated probability is stored. This function allows adding
probabilities stored in such a fashion.

This op should be disambiguated with :func:`torch.logsumexp` which performs a
reduction on a single tensor.

Args:
    {input}
    other (Tensor): the second input tensor

Keyword arguments:
    {out}

Example::

    >>> torch.logaddexp(torch.tensor([-1.0]), torch.tensor([-1.0, -2, -3]))
    tensor([-0.3069, -0.6867, -0.8731])
    >>> torch.logaddexp(torch.tensor([-100.0, -200, -300]), torch.tensor([-1.0, -2, -3]))
    tensor([-1., -2., -3.])
    >>> torch.logaddexp(torch.tensor([1.0, 2000, 30000]), torch.tensor([-1.0, -2, -3]))
    tensor([1.1269e+00, 2.0000e+03, 3.0000e+04])
""".format(**common_args))

add_docstr(torch.logaddexp2,
           r"""
logaddexp2(input, other, *, out=None) -> Tensor

Logarithm of the sum of exponentiations of the inputs in base-2.

Calculates pointwise :math:`\log_2\left(2^x + 2^y\right)`. See
:func:`torch.logaddexp` for more details.

Args:
    {input}
    other (Tensor): the second input tensor

Keyword arguments:
    {out}
""".format(**common_args))

add_docstr(torch.logical_and,
           r"""
logical_and(input, other, *, out=None) -> Tensor

Computes the element-wise logical AND of the given input tensors. Zeros are treated as ``False`` and nonzeros are
treated as ``True``.

Args:
    {input}
    other (Tensor): the tensor to compute AND with

Keyword args:
    {out}

Example::

    >>> torch.logical_and(torch.tensor([True, False, True]), torch.tensor([True, False, False]))
    tensor([ True, False, False])
    >>> a = torch.tensor([0, 1, 10, 0], dtype=torch.int8)
    >>> b = torch.tensor([4, 0, 1, 0], dtype=torch.int8)
    >>> torch.logical_and(a, b)
    tensor([False, False,  True, False])
    >>> torch.logical_and(a.double(), b.double())
    tensor([False, False,  True, False])
    >>> torch.logical_and(a.double(), b)
    tensor([False, False,  True, False])
    >>> torch.logical_and(a, b, out=torch.empty(4, dtype=torch.bool))
    tensor([False, False,  True, False])
""".format(**common_args))

add_docstr(torch.logical_not,
           r"""
logical_not(input, *, out=None) -> Tensor

Computes the element-wise logical NOT of the given input tensor. If not specified, the output tensor will have the bool
dtype. If the input tensor is not a bool tensor, zeros are treated as ``False`` and non-zeros are treated as ``True``.

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> torch.logical_not(torch.tensor([True, False]))
    tensor([False,  True])
    >>> torch.logical_not(torch.tensor([0, 1, -10], dtype=torch.int8))
    tensor([ True, False, False])
    >>> torch.logical_not(torch.tensor([0., 1.5, -10.], dtype=torch.double))
    tensor([ True, False, False])
    >>> torch.logical_not(torch.tensor([0., 1., -10.], dtype=torch.double), out=torch.empty(3, dtype=torch.int16))
    tensor([1, 0, 0], dtype=torch.int16)
""".format(**common_args))

add_docstr(torch.logical_or,
           r"""
logical_or(input, other, *, out=None) -> Tensor

Computes the element-wise logical OR of the given input tensors. Zeros are treated as ``False`` and nonzeros are
treated as ``True``.

Args:
    {input}
    other (Tensor): the tensor to compute OR with

Keyword args:
    {out}

Example::

    >>> torch.logical_or(torch.tensor([True, False, True]), torch.tensor([True, False, False]))
    tensor([ True, False,  True])
    >>> a = torch.tensor([0, 1, 10, 0], dtype=torch.int8)
    >>> b = torch.tensor([4, 0, 1, 0], dtype=torch.int8)
    >>> torch.logical_or(a, b)
    tensor([ True,  True,  True, False])
    >>> torch.logical_or(a.double(), b.double())
    tensor([ True,  True,  True, False])
    >>> torch.logical_or(a.double(), b)
    tensor([ True,  True,  True, False])
    >>> torch.logical_or(a, b, out=torch.empty(4, dtype=torch.bool))
    tensor([ True,  True,  True, False])
""".format(**common_args))

add_docstr(torch.logical_xor,
           r"""
logical_xor(input, other, *, out=None) -> Tensor

Computes the element-wise logical XOR of the given input tensors. Zeros are treated as ``False`` and nonzeros are
treated as ``True``.

Args:
    {input}
    other (Tensor): the tensor to compute XOR with

Keyword args:
    {out}

Example::

    >>> torch.logical_xor(torch.tensor([True, False, True]), torch.tensor([True, False, False]))
    tensor([False, False,  True])
    >>> a = torch.tensor([0, 1, 10, 0], dtype=torch.int8)
    >>> b = torch.tensor([4, 0, 1, 0], dtype=torch.int8)
    >>> torch.logical_xor(a, b)
    tensor([ True,  True, False, False])
    >>> torch.logical_xor(a.double(), b.double())
    tensor([ True,  True, False, False])
    >>> torch.logical_xor(a.double(), b)
    tensor([ True,  True, False, False])
    >>> torch.logical_xor(a, b, out=torch.empty(4, dtype=torch.bool))
    tensor([ True,  True, False, False])
""".format(**common_args))

# TODO: update kwargs formatting (see https://github.com/pytorch/pytorch/issues/43667)
add_docstr(torch.logspace, """
logspace(start, end, steps, base=10.0, *, \
         out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor
""" + r"""

Creates a one-dimensional tensor of size :attr:`steps` whose values are evenly
spaced from :math:`{{\text{{base}}}}^{{\text{{start}}}}` to
:math:`{{\text{{base}}}}^{{\text{{end}}}}`, inclusive, on a logarithmic scale
with base :attr:`base`. That is, the values are:

.. math::
    (\text{base}^{\text{start}},
    \text{base}^{(\text{start} + \frac{\text{end} - \text{start}}{ \text{steps} - 1})},
    \ldots,
    \text{base}^{(\text{start} + (\text{steps} - 2) * \frac{\text{end} - \text{start}}{ \text{steps} - 1})},
    \text{base}^{\text{end}})
""" + """

.. warning::
    Not providing a value for :attr:`steps` is deprecated. For backwards
    compatibility, not providing a value for :attr:`steps` will create a tensor
    with 100 elements. Note that this behavior is not reflected in the
    documented function signature and should not be relied on. In a future
    PyTorch release, failing to provide a value for :attr:`steps` will throw a
    runtime error.

Args:
    start (float): the starting value for the set of points
    end (float): the ending value for the set of points
    steps (int): size of the constructed tensor
    base (float): base of the logarithm function. Default: ``10.0``.
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.logspace(start=-10, end=10, steps=5)
    tensor([ 1.0000e-10,  1.0000e-05,  1.0000e+00,  1.0000e+05,  1.0000e+10])
    >>> torch.logspace(start=0.1, end=1.0, steps=5)
    tensor([  1.2589,   2.1135,   3.5481,   5.9566,  10.0000])
    >>> torch.logspace(start=0.1, end=1.0, steps=1)
    tensor([1.2589])
    >>> torch.logspace(start=2, end=2, steps=1, base=2)
    tensor([4.0])
""".format(**factory_common_args))

add_docstr(torch.logsumexp,
           r"""
logsumexp(input, dim, keepdim=False, *, out=None)

Returns the log of summed exponentials of each row of the :attr:`input`
tensor in the given dimension :attr:`dim`. The computation is numerically
stabilized.

For summation index :math:`j` given by `dim` and other indices :math:`i`, the result is

    .. math::
        \text{{logsumexp}}(x)_{{i}} = \log \sum_j \exp(x_{{ij}})

{keepdim_details}

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    {out}


Example::
    >>> a = torch.randn(3, 3)
    >>> torch.logsumexp(a, 1)
    tensor([ 0.8442,  1.4322,  0.8711])
""".format(**multi_dim_common))

add_docstr(torch.lstsq,
           r"""
lstsq(input, A, *, out=None) -> Tensor

Computes the solution to the least squares and least norm problems for a full
rank matrix :math:`A` of size :math:`(m \times n)` and a matrix :math:`B` of
size :math:`(m \times k)`.

If :math:`m \geq n`, :func:`lstsq` solves the least-squares problem:

.. math::

   \begin{array}{ll}
   \min_X & \|AX-B\|_2.
   \end{array}

If :math:`m < n`, :func:`lstsq` solves the least-norm problem:

.. math::

   \begin{array}{ll}
   \min_X & \|X\|_2 & \text{subject to} & AX = B.
   \end{array}

Returned tensor :math:`X` has shape :math:`(\max(m, n) \times k)`. The first :math:`n`
rows of :math:`X` contains the solution. If :math:`m \geq n`, the residual sum of squares
for the solution in each column is given by the sum of squares of elements in the
remaining :math:`m - n` rows of that column.

.. note::
    The case when :math:`m < n` is not supported on the GPU.

Args:
    input (Tensor): the matrix :math:`B`
    A (Tensor): the :math:`m` by :math:`n` matrix :math:`A`

Keyword args:
    out (tuple, optional): the optional destination tensor

Returns:
    (Tensor, Tensor): A namedtuple (solution, QR) containing:

        - **solution** (*Tensor*): the least squares solution
        - **QR** (*Tensor*): the details of the QR factorization

.. note::

    The returned matrices will always be transposed, irrespective of the strides
    of the input matrices. That is, they will have stride `(1, m)` instead of
    `(m, 1)`.

Example::

    >>> A = torch.tensor([[1., 1, 1],
                          [2, 3, 4],
                          [3, 5, 2],
                          [4, 2, 5],
                          [5, 4, 3]])
    >>> B = torch.tensor([[-10., -3],
                          [ 12, 14],
                          [ 14, 12],
                          [ 16, 16],
                          [ 18, 16]])
    >>> X, _ = torch.lstsq(B, A)
    >>> X
    tensor([[  2.0000,   1.0000],
            [  1.0000,   1.0000],
            [  1.0000,   2.0000],
            [ 10.9635,   4.8501],
            [  8.9332,   5.2418]])
""")

add_docstr(torch.lt, r"""
lt(input, other, *, out=None) -> Tensor

Computes :math:`\text{input} < \text{other}` element-wise.
""" + r"""

The second argument can be a number or a tensor whose shape is
:ref:`broadcastable <broadcasting-semantics>` with the first argument.

Args:
    input (Tensor): the tensor to compare
    other (Tensor or float): the tensor or value to compare

Keyword args:
    {out}

Returns:
    A boolean tensor that is True where :attr:`input` is less than :attr:`other` and False elsewhere

Example::

    >>> torch.lt(torch.tensor([[1, 2], [3, 4]]), torch.tensor([[1, 1], [4, 4]]))
    tensor([[False, False], [True, False]])
""".format(**common_args))

add_docstr(torch.less, r"""
less(input, other, *, out=None) -> Tensor

Alias for :func:`torch.lt`.
""")

add_docstr(torch.lu_solve,
           r"""
lu_solve(input, LU_data, LU_pivots, *, out=None) -> Tensor

Returns the LU solve of the linear system :math:`Ax = b` using the partially pivoted
LU factorization of A from :meth:`torch.lu`.

Arguments:
    b (Tensor): the RHS tensor of size :math:`(*, m, k)`, where :math:`*`
                is zero or more batch dimensions.
    LU_data (Tensor): the pivoted LU factorization of A from :meth:`torch.lu` of size :math:`(*, m, m)`,
                       where :math:`*` is zero or more batch dimensions.
    LU_pivots (IntTensor): the pivots of the LU factorization from :meth:`torch.lu` of size :math:`(*, m)`,
                           where :math:`*` is zero or more batch dimensions.
                           The batch dimensions of :attr:`LU_pivots` must be equal to the batch dimensions of
                           :attr:`LU_data`.

Keyword args:
    {out}

Example::

    >>> A = torch.randn(2, 3, 3)
    >>> b = torch.randn(2, 3, 1)
    >>> A_LU = torch.lu(A)
    >>> x = torch.lu_solve(b, *A_LU)
    >>> torch.norm(torch.bmm(A, x) - b)
    tensor(1.00000e-07 *
           2.8312)
""".format(**common_args))

add_docstr(torch.masked_select,
           r"""
masked_select(input, mask, *, out=None) -> Tensor

Returns a new 1-D tensor which indexes the :attr:`input` tensor according to
the boolean mask :attr:`mask` which is a `BoolTensor`.

The shapes of the :attr:`mask` tensor and the :attr:`input` tensor don't need
to match, but they must be :ref:`broadcastable <broadcasting-semantics>`.

.. note:: The returned tensor does **not** use the same storage
          as the original tensor

Args:
    {input}
    mask  (BoolTensor): the tensor containing the binary mask to index with

Keyword args:
    {out}

Example::

    >>> x = torch.randn(3, 4)
    >>> x
    tensor([[ 0.3552, -2.3825, -0.8297,  0.3477],
            [-1.2035,  1.2252,  0.5002,  0.6248],
            [ 0.1307, -2.0608,  0.1244,  2.0139]])
    >>> mask = x.ge(0.5)
    >>> mask
    tensor([[False, False, False, False],
            [False, True, True, True],
            [False, False, False, True]])
    >>> torch.masked_select(x, mask)
    tensor([ 1.2252,  0.5002,  0.6248,  2.0139])
""".format(**common_args))

add_docstr(torch.matrix_rank,
           r"""
matrix_rank(input, tol=None, symmetric=False) -> Tensor

Returns the numerical rank of a 2-D tensor. The method to compute the
matrix rank is done using SVD by default. If :attr:`symmetric` is ``True``,
then :attr:`input` is assumed to be symmetric, and the computation of the
rank is done by obtaining the eigenvalues.

:attr:`tol` is the threshold below which the singular values (or the eigenvalues
when :attr:`symmetric` is ``True``) are considered to be 0. If :attr:`tol` is not
specified, :attr:`tol` is set to ``S.max() * max(S.size()) * eps`` where `S` is the
singular values (or the eigenvalues when :attr:`symmetric` is ``True``), and ``eps``
is the epsilon value for the datatype of :attr:`input`.

Args:
    input (Tensor): the input 2-D tensor
    tol (float, optional): the tolerance value. Default: ``None``
    symmetric(bool, optional): indicates whether :attr:`input` is symmetric.
                               Default: ``False``

Example::

    >>> a = torch.eye(10)
    >>> torch.matrix_rank(a)
    tensor(10)
    >>> b = torch.eye(10)
    >>> b[0, 0] = 0
    >>> torch.matrix_rank(b)
    tensor(9)
""")

add_docstr(torch.matrix_power,
           r"""
matrix_power(input, n) -> Tensor

Returns the matrix raised to the power :attr:`n` for square matrices.
For batch of matrices, each individual matrix is raised to the power :attr:`n`.

If :attr:`n` is negative, then the inverse of the matrix (if invertible) is
raised to the power :attr:`n`.  For a batch of matrices, the batched inverse
(if invertible) is raised to the power :attr:`n`. If :attr:`n` is 0, then an identity matrix
is returned.

Args:
    {input}
    n (int): the power to raise the matrix to

Example::

    >>> a = torch.randn(2, 2, 2)
    >>> a
    tensor([[[-1.9975, -1.9610],
             [ 0.9592, -2.3364]],

            [[-1.2534, -1.3429],
             [ 0.4153, -1.4664]]])
    >>> torch.matrix_power(a, 3)
    tensor([[[  3.9392, -23.9916],
             [ 11.7357,  -0.2070]],

            [[  0.2468,  -6.7168],
             [  2.0774,  -0.8187]]])
""".format(**common_args))

add_docstr(torch.matrix_exp,
           r"""
Returns the matrix exponential. Supports batched input.
For a matrix ``A``, the matrix exponential is defined as

.. math::
    \mathrm{e}^A = \sum_{k=0}^\infty A^k / k!

""" + r"""
The implementation is based on:

Bader, P.; Blanes, S.; Casas, F.
Computing the Matrix Exponential with an Optimized Taylor Polynomial Approximation.
Mathematics 2019, 7, 1174.

Args:
    {input}

Example::

    >>> a = torch.randn(2, 2, 2)
    >>> a[0, :, :] = torch.eye(2, 2)
    >>> a[1, :, :] = 2 * torch.eye(2, 2)
    >>> a
    tensor([[[1., 0.],
             [0., 1.]],

            [[2., 0.],
             [0., 2.]]])
    >>> torch.matrix_exp(a)
    tensor([[[2.7183, 0.0000],
             [0.0000, 2.7183]],

             [[7.3891, 0.0000],
              [0.0000, 7.3891]]])

    >>> import math
    >>> x = torch.tensor([[0, math.pi/3], [-math.pi/3, 0]])
    >>> x.matrix_exp() # should be [[cos(pi/3), sin(pi/3)], [-sin(pi/3), cos(pi/3)]]
    tensor([[ 0.5000,  0.8660],
            [-0.8660,  0.5000]])
""".format(**common_args))

add_docstr(torch.max,
           r"""
max(input) -> Tensor

Returns the maximum value of all elements in the ``input`` tensor.

.. warning::
    This function produces deterministic (sub)gradients unlike ``max(dim=0)``

Args:
    {input}

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[ 0.6763,  0.7445, -2.2369]])
    >>> torch.max(a)
    tensor(0.7445)

.. function:: max(input, dim, keepdim=False, *, out=None) -> (Tensor, LongTensor)

Returns a namedtuple ``(values, indices)`` where ``values`` is the maximum
value of each row of the :attr:`input` tensor in the given dimension
:attr:`dim`. And ``indices`` is the index location of each maximum value found
(argmax).

If ``keepdim`` is ``True``, the output tensors are of the same size
as ``input`` except in the dimension ``dim`` where they are of size 1.
Otherwise, ``dim`` is squeezed (see :func:`torch.squeeze`), resulting
in the output tensors having 1 fewer dimension than ``input``.

.. note:: If there are multiple maximal values in a reduced row then
          the indices of the first maximal value are returned.

Args:
    {input}
    {dim}
    {keepdim} Default: ``False``.

Keyword args:
    out (tuple, optional): the result tuple of two output tensors (max, max_indices)

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[-1.2360, -0.2942, -0.1222,  0.8475],
            [ 1.1949, -1.1127, -2.2379, -0.6702],
            [ 1.5717, -0.9207,  0.1297, -1.8768],
            [-0.6172,  1.0036, -0.6060, -0.2432]])
    >>> torch.max(a, 1)
    torch.return_types.max(values=tensor([0.8475, 1.1949, 1.5717, 1.0036]), indices=tensor([3, 0, 0, 1]))

.. function:: max(input, other, *, out=None) -> Tensor

See :func:`torch.maximum`.

""".format(**single_dim_common))

add_docstr(torch.maximum, r"""
maximum(input, other, *, out=None) -> Tensor

Computes the element-wise maximum of :attr:`input` and :attr:`other`.

.. note::
    If one of the elements being compared is a NaN, then that element is returned.
    :func:`maximum` is not supported for tensors with complex dtypes.

Args:
    {input}
    other (Tensor): the second input tensor

Keyword args:
    {out}

Example::

    >>> a = torch.tensor((1, 2, -1))
    >>> b = torch.tensor((3, 0, 4))
    >>> torch.maximum(a, b)
    tensor([3, 2, 4])
""".format(**common_args))

add_docstr(torch.amax,
           r"""
amax(input, dim, keepdim=False, *, out=None) -> Tensor

Returns the maximum value of each slice of the :attr:`input` tensor in the given
dimension(s) :attr:`dim`.

.. note::
    The difference between ``max``/``min`` and ``amax``/``amin`` is:
        - ``amax``/``amin`` supports reducing on multiple dimensions,
        - ``amax``/``amin`` does not return indices,
        - ``amax``/``amin`` evenly distributes gradient between equal values,
          while ``max(dim)``/``min(dim)`` propagates gradient only to a single
          index in the source tensor.

If :attr:`keepdim is ``True``, the output tensors are of the same size
as :attr:`input` except in the dimension(s) :attr:`dim` where they are of size 1.
Otherwise, :attr:`dim`s are squeezed (see :func:`torch.squeeze`), resulting
in the output tensors having fewer dimension than :attr:`input`.

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
  {out}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.8177,  1.4878, -0.2491,  0.9130],
            [-0.7158,  1.1775,  2.0992,  0.4817],
            [-0.0053,  0.0164, -1.3738, -0.0507],
            [ 1.9700,  1.1106, -1.0318, -1.0816]])
    >>> torch.amax(a, 1)
    tensor([1.4878, 2.0992, 0.0164, 1.9700])
""".format(**multi_dim_common))

add_docstr(torch.argmax,
           r"""
argmax(input) -> LongTensor

Returns the indices of the maximum value of all elements in the :attr:`input` tensor.

This is the second value returned by :meth:`torch.max`. See its
documentation for the exact semantics of this method.

.. note:: If there are multiple minimal values then the indices of the first minimal value are returned.

Args:
    {input}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 1.3398,  0.2663, -0.2686,  0.2450],
            [-0.7401, -0.8805, -0.3402, -1.1936],
            [ 0.4907, -1.3948, -1.0691, -0.3132],
            [-1.6092,  0.5419, -0.2993,  0.3195]])
    >>> torch.argmax(a)
    tensor(0)

.. function:: argmax(input, dim, keepdim=False) -> LongTensor

Returns the indices of the maximum values of a tensor across a dimension.

This is the second value returned by :meth:`torch.max`. See its
documentation for the exact semantics of this method.

Args:
    {input}
    {dim} If ``None``, the argmax of the flattened input is returned.
    {keepdim} Ignored if ``dim=None``.

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 1.3398,  0.2663, -0.2686,  0.2450],
            [-0.7401, -0.8805, -0.3402, -1.1936],
            [ 0.4907, -1.3948, -1.0691, -0.3132],
            [-1.6092,  0.5419, -0.2993,  0.3195]])
    >>> torch.argmax(a, dim=1)
    tensor([ 0,  2,  0,  1])
""".format(**single_dim_common))

add_docstr(torch.mean,
           r"""
mean(input) -> Tensor

Returns the mean value of all elements in the :attr:`input` tensor.

Args:
    {input}

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[ 0.2294, -0.5481,  1.3288]])
    >>> torch.mean(a)
    tensor(0.3367)

.. function:: mean(input, dim, keepdim=False, *, out=None) -> Tensor

Returns the mean value of each row of the :attr:`input` tensor in the given
dimension :attr:`dim`. If :attr:`dim` is a list of dimensions,
reduce over all of them.

{keepdim_details}

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[-0.3841,  0.6320,  0.4254, -0.7384],
            [-0.9644,  1.0131, -0.6549, -1.4279],
            [-0.2951, -1.3350, -0.7694,  0.5600],
            [ 1.0842, -0.9580,  0.3623,  0.2343]])
    >>> torch.mean(a, 1)
    tensor([-0.0163, -0.5085, -0.4599,  0.1807])
    >>> torch.mean(a, 1, True)
    tensor([[-0.0163],
            [-0.5085],
            [-0.4599],
            [ 0.1807]])
""".format(**multi_dim_common))

add_docstr(torch.median,
           r"""
median(input) -> Tensor

Returns the median of the values in :attr:`input`.

.. note::
    The median is not unique for :attr:`input` tensors with an even number
    of elements. In this case the lower of the two medians is returned. To
    compute the mean of both medians, use :func:`torch.quantile` with ``q=0.5`` instead.

.. warning::
    This function produces deterministic (sub)gradients unlike ``median(dim=0)``

Args:
    {input}

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[ 1.5219, -1.5212,  0.2202]])
    >>> torch.median(a)
    tensor(0.2202)

.. function:: median(input, dim=-1, keepdim=False, *, out=None) -> (Tensor, LongTensor)

Returns a namedtuple ``(values, indices)`` where ``values`` contains the median of each row of :attr:`input`
in the dimension :attr:`dim`, and ``indices`` contains the index of the median values found in the dimension :attr:`dim`.

By default, :attr:`dim` is the last dimension of the :attr:`input` tensor.

If :attr:`keepdim` is ``True``, the output tensors are of the same size
as :attr:`input` except in the dimension :attr:`dim` where they are of size 1.
Otherwise, :attr:`dim` is squeezed (see :func:`torch.squeeze`), resulting in
the outputs tensor having 1 fewer dimension than :attr:`input`.

.. note::
    The median is not unique for :attr:`input` tensors with an even number
    of elements in the dimension :attr:`dim`. In this case the lower of the
    two medians is returned. To compute the mean of both medians in
    :attr:`input`, use :func:`torch.quantile` with ``q=0.5`` instead.

.. warning::
    ``indices`` does not necessarily contain the first occurrence of each
    median value found, unless it is unique.
    The exact implementation details are device-specific.
    Do not expect the same result when run on CPU and GPU in general.
    For the same reason do not expect the gradients to be deterministic.

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    out ((Tensor, Tensor), optional): The first tensor will be populated with the median values and the second
                                      tensor, which must have dtype long, with their indices in the dimension
                                      :attr:`dim` of :attr:`input`.

Example::

    >>> a = torch.randn(4, 5)
    >>> a
    tensor([[ 0.2505, -0.3982, -0.9948,  0.3518, -1.3131],
            [ 0.3180, -0.6993,  1.0436,  0.0438,  0.2270],
            [-0.2751,  0.7303,  0.2192,  0.3321,  0.2488],
            [ 1.0778, -1.9510,  0.7048,  0.4742, -0.7125]])
    >>> torch.median(a, 1)
    torch.return_types.median(values=tensor([-0.3982,  0.2270,  0.2488,  0.4742]), indices=tensor([1, 4, 4, 3]))
""".format(**single_dim_common))

add_docstr(torch.nanmedian,
           r"""
nanmedian(input) -> Tensor

Returns the median of the values in :attr:`input`, ignoring ``NaN`` values.

This function is identical to :func:`torch.median` when there are no ``NaN`` values in :attr:`input`.
When :attr:`input` has one or more ``NaN`` values, :func:`torch.median` will always return ``NaN``,
while this function will return the median of the non-``NaN`` elements in :attr:`input`.
If all the elements in :attr:`input` are ``NaN`` it will also return ``NaN``.

Args:
    {input}

Example::

    >>> a = torch.tensor([1, float('nan'), 3, 2])
    >>> a.median()
    tensor(nan)
    >>> a.nanmedian()
    tensor(2.)

.. function:: nanmedian(input, dim=-1, keepdim=False, *, out=None) -> (Tensor, LongTensor)

Returns a namedtuple ``(values, indices)`` where ``values`` contains the median of each row of :attr:`input`
in the dimension :attr:`dim`, ignoring ``NaN`` values, and ``indices`` contains the index of the median values
found in the dimension :attr:`dim`.

This function is identical to :func:`torch.median` when there are no ``NaN`` values in a reduced row. When a reduced row has
one or more ``NaN`` values, :func:`torch.median` will always reduce it to ``NaN``, while this function will reduce it to the
median of the non-``NaN`` elements. If all the elements in a reduced row are ``NaN`` then it will be reduced to ``NaN``, too.

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    out ((Tensor, Tensor), optional): The first tensor will be populated with the median values and the second
                                      tensor, which must have dtype long, with their indices in the dimension
                                      :attr:`dim` of :attr:`input`.

Example::

    >>> a = torch.tensor([[2, 3, 1], [float('nan'), 1, float('nan')]])
    >>> a
    tensor([[2., 3., 1.],
            [nan, 1., nan]])
    >>> a.median(0)
    torch.return_types.median(values=tensor([nan, 1., nan]), indices=tensor([1, 1, 1]))
    >>> a.nanmedian(0)
    torch.return_types.nanmedian(values=tensor([2., 1., 1.]), indices=tensor([0, 1, 0]))
""".format(**single_dim_common))

add_docstr(torch.quantile,
           r"""
quantile(input, q) -> Tensor

Returns the q-th quantiles of all elements in the :attr:`input` tensor, doing a linear
interpolation when the q-th quantile lies between two data points.

Args:
    {input}
    q (float or Tensor): a scalar or 1D tensor of quantile values in the range [0, 1]

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[ 0.0700, -0.5446,  0.9214]])
    >>> q = torch.tensor([0, 0.5, 1])
    >>> torch.quantile(a, q)
    tensor([-0.5446,  0.0700,  0.9214])

.. function:: quantile(input, q, dim=None, keepdim=False, *, out=None) -> Tensor

Returns the q-th quantiles of each row of the :attr:`input` tensor along the dimension
:attr:`dim`, doing a linear interpolation when the q-th quantile lies between two
data points. By default, :attr:`dim` is ``None`` resulting in the :attr:`input` tensor
being flattened before computation.

If :attr:`keepdim` is ``True``, the output dimensions are of the same size as :attr:`input`
except in the dimensions being reduced (:attr:`dim` or all if :attr:`dim` is ``None``) where they
have size 1. Otherwise, the dimensions being reduced are squeezed (see :func:`torch.squeeze`).
If :attr:`q` is a 1D tensor, an extra dimension is prepended to the output tensor with the same
size as :attr:`q` which represents the quantiles.

Args:
    {input}
    q (float or Tensor): a scalar or 1D tensor of quantile values in the range [0, 1]
    {dim}
    {keepdim}

Keyword arguments:
    {out}

Example::

    >>> a = torch.randn(2, 3)
    >>> a
    tensor([[ 0.0795, -1.2117,  0.9765],
            [ 1.1707,  0.6706,  0.4884]])
    >>> q = torch.tensor([0.25, 0.5, 0.75])
    >>> torch.quantile(a, q, dim=1, keepdim=True)
    tensor([[[-0.5661],
            [ 0.5795]],

            [[ 0.0795],
            [ 0.6706]],

            [[ 0.5280],
            [ 0.9206]]])
    >>> torch.quantile(a, q, dim=1, keepdim=True).shape
    torch.Size([3, 2, 1])
""".format(**single_dim_common))

add_docstr(torch.nanquantile,
           r"""
nanquantile(input, q, dim=None, keepdim=False, *, out=None) -> Tensor

This is a variant of :func:`torch.quantile` that "ignores" ``NaN`` values,
computing the quantiles :attr:`q` as if ``NaN`` values in :attr:`input` did
not exist. If all values in a reduced row are ``NaN`` then the quantiles for
that reduction will be ``NaN``. See the documentation for :func:`torch.quantile`.

Args:
    {input}
    q (float or Tensor): a scalar or 1D tensor of quantile values in the range [0, 1]
    {dim}
    {keepdim}

Keyword arguments:
    {out}

Example::

    >>> t = torch.tensor([float('nan'), 1, 2])
    >>> t.quantile(0.5)
    tensor(nan)
    >>> t.nanquantile(0.5)
    tensor(1.5000)

    >>> t = torch.tensor([[float('nan'), float('nan')], [1, 2]])
    >>> t
    tensor([[nan, nan],
            [1., 2.]])
    >>> t.nanquantile(0.5, dim=0)
    tensor([1., 2.])
    >>> t.nanquantile(0.5, dim=1)
    tensor([   nan, 1.5000])
""".format(**single_dim_common))

add_docstr(torch.min,
           r"""
min(input) -> Tensor

Returns the minimum value of all elements in the :attr:`input` tensor.

.. warning::
    This function produces deterministic (sub)gradients unlike ``min(dim=0)``

Args:
    {input}

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[ 0.6750,  1.0857,  1.7197]])
    >>> torch.min(a)
    tensor(0.6750)

.. function:: min(input, dim, keepdim=False, *, out=None) -> (Tensor, LongTensor)

Returns a namedtuple ``(values, indices)`` where ``values`` is the minimum
value of each row of the :attr:`input` tensor in the given dimension
:attr:`dim`. And ``indices`` is the index location of each minimum value found
(argmin).

If :attr:`keepdim` is ``True``, the output tensors are of the same size as
:attr:`input` except in the dimension :attr:`dim` where they are of size 1.
Otherwise, :attr:`dim` is squeezed (see :func:`torch.squeeze`), resulting in
the output tensors having 1 fewer dimension than :attr:`input`.

.. note:: If there are multiple minimal values in a reduced row then
          the indices of the first minimal value are returned.

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    out (tuple, optional): the tuple of two output tensors (min, min_indices)

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[-0.6248,  1.1334, -1.1899, -0.2803],
            [-1.4644, -0.2635, -0.3651,  0.6134],
            [ 0.2457,  0.0384,  1.0128,  0.7015],
            [-0.1153,  2.9849,  2.1458,  0.5788]])
    >>> torch.min(a, 1)
    torch.return_types.min(values=tensor([-1.1899, -1.4644,  0.0384, -0.1153]), indices=tensor([2, 0, 1, 0]))

.. function:: min(input, other, *, out=None) -> Tensor

See :func:`torch.minimum`.
""".format(**single_dim_common))

add_docstr(torch.minimum, r"""
minimum(input, other, *, out=None) -> Tensor

Computes the element-wise minimum of :attr:`input` and :attr:`other`.

.. note::
    If one of the elements being compared is a NaN, then that element is returned.
    :func:`minimum` is not supported for tensors with complex dtypes.

Args:
    {input}
    other (Tensor): the second input tensor

Keyword args:
    {out}

Example::

    >>> a = torch.tensor((1, 2, -1))
    >>> b = torch.tensor((3, 0, 4))
    >>> torch.minimum(a, b)
    tensor([1, 0, -1])
""".format(**common_args))

add_docstr(torch.amin,
           r"""
amin(input, dim, keepdim=False, *, out=None) -> Tensor

Returns the minimum value of each slice of the :attr:`input` tensor in the given
dimension(s) :attr:`dim`.

.. note::
    The difference between ``max``/``min`` and ``amax``/``amin`` is:
        - ``amax``/``amin`` supports reducing on multiple dimensions,
        - ``amax``/``amin`` does not return indices,
        - ``amax``/``amin`` evenly distributes gradient between equal values,
          while ``max(dim)``/``min(dim)`` propagates gradient only to a single
          index in the source tensor.

If :attr:`keepdim` is ``True``, the output tensors are of the same size as
:attr:`input` except in the dimension(s) :attr:`dim` where they are of size 1.
Otherwise, :attr:`dim`s are squeezed (see :func:`torch.squeeze`), resulting in
the output tensors having fewer dimensions than :attr:`input`.

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
  {out}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.6451, -0.4866,  0.2987, -1.3312],
            [-0.5744,  1.2980,  1.8397, -0.2713],
            [ 0.9128,  0.9214, -1.7268, -0.2995],
            [ 0.9023,  0.4853,  0.9075, -1.6165]])
    >>> torch.amin(a, 1)
    tensor([-1.3312, -0.5744, -1.7268, -1.6165])
""".format(**multi_dim_common))

add_docstr(torch.argmin,
           r"""
argmin(input) -> LongTensor

Returns the indices of the minimum value of all elements in the :attr:`input` tensor.

This is the second value returned by :meth:`torch.min`. See its
documentation for the exact semantics of this method.

.. note:: If there are multiple minimal values then the indices of the first minimal value are returned.

Args:
    {input}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.1139,  0.2254, -0.1381,  0.3687],
            [ 1.0100, -1.1975, -0.0102, -0.4732],
            [-0.9240,  0.1207, -0.7506, -1.0213],
            [ 1.7809, -1.2960,  0.9384,  0.1438]])
    >>> torch.argmin(a)
    tensor(13)

.. function:: argmin(input, dim, keepdim=False) -> LongTensor

Returns the indices of the minimum values of a tensor across a dimension.

This is the second value returned by :meth:`torch.min`. See its
documentation for the exact semantics of this method.

Args:
    {input}
    {dim} If ``None``, the argmin of the flattened input is returned.
    {keepdim} Ignored if ``dim=None``.

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.1139,  0.2254, -0.1381,  0.3687],
            [ 1.0100, -1.1975, -0.0102, -0.4732],
            [-0.9240,  0.1207, -0.7506, -1.0213],
            [ 1.7809, -1.2960,  0.9384,  0.1438]])
    >>> torch.argmin(a, dim=1)
    tensor([ 2,  1,  3,  1])
""".format(**single_dim_common))

add_docstr(torch.mm,
           r"""
mm(input, mat2, *, out=None) -> Tensor

Performs a matrix multiplication of the matrices :attr:`input` and :attr:`mat2`.

If :attr:`input` is a :math:`(n \times m)` tensor, :attr:`mat2` is a
:math:`(m \times p)` tensor, :attr:`out` will be a :math:`(n \times p)` tensor.

.. note:: This function does not :ref:`broadcast <broadcasting-semantics>`.
          For broadcasting matrix products, see :func:`torch.matmul`.

Supports strided and sparse 2-D tensors as inputs, autograd with
respect to strided inputs.

{tf32_note}

Args:
    input (Tensor): the first matrix to be matrix multiplied
    mat2 (Tensor): the second matrix to be matrix multiplied

Keyword args:
    {out}

Example::

    >>> mat1 = torch.randn(2, 3)
    >>> mat2 = torch.randn(3, 3)
    >>> torch.mm(mat1, mat2)
    tensor([[ 0.4851,  0.5037, -0.3633],
            [-0.0760, -3.6705,  2.4784]])
""".format(**common_args, **tf32_notes))

add_docstr(torch.hspmm,
           r"""
hspmm(mat1, mat2, *, out=None) -> Tensor

Performs a matrix multiplication of a :ref:`sparse COO matrix
<sparse-coo-docs>` :attr:`mat1` and a strided matrix :attr:`mat2`. The
result is a (1 + 1)-dimensional :ref:`hybrid COO matrix
<sparse-hybrid-coo-docs>`.

Args:
    mat1 (Tensor): the first sparse matrix to be matrix multiplied
    mat2 (Tensor): the second strided matrix to be matrix multiplied

Keyword args:
    {out}
""")

add_docstr(torch.matmul,
           r"""
matmul(input, other, *, out=None) -> Tensor

Matrix product of two tensors.

The behavior depends on the dimensionality of the tensors as follows:

- If both tensors are 1-dimensional, the dot product (scalar) is returned.
- If both arguments are 2-dimensional, the matrix-matrix product is returned.
- If the first argument is 1-dimensional and the second argument is 2-dimensional,
  a 1 is prepended to its dimension for the purpose of the matrix multiply.
  After the matrix multiply, the prepended dimension is removed.
- If the first argument is 2-dimensional and the second argument is 1-dimensional,
  the matrix-vector product is returned.
- If both arguments are at least 1-dimensional and at least one argument is
  N-dimensional (where N > 2), then a batched matrix multiply is returned.  If the first
  argument is 1-dimensional, a 1 is prepended to its dimension for the purpose of the
  batched matrix multiply and removed after.  If the second argument is 1-dimensional, a
  1 is appended to its dimension for the purpose of the batched matrix multiple and removed after.
  The non-matrix (i.e. batch) dimensions are :ref:`broadcasted <broadcasting-semantics>` (and thus
  must be broadcastable).  For example, if :attr:`input` is a
  :math:`(j \times 1 \times n \times n)` tensor and :attr:`other` is a :math:`(k \times n \times n)`
  tensor, :attr:`out` will be a :math:`(j \times k \times n \times n)` tensor.

  Note that the broadcasting logic only looks at the batch dimensions when determining if the inputs
  are broadcastable, and not the matrix dimensions. For example, if :attr:`input` is a
  :math:`(j \times 1 \times n \times m)` tensor and :attr:`other` is a :math:`(k \times m \times p)`
  tensor, these inputs are valid for broadcasting even though the final two dimensions (i.e. the
  matrix dimensions) are different. :attr:`out` will be a :math:`(j \times k \times n \times p)` tensor.

{tf32_note}

.. note::

    The 1-dimensional dot product version of this function does not support an :attr:`out` parameter.

Arguments:
    input (Tensor): the first tensor to be multiplied
    other (Tensor): the second tensor to be multiplied

Keyword args:
    {out}

Example::

    >>> # vector x vector
    >>> tensor1 = torch.randn(3)
    >>> tensor2 = torch.randn(3)
    >>> torch.matmul(tensor1, tensor2).size()
    torch.Size([])
    >>> # matrix x vector
    >>> tensor1 = torch.randn(3, 4)
    >>> tensor2 = torch.randn(4)
    >>> torch.matmul(tensor1, tensor2).size()
    torch.Size([3])
    >>> # batched matrix x broadcasted vector
    >>> tensor1 = torch.randn(10, 3, 4)
    >>> tensor2 = torch.randn(4)
    >>> torch.matmul(tensor1, tensor2).size()
    torch.Size([10, 3])
    >>> # batched matrix x batched matrix
    >>> tensor1 = torch.randn(10, 3, 4)
    >>> tensor2 = torch.randn(10, 4, 5)
    >>> torch.matmul(tensor1, tensor2).size()
    torch.Size([10, 3, 5])
    >>> # batched matrix x broadcasted matrix
    >>> tensor1 = torch.randn(10, 3, 4)
    >>> tensor2 = torch.randn(4, 5)
    >>> torch.matmul(tensor1, tensor2).size()
    torch.Size([10, 3, 5])

""".format(**common_args, **tf32_notes))

add_docstr(torch.mode,
           r"""
mode(input, dim=-1, keepdim=False, *, out=None) -> (Tensor, LongTensor)

Returns a namedtuple ``(values, indices)`` where ``values`` is the mode
value of each row of the :attr:`input` tensor in the given dimension
:attr:`dim`, i.e. a value which appears most often
in that row, and ``indices`` is the index location of each mode value found.

By default, :attr:`dim` is the last dimension of the :attr:`input` tensor.

If :attr:`keepdim` is ``True``, the output tensors are of the same size as
:attr:`input` except in the dimension :attr:`dim` where they are of size 1.
Otherwise, :attr:`dim` is squeezed (see :func:`torch.squeeze`), resulting
in the output tensors having 1 fewer dimension than :attr:`input`.

.. note:: This function is not defined for ``torch.cuda.Tensor`` yet.

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    out (tuple, optional): the result tuple of two output tensors (values, indices)

Example::

    >>> a = torch.randint(10, (5,))
    >>> a
    tensor([6, 5, 1, 0, 2])
    >>> b = a + (torch.randn(50, 1) * 5).long()
    >>> torch.mode(b, 0)
    torch.return_types.mode(values=tensor([6, 5, 1, 0, 2]), indices=tensor([2, 2, 2, 2, 2]))
""".format(**single_dim_common))

add_docstr(torch.mul, r"""
mul(input, other, *, out=None)

Multiplies each element of the input :attr:`input` with the scalar
:attr:`other` and returns a new resulting tensor.

.. math::
    \text{out}_i = \text{other} \times \text{input}_i
""" + r"""
If :attr:`input` is of type `FloatTensor` or `DoubleTensor`, :attr:`other`
should be a real number, otherwise it should be an integer

Args:
    {input}
    other (Number): the number to be multiplied to each element of :attr:`input`

Keyword args:
    {out}

Example::

    >>> a = torch.randn(3)
    >>> a
    tensor([ 0.2015, -0.4255,  2.6087])
    >>> torch.mul(a, 100)
    tensor([  20.1494,  -42.5491,  260.8663])

.. function:: mul(input, other, *, out=None)

Each element of the tensor :attr:`input` is multiplied by the corresponding
element of the Tensor :attr:`other`. The resulting tensor is returned.

The shapes of :attr:`input` and :attr:`other` must be
:ref:`broadcastable <broadcasting-semantics>`.

.. math::
    \text{out}_i = \text{input}_i \times \text{other}_i
""" + r"""

Args:
    input (Tensor): the first multiplicand tensor
    other (Tensor): the second multiplicand tensor

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4, 1)
    >>> a
    tensor([[ 1.1207],
            [-0.3137],
            [ 0.0700],
            [ 0.8378]])
    >>> b = torch.randn(1, 4)
    >>> b
    tensor([[ 0.5146,  0.1216, -0.5244,  2.2382]])
    >>> torch.mul(a, b)
    tensor([[ 0.5767,  0.1363, -0.5877,  2.5083],
            [-0.1614, -0.0382,  0.1645, -0.7021],
            [ 0.0360,  0.0085, -0.0367,  0.1567],
            [ 0.4312,  0.1019, -0.4394,  1.8753]])
""".format(**common_args))

add_docstr(torch.multiply, r"""
multiply(input, other, *, out=None)

Alias for :func:`torch.mul`.
""".format(**common_args))

add_docstr(torch.multinomial,
           r"""
multinomial(input, num_samples, replacement=False, *, generator=None, out=None) -> LongTensor

Returns a tensor where each row contains :attr:`num_samples` indices sampled
from the multinomial probability distribution located in the corresponding row
of tensor :attr:`input`.

.. note::
    The rows of :attr:`input` do not need to sum to one (in which case we use
    the values as weights), but must be non-negative, finite and have
    a non-zero sum.

Indices are ordered from left to right according to when each was sampled
(first samples are placed in first column).

If :attr:`input` is a vector, :attr:`out` is a vector of size :attr:`num_samples`.

If :attr:`input` is a matrix with `m` rows, :attr:`out` is an matrix of shape
:math:`(m \times \text{{num\_samples}})`.

If replacement is ``True``, samples are drawn with replacement.

If not, they are drawn without replacement, which means that when a
sample index is drawn for a row, it cannot be drawn again for that row.

.. note::
    When drawn without replacement, :attr:`num_samples` must be lower than
    number of non-zero elements in :attr:`input` (or the min number of non-zero
    elements in each row of :attr:`input` if it is a matrix).

Args:
    input (Tensor): the input tensor containing probabilities
    num_samples (int): number of samples to draw
    replacement (bool, optional): whether to draw with replacement or not

Keyword args:
    {generator}
    {out}

Example::

    >>> weights = torch.tensor([0, 10, 3, 0], dtype=torch.float) # create a tensor of weights
    >>> torch.multinomial(weights, 2)
    tensor([1, 2])
    >>> torch.multinomial(weights, 4) # ERROR!
    RuntimeError: invalid argument 2: invalid multinomial distribution (with replacement=False,
    not enough non-negative category to sample) at ../aten/src/TH/generic/THTensorRandom.cpp:320
    >>> torch.multinomial(weights, 4, replacement=True)
    tensor([ 2,  1,  1,  1])
""".format(**common_args))

add_docstr(torch.mv,
           r"""
mv(input, vec, *, out=None) -> Tensor

Performs a matrix-vector product of the matrix :attr:`input` and the vector
:attr:`vec`.

If :attr:`input` is a :math:`(n \times m)` tensor, :attr:`vec` is a 1-D tensor of
size :math:`m`, :attr:`out` will be 1-D of size :math:`n`.

.. note:: This function does not :ref:`broadcast <broadcasting-semantics>`.

Args:
    input (Tensor): matrix to be multiplied
    vec (Tensor): vector to be multiplied

Keyword args:
    {out}

Example::

    >>> mat = torch.randn(2, 3)
    >>> vec = torch.randn(3)
    >>> torch.mv(mat, vec)
    tensor([ 1.0404, -0.6361])
""".format(**common_args))

add_docstr(torch.mvlgamma,
           r"""
mvlgamma(input, p) -> Tensor

Computes the `multivariate log-gamma function
<https://en.wikipedia.org/wiki/Multivariate_gamma_function>`_) with dimension
:math:`p` element-wise, given by

.. math::
    \log(\Gamma_{p}(a)) = C + \displaystyle \sum_{i=1}^{p} \log\left(\Gamma\left(a - \frac{i - 1}{2}\right)\right)

where :math:`C = \log(\pi) \times \frac{p (p - 1)}{4}` and :math:`\Gamma(\cdot)` is the Gamma function.

All elements must be greater than :math:`\frac{p - 1}{2}`, otherwise an error would be thrown.

Args:
    input (Tensor): the tensor to compute the multivariate log-gamma function
    p (int): the number of dimensions

Example::

    >>> a = torch.empty(2, 3).uniform_(1, 2)
    >>> a
    tensor([[1.6835, 1.8474, 1.1929],
            [1.0475, 1.7162, 1.4180]])
    >>> torch.mvlgamma(a, 2)
    tensor([[0.3928, 0.4007, 0.7586],
            [1.0311, 0.3901, 0.5049]])
""")

add_docstr(torch.movedim, r"""
movedim(input, source, destination) -> Tensor

Moves the dimension(s) of :attr:`input` at the position(s) in :attr:`source`
to the position(s) in :attr:`destination`.

Other dimensions of :attr:`input` that are not explicitly moved remain in
their original order and appear at the positions not specified in :attr:`destination`.

Args:
    {input}
    source (int or tuple of ints): Original positions of the dims to move. These must be unique.
    destination (int or tuple of ints): Destination positions for each of the original dims. These must also be unique.

Examples::

    >>> t = torch.randn(3,2,1)
    >>> t
    tensor([[[-0.3362],
            [-0.8437]],

            [[-0.9627],
            [ 0.1727]],

            [[ 0.5173],
            [-0.1398]]])
    >>> torch.movedim(t, 1, 0).shape
    torch.Size([2, 3, 1])
    >>> torch.movedim(t, 1, 0)
    tensor([[[-0.3362],
            [-0.9627],
            [ 0.5173]],

            [[-0.8437],
            [ 0.1727],
            [-0.1398]]])
    >>> torch.movedim(t, (1, 2), (0, 1)).shape
    torch.Size([2, 1, 3])
    >>> torch.movedim(t, (1, 2), (0, 1))
    tensor([[[-0.3362, -0.9627,  0.5173]],

            [[-0.8437,  0.1727, -0.1398]]])
""".format(**common_args))

add_docstr(torch.narrow,
           r"""
narrow(input, dim, start, length) -> Tensor

Returns a new tensor that is a narrowed version of :attr:`input` tensor. The
dimension :attr:`dim` is input from :attr:`start` to :attr:`start + length`. The
returned tensor and :attr:`input` tensor share the same underlying storage.

Args:
    input (Tensor): the tensor to narrow
    dim (int): the dimension along which to narrow
    start (int): the starting dimension
    length (int): the distance to the ending dimension

Example::

    >>> x = torch.tensor([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    >>> torch.narrow(x, 0, 0, 2)
    tensor([[ 1,  2,  3],
            [ 4,  5,  6]])
    >>> torch.narrow(x, 1, 1, 2)
    tensor([[ 2,  3],
            [ 5,  6],
            [ 8,  9]])
""")

add_docstr(torch.nan_to_num,
           r"""
nan_to_num(input, nan=0.0, posinf=None, neginf=None, *, out=None) -> Tensor

Replaces :literal:`NaN`, positive infinity, and negative infinity values in :attr:`input`
with the values specified by :attr:`nan`, :attr:`posinf`, and :attr:`neginf`, respectively.
By default, :literal:`NaN`s are replaced with zero, positive infinity is replaced with the
greatest finite value representable by :attr:`input`'s dtype, and negative infinity
is replaced with the least finite value representable by :attr:`input`'s dtype.

Args:
    {input}
    nan (Number, optional): the value to replace :literal:`NaN`\s with. Default is zero.
    posinf (Number, optional): if a Number, the value to replace positive infinity values with.
        If None, positive infinity values are replaced with the greatest finite value representable by :attr:`input`'s dtype.
        Default is None.
    neginf (Number, optional): if a Number, the value to replace negative infinity values with.
        If None, negative infinity values are replaced with the lowest finite value representable by :attr:`input`'s dtype.
        Default is None.

Keyword args:
    {out}

Example::

    >>> x = torch.tensor([float('nan'), float('inf'), -float('inf'), 3.14])
    >>> torch.nan_to_num(x)
    tensor([ 0.0000e+00,  3.4028e+38, -3.4028e+38,  3.1400e+00])
    >>> torch.nan_to_num(x, nan=2.0)
    tensor([ 2.0000e+00,  3.4028e+38, -3.4028e+38,  3.1400e+00])
    >>> torch.nan_to_num(x, nan=2.0, posinf=1.0)
    tensor([ 2.0000e+00,  1.0000e+00, -3.4028e+38,  3.1400e+00])

""".format(**common_args))

add_docstr(torch.ne, r"""
ne(input, other, *, out=None) -> Tensor

Computes :math:`\text{input} \neq \text{other}` element-wise.
""" + r"""

The second argument can be a number or a tensor whose shape is
:ref:`broadcastable <broadcasting-semantics>` with the first argument.

Args:
    input (Tensor): the tensor to compare
    other (Tensor or float): the tensor or value to compare

Keyword args:
    {out}

Returns:
    A boolean tensor that is True where :attr:`input` is not equal to :attr:`other` and False elsewhere

Example::

    >>> torch.ne(torch.tensor([[1, 2], [3, 4]]), torch.tensor([[1, 1], [4, 4]]))
    tensor([[False, True], [True, False]])
""".format(**common_args))

add_docstr(torch.not_equal, r"""
not_equal(input, other, *, out=None) -> Tensor

Alias for :func:`torch.ne`.
""")

add_docstr(torch.neg,
           r"""
neg(input, *, out=None) -> Tensor

Returns a new tensor with the negative of the elements of :attr:`input`.

.. math::
    \text{out} = -1 \times \text{input}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(5)
    >>> a
    tensor([ 0.0090, -0.2262, -0.0682, -0.2866,  0.3940])
    >>> torch.neg(a)
    tensor([-0.0090,  0.2262,  0.0682,  0.2866, -0.3940])
""".format(**common_args))

add_docstr(torch.negative,
           r"""
negative(input, *, out=None) -> Tensor

Alias for :func:`torch.neg`
""".format(**common_args))

add_docstr(torch.nextafter,
           r"""
nextafter(input, other, *, out=None) -> Tensor

Return the next floating-point value after :attr:`input` towards :attr:`other`, elementwise.

The shapes of ``input`` and ``other`` must be
:ref:`broadcastable <broadcasting-semantics>`.

Args:
    input (Tensor): the first input tensor
    other (Tensor): the second input tensor

Keyword args:
    {out}

Example::
    >>> eps = torch.finfo(torch.float32).eps
    >>> torch.nextafter(torch.Tensor([1, 2]), torch.Tensor([2, 1])) == torch.Tensor([eps + 1, 2 - eps])
    tensor([True, True])

""".format(**common_args))

add_docstr(torch.nonzero,
           r"""
nonzero(input, *, out=None, as_tuple=False) -> LongTensor or tuple of LongTensors

.. note::
    :func:`torch.nonzero(..., as_tuple=False) <torch.nonzero>` (default) returns a
    2-D tensor where each row is the index for a nonzero value.

    :func:`torch.nonzero(..., as_tuple=True) <torch.nonzero>` returns a tuple of 1-D
    index tensors, allowing for advanced indexing, so ``x[x.nonzero(as_tuple=True)]``
    gives all nonzero values of tensor ``x``. Of the returned tuple, each index tensor
    contains nonzero indices for a certain dimension.

    See below for more details on the two behaviors.

    When :attr:`input` is on CUDA, :func:`torch.nonzero() <torch.nonzero>` causes
    host-device synchronization.

**When** :attr:`as_tuple` **is ``False`` (default)**:

Returns a tensor containing the indices of all non-zero elements of
:attr:`input`.  Each row in the result contains the indices of a non-zero
element in :attr:`input`. The result is sorted lexicographically, with
the last index changing the fastest (C-style).

If :attr:`input` has :math:`n` dimensions, then the resulting indices tensor
:attr:`out` is of size :math:`(z \times n)`, where :math:`z` is the total number of
non-zero elements in the :attr:`input` tensor.

**When** :attr:`as_tuple` **is ``True``**:

Returns a tuple of 1-D tensors, one for each dimension in :attr:`input`,
each containing the indices (in that dimension) of all non-zero elements of
:attr:`input` .

If :attr:`input` has :math:`n` dimensions, then the resulting tuple contains :math:`n`
tensors of size :math:`z`, where :math:`z` is the total number of
non-zero elements in the :attr:`input` tensor.

As a special case, when :attr:`input` has zero dimensions and a nonzero scalar
value, it is treated as a one-dimensional tensor with one element.

Args:
    {input}

Keyword args:
    out (LongTensor, optional): the output tensor containing indices

Returns:
    LongTensor or tuple of LongTensor: If :attr:`as_tuple` is ``False``, the output
    tensor containing indices. If :attr:`as_tuple` is ``True``, one 1-D tensor for
    each dimension, containing the indices of each nonzero element along that
    dimension.

Example::

    >>> torch.nonzero(torch.tensor([1, 1, 1, 0, 1]))
    tensor([[ 0],
            [ 1],
            [ 2],
            [ 4]])
    >>> torch.nonzero(torch.tensor([[0.6, 0.0, 0.0, 0.0],
                                    [0.0, 0.4, 0.0, 0.0],
                                    [0.0, 0.0, 1.2, 0.0],
                                    [0.0, 0.0, 0.0,-0.4]]))
    tensor([[ 0,  0],
            [ 1,  1],
            [ 2,  2],
            [ 3,  3]])
    >>> torch.nonzero(torch.tensor([1, 1, 1, 0, 1]), as_tuple=True)
    (tensor([0, 1, 2, 4]),)
    >>> torch.nonzero(torch.tensor([[0.6, 0.0, 0.0, 0.0],
                                    [0.0, 0.4, 0.0, 0.0],
                                    [0.0, 0.0, 1.2, 0.0],
                                    [0.0, 0.0, 0.0,-0.4]]), as_tuple=True)
    (tensor([0, 1, 2, 3]), tensor([0, 1, 2, 3]))
    >>> torch.nonzero(torch.tensor(5), as_tuple=True)
    (tensor([0]),)
""".format(**common_args))

add_docstr(torch.normal,
           r"""
normal(mean, std, *, generator=None, out=None) -> Tensor

Returns a tensor of random numbers drawn from separate normal distributions
whose mean and standard deviation are given.

The :attr:`mean` is a tensor with the mean of
each output element's normal distribution

The :attr:`std` is a tensor with the standard deviation of
each output element's normal distribution

The shapes of :attr:`mean` and :attr:`std` don't need to match, but the
total number of elements in each tensor need to be the same.

.. note:: When the shapes do not match, the shape of :attr:`mean`
          is used as the shape for the returned output tensor

Args:
    mean (Tensor): the tensor of per-element means
    std (Tensor): the tensor of per-element standard deviations

Keyword args:
    {generator}
    {out}

Example::

    >>> torch.normal(mean=torch.arange(1., 11.), std=torch.arange(1, 0, -0.1))
    tensor([  1.0425,   3.5672,   2.7969,   4.2925,   4.7229,   6.2134,
              8.0505,   8.1408,   9.0563,  10.0566])

.. function:: normal(mean=0.0, std, *, out=None) -> Tensor

Similar to the function above, but the means are shared among all drawn
elements.

Args:
    mean (float, optional): the mean for all distributions
    std (Tensor): the tensor of per-element standard deviations

Keyword args:
    {out}

Example::

    >>> torch.normal(mean=0.5, std=torch.arange(1., 6.))
    tensor([-1.2793, -1.0732, -2.0687,  5.1177, -1.2303])

.. function:: normal(mean, std=1.0, *, out=None) -> Tensor

Similar to the function above, but the standard-deviations are shared among
all drawn elements.

Args:
    mean (Tensor): the tensor of per-element means
    std (float, optional): the standard deviation for all distributions

Keyword args:
    out (Tensor, optional): the output tensor

Example::

    >>> torch.normal(mean=torch.arange(1., 6.))
    tensor([ 1.1552,  2.6148,  2.6535,  5.8318,  4.2361])

.. function:: normal(mean, std, size, *, out=None) -> Tensor

Similar to the function above, but the means and standard deviations are shared
among all drawn elements. The resulting tensor has size given by :attr:`size`.

Args:
    mean (float): the mean for all distributions
    std (float): the standard deviation for all distributions
    size (int...): a sequence of integers defining the shape of the output tensor.

Keyword args:
    {out}

Example::

    >>> torch.normal(2, 3, size=(1, 4))
    tensor([[-1.3987, -1.9544,  3.6048,  0.7909]])
""".format(**common_args))

add_docstr(torch.numel,
           r"""
numel(input) -> int

Returns the total number of elements in the :attr:`input` tensor.

Args:
    {input}

Example::

    >>> a = torch.randn(1, 2, 3, 4, 5)
    >>> torch.numel(a)
    120
    >>> a = torch.zeros(4,4)
    >>> torch.numel(a)
    16

""".format(**common_args))

# TODO: see https://github.com/pytorch/pytorch/issues/43667
add_docstr(torch.ones,
           r"""
ones(*size, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a tensor filled with the scalar value `1`, with the shape defined
by the variable argument :attr:`size`.

Args:
    size (int...): a sequence of integers defining the shape of the output tensor.
        Can be a variable number of arguments or a collection like a list or tuple.
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.ones(2, 3)
    tensor([[ 1.,  1.,  1.],
            [ 1.,  1.,  1.]])

    >>> torch.ones(5)
    tensor([ 1.,  1.,  1.,  1.,  1.])

""".format(**factory_common_args))

# TODO: see https://github.com/pytorch/pytorch/issues/43667
add_docstr(torch.ones_like,
           r"""
ones_like(input, *, dtype=None, layout=None, device=None, requires_grad=False, memory_format=torch.preserve_format) -> Tensor

Returns a tensor filled with the scalar value `1`, with the same size as
:attr:`input`. ``torch.ones_like(input)`` is equivalent to
``torch.ones(input.size(), dtype=input.dtype, layout=input.layout, device=input.device)``.

.. warning::
    As of 0.4, this function does not support an :attr:`out` keyword. As an alternative,
    the old ``torch.ones_like(input, out=output)`` is equivalent to
    ``torch.ones(input.size(), out=output)``.

Args:
    {input}
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}

Example::

    >>> input = torch.empty(2, 3)
    >>> torch.ones_like(input)
    tensor([[ 1.,  1.,  1.],
            [ 1.,  1.,  1.]])
""".format(**factory_like_common_args))

add_docstr(torch.orgqr,
           r"""
orgqr(input, input2) -> Tensor

Computes the orthogonal matrix `Q` of a QR factorization, from the `(input, input2)`
tuple returned by :func:`torch.geqrf`.

This directly calls the underlying LAPACK function `?orgqr`.
See `LAPACK documentation for orgqr`_ for further details.

Args:
    input (Tensor): the `a` from :func:`torch.geqrf`.
    input2 (Tensor): the `tau` from :func:`torch.geqrf`.

.. _LAPACK documentation for orgqr:
    https://software.intel.com/en-us/mkl-developer-reference-c-orgqr

""")

add_docstr(torch.ormqr,
           r"""
ormqr(input, input2, input3, left=True, transpose=False) -> Tensor

Multiplies `mat` (given by :attr:`input3`) by the orthogonal `Q` matrix of the QR factorization
formed by :func:`torch.geqrf` that is represented by `(a, tau)` (given by (:attr:`input`, :attr:`input2`)).

This directly calls the underlying LAPACK function `?ormqr`.
See `LAPACK documentation for ormqr`_ for further details.

Args:
    input (Tensor): the `a` from :func:`torch.geqrf`.
    input2 (Tensor): the `tau` from :func:`torch.geqrf`.
    input3 (Tensor): the matrix to be multiplied.

.. _LAPACK documentation for ormqr:
    https://software.intel.com/en-us/mkl-developer-reference-c-ormqr

""")

add_docstr(torch.poisson,
           r"""
poisson(input, generator=None) -> Tensor

Returns a tensor of the same size as :attr:`input` with each element
sampled from a Poisson distribution with rate parameter given by the corresponding
element in :attr:`input` i.e.,

.. math::
    \text{{out}}_i \sim \text{{Poisson}}(\text{{input}}_i)

Args:
    input (Tensor): the input tensor containing the rates of the Poisson distribution

Keyword args:
    {generator}

Example::

    >>> rates = torch.rand(4, 4) * 5  # rate parameter between 0 and 5
    >>> torch.poisson(rates)
    tensor([[9., 1., 3., 5.],
            [8., 6., 6., 0.],
            [0., 4., 5., 3.],
            [2., 1., 4., 2.]])
""".format(**common_args))

add_docstr(torch.polygamma,
           r"""
polygamma(n, input, *, out=None) -> Tensor

Computes the :math:`n^{th}` derivative of the digamma function on :attr:`input`.
:math:`n \geq 0` is called the order of the polygamma function.

.. math::
    \psi^{(n)}(x) = \frac{d^{(n)}}{dx^{(n)}} \psi(x)

.. note::
    This function is implemented only for nonnegative integers :math:`n \geq 0`.
""" + """
Args:
    n (int): the order of the polygamma function
    {input}

Keyword args:
    {out}

Example::
    >>> a = torch.tensor([1, 0.5])
    >>> torch.polygamma(1, a)
    tensor([1.64493, 4.9348])
    >>> torch.polygamma(2, a)
    tensor([ -2.4041, -16.8288])
    >>> torch.polygamma(3, a)
    tensor([ 6.4939, 97.4091])
    >>> torch.polygamma(4, a)
    tensor([ -24.8863, -771.4742])
""".format(**common_args))

add_docstr(torch.pow,
           r"""
pow(input, exponent, *, out=None) -> Tensor

Takes the power of each element in :attr:`input` with :attr:`exponent` and
returns a tensor with the result.

:attr:`exponent` can be either a single ``float`` number or a `Tensor`
with the same number of elements as :attr:`input`.

When :attr:`exponent` is a scalar value, the operation applied is:

.. math::
    \text{out}_i = x_i ^ \text{exponent}

When :attr:`exponent` is a tensor, the operation applied is:

.. math::
    \text{out}_i = x_i ^ {\text{exponent}_i}
""" + r"""
When :attr:`exponent` is a tensor, the shapes of :attr:`input`
and :attr:`exponent` must be :ref:`broadcastable <broadcasting-semantics>`.

Args:
    {input}
    exponent (float or tensor): the exponent value

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.4331,  1.2475,  0.6834, -0.2791])
    >>> torch.pow(a, 2)
    tensor([ 0.1875,  1.5561,  0.4670,  0.0779])
    >>> exp = torch.arange(1., 5.)

    >>> a = torch.arange(1., 5.)
    >>> a
    tensor([ 1.,  2.,  3.,  4.])
    >>> exp
    tensor([ 1.,  2.,  3.,  4.])
    >>> torch.pow(a, exp)
    tensor([   1.,    4.,   27.,  256.])

.. function:: pow(self, exponent, *, out=None) -> Tensor

:attr:`self` is a scalar ``float`` value, and :attr:`exponent` is a tensor.
The returned tensor :attr:`out` is of the same shape as :attr:`exponent`

The operation applied is:

.. math::
    \text{{out}}_i = \text{{self}} ^ {{\text{{exponent}}_i}}

Args:
    self (float): the scalar base value for the power operation
    exponent (Tensor): the exponent tensor

Keyword args:
    {out}

Example::

    >>> exp = torch.arange(1., 5.)
    >>> base = 2
    >>> torch.pow(base, exp)
    tensor([  2.,   4.,   8.,  16.])
""".format(**common_args))

add_docstr(torch.prod,
           r"""
prod(input, *, dtype=None) -> Tensor

Returns the product of all elements in the :attr:`input` tensor.

Args:
    {input}

Keyword args:
    {dtype}

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[-0.8020,  0.5428, -1.5854]])
    >>> torch.prod(a)
    tensor(0.6902)

.. function:: prod(input, dim, keepdim=False, *, dtype=None) -> Tensor

Returns the product of each row of the :attr:`input` tensor in the given
dimension :attr:`dim`.

{keepdim_details}

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    {dtype}

Example::

    >>> a = torch.randn(4, 2)
    >>> a
    tensor([[ 0.5261, -0.3837],
            [ 1.1857, -0.2498],
            [-1.1646,  0.0705],
            [ 1.1131, -1.0629]])
    >>> torch.prod(a, 1)
    tensor([-0.2018, -0.2962, -0.0821, -1.1831])
""".format(**single_dim_common))

add_docstr(torch.promote_types,
           r"""
promote_types(type1, type2) -> dtype

Returns the :class:`torch.dtype` with the smallest size and scalar kind that is
not smaller nor of lower kind than either `type1` or `type2`. See type promotion
:ref:`documentation <type-promotion-doc>` for more information on the type
promotion logic.

Args:
    type1 (:class:`torch.dtype`)
    type2 (:class:`torch.dtype`)

Example::

    >>> torch.promote_types(torch.int32, torch.float32))
    torch.float32
    >>> torch.promote_types(torch.uint8, torch.long)
    torch.long
""")

add_docstr(torch.qr,
           r"""
qr(input, some=True, *, out=None) -> (Tensor, Tensor)

Computes the QR decomposition of a matrix or a batch of matrices :attr:`input`,
and returns a namedtuple (Q, R) of tensors such that :math:`\text{input} = Q R`
with :math:`Q` being an orthogonal matrix or batch of orthogonal matrices and
:math:`R` being an upper triangular matrix or batch of upper triangular matrices.

If :attr:`some` is ``True``, then this function returns the thin (reduced) QR factorization.
Otherwise, if :attr:`some` is ``False``, this function returns the complete QR factorization.

.. warning::
          If you plan to backpropagate through QR, note that the current backward implementation
          is only well-defined when the first :math:`\min(input.size(-1), input.size(-2))`
          columns of :attr:`input` are linearly independent.
          This behavior will propably change once QR supports pivoting.

.. note:: precision may be lost if the magnitudes of the elements of :attr:`input`
          are large

.. note:: While it should always give you a valid decomposition, it may not
          give you the same one across platforms - it will depend on your
          LAPACK implementation.

Args:
    input (Tensor): the input tensor of size :math:`(*, m, n)` where `*` is zero or more
                batch dimensions consisting of matrices of dimension :math:`m \times n`.
    some (bool, optional): Set to ``True`` for reduced QR decomposition and ``False`` for
                complete QR decomposition.

Keyword args:
    out (tuple, optional): tuple of `Q` and `R` tensors
                satisfying :code:`input = torch.matmul(Q, R)`.
                The dimensions of `Q` and `R` are :math:`(*, m, k)` and :math:`(*, k, n)`
                respectively, where :math:`k = \min(m, n)` if :attr:`some:` is ``True`` and
                :math:`k = m` otherwise.

Example::

    >>> a = torch.tensor([[12., -51, 4], [6, 167, -68], [-4, 24, -41]])
    >>> q, r = torch.qr(a)
    >>> q
    tensor([[-0.8571,  0.3943,  0.3314],
            [-0.4286, -0.9029, -0.0343],
            [ 0.2857, -0.1714,  0.9429]])
    >>> r
    tensor([[ -14.0000,  -21.0000,   14.0000],
            [   0.0000, -175.0000,   70.0000],
            [   0.0000,    0.0000,  -35.0000]])
    >>> torch.mm(q, r).round()
    tensor([[  12.,  -51.,    4.],
            [   6.,  167.,  -68.],
            [  -4.,   24.,  -41.]])
    >>> torch.mm(q.t(), q).round()
    tensor([[ 1.,  0.,  0.],
            [ 0.,  1., -0.],
            [ 0., -0.,  1.]])
    >>> a = torch.randn(3, 4, 5)
    >>> q, r = torch.qr(a, some=False)
    >>> torch.allclose(torch.matmul(q, r), a)
    True
    >>> torch.allclose(torch.matmul(q.transpose(-2, -1), q), torch.eye(5))
    True
""")

add_docstr(torch.rad2deg,
           r"""
rad2deg(input, *, out=None) -> Tensor

Returns a new tensor with each of the elements of :attr:`input`
converted from angles in radians to degrees.

Args:
    {input}

Keyword arguments:
    {out}

Example::

    >>> a = torch.tensor([[3.142, -3.142], [6.283, -6.283], [1.570, -1.570]])
    >>> torch.rad2deg(a)
    tensor([[ 180.0233, -180.0233],
            [ 359.9894, -359.9894],
            [  89.9544,  -89.9544]])

""".format(**common_args))

add_docstr(torch.deg2rad,
           r"""
deg2rad(input, *, out=None) -> Tensor

Returns a new tensor with each of the elements of :attr:`input`
converted from angles in degrees to radians.

Args:
    {input}

Keyword arguments:
    {out}

Example::

    >>> a = torch.tensor([[180.0, -180.0], [360.0, -360.0], [90.0, -90.0]])
    >>> torch.deg2rad(a)
    tensor([[ 3.1416, -3.1416],
            [ 6.2832, -6.2832],
            [ 1.5708, -1.5708]])

""".format(**common_args))

add_docstr(torch.heaviside,
           r"""
heaviside(input, values, *, out=None) -> Tensor

Computes the Heaviside step function for each element in :attr:`input`.
The Heaviside step function is defined as:

.. math::
    \text{{heaviside}}(input, values) = \begin{cases}
        0, & \text{if input < 0}\\
        values, & \text{if input == 0}\\
        1, & \text{if input > 0}
    \end{cases}
""" + r"""

Args:
    {input}
    values (Tensor): The values to use where :attr:`input` is zero.

Keyword arguments:
    {out}

Example::

    >>> input = torch.tensor([-1.5, 0, 2.0])
    >>> values = torch.tensor([0.5])
    >>> torch.heaviside(input, values)
    tensor([0.0000, 0.5000, 1.0000])
    >>> values = torch.tensor([1.2, -2.0, 3.5])
    >>> torch.heaviside(input, values)
    tensor([0., -2., 1.])

""".format(**common_args))

add_docstr(torch.rand,
           r"""
rand(*size, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a tensor filled with random numbers from a uniform distribution
on the interval :math:`[0, 1)`

The shape of the tensor is defined by the variable argument :attr:`size`.

Args:
    size (int...): a sequence of integers defining the shape of the output tensor.
        Can be a variable number of arguments or a collection like a list or tuple.

Keyword args:
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.rand(4)
    tensor([ 0.5204,  0.2503,  0.3525,  0.5673])
    >>> torch.rand(2, 3)
    tensor([[ 0.8237,  0.5781,  0.6879],
            [ 0.3816,  0.7249,  0.0998]])
""".format(**factory_common_args))

add_docstr(torch.rand_like,
           r"""
rand_like(input, *, dtype=None, layout=None, device=None, requires_grad=False, memory_format=torch.preserve_format) -> Tensor

Returns a tensor with the same size as :attr:`input` that is filled with
random numbers from a uniform distribution on the interval :math:`[0, 1)`.
``torch.rand_like(input)`` is equivalent to
``torch.rand(input.size(), dtype=input.dtype, layout=input.layout, device=input.device)``.

Args:
    {input}

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}

""".format(**factory_like_common_args))

add_docstr(torch.randint,
           """
randint(low=0, high, size, \\*, generator=None, out=None, \
dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a tensor filled with random integers generated uniformly
between :attr:`low` (inclusive) and :attr:`high` (exclusive).

The shape of the tensor is defined by the variable argument :attr:`size`.

.. note:
    With the global dtype default (``torch.float32``), this function returns
    a tensor with dtype ``torch.int64``.

Args:
    low (int, optional): Lowest integer to be drawn from the distribution. Default: 0.
    high (int): One above the highest integer to be drawn from the distribution.
    size (tuple): a tuple defining the shape of the output tensor.

Keyword args:
    {generator}
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.randint(3, 5, (3,))
    tensor([4, 3, 4])


    >>> torch.randint(10, (2, 2))
    tensor([[0, 2],
            [5, 5]])


    >>> torch.randint(3, 10, (2, 2))
    tensor([[4, 5],
            [6, 7]])


""".format(**factory_common_args))

add_docstr(torch.randint_like,
           """
randint_like(input, low=0, high, \\*, dtype=None, layout=torch.strided, device=None, requires_grad=False, \
memory_format=torch.preserve_format) -> Tensor

Returns a tensor with the same shape as Tensor :attr:`input` filled with
random integers generated uniformly between :attr:`low` (inclusive) and
:attr:`high` (exclusive).

.. note:
    With the global dtype default (``torch.float32``), this function returns
    a tensor with dtype ``torch.int64``.

Args:
    {input}
    low (int, optional): Lowest integer to be drawn from the distribution. Default: 0.
    high (int): One above the highest integer to be drawn from the distribution.

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}

""".format(**factory_like_common_args))

add_docstr(torch.randn,
           r"""
randn(*size, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a tensor filled with random numbers from a normal distribution
with mean `0` and variance `1` (also called the standard normal
distribution).

.. math::
    \text{{out}}_{{i}} \sim \mathcal{{N}}(0, 1)

The shape of the tensor is defined by the variable argument :attr:`size`.

Args:
    size (int...): a sequence of integers defining the shape of the output tensor.
        Can be a variable number of arguments or a collection like a list or tuple.

Keyword args:
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.randn(4)
    tensor([-2.1436,  0.9966,  2.3426, -0.6366])
    >>> torch.randn(2, 3)
    tensor([[ 1.5954,  2.8929, -1.0923],
            [ 1.1719, -0.4709, -0.1996]])
""".format(**factory_common_args))

add_docstr(torch.randn_like,
           r"""
randn_like(input, *, dtype=None, layout=None, device=None, requires_grad=False, memory_format=torch.preserve_format) -> Tensor

Returns a tensor with the same size as :attr:`input` that is filled with
random numbers from a normal distribution with mean 0 and variance 1.
``torch.randn_like(input)`` is equivalent to
``torch.randn(input.size(), dtype=input.dtype, layout=input.layout, device=input.device)``.

Args:
    {input}

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}

""".format(**factory_like_common_args))

add_docstr(torch.randperm,
           r"""
randperm(n, *, out=None, dtype=torch.int64, layout=torch.strided, device=None, requires_grad=False) -> LongTensor

Returns a random permutation of integers from ``0`` to ``n - 1``.

Args:
    n (int): the upper bound (exclusive)

Keyword args:
    {out}
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        Default: ``torch.int64``.
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.randperm(4)
    tensor([2, 1, 0, 3])
""".format(**factory_common_args))

add_docstr(torch.tensor,
           r"""
tensor(data, *, dtype=None, device=None, requires_grad=False, pin_memory=False) -> Tensor

Constructs a tensor with :attr:`data`.

.. warning::

    :func:`torch.tensor` always copies :attr:`data`. If you have a Tensor
    ``data`` and want to avoid a copy, use :func:`torch.Tensor.requires_grad_`
    or :func:`torch.Tensor.detach`.
    If you have a NumPy ``ndarray`` and want to avoid a copy, use
    :func:`torch.as_tensor`.

.. warning::

    When data is a tensor `x`, :func:`torch.tensor` reads out 'the data' from whatever it is passed,
    and constructs a leaf variable. Therefore ``torch.tensor(x)`` is equivalent to ``x.clone().detach()``
    and ``torch.tensor(x, requires_grad=True)`` is equivalent to ``x.clone().detach().requires_grad_(True)``.
    The equivalents using ``clone()`` and ``detach()`` are recommended.

Args:
    {data}

Keyword args:
    {dtype}
    {device}
    {requires_grad}
    {pin_memory}


Example::

    >>> torch.tensor([[0.1, 1.2], [2.2, 3.1], [4.9, 5.2]])
    tensor([[ 0.1000,  1.2000],
            [ 2.2000,  3.1000],
            [ 4.9000,  5.2000]])

    >>> torch.tensor([0, 1])  # Type inference on data
    tensor([ 0,  1])

    >>> torch.tensor([[0.11111, 0.222222, 0.3333333]],
                     dtype=torch.float64,
                     device=torch.device('cuda:0'))  # creates a torch.cuda.DoubleTensor
    tensor([[ 0.1111,  0.2222,  0.3333]], dtype=torch.float64, device='cuda:0')

    >>> torch.tensor(3.14159)  # Create a scalar (zero-dimensional tensor)
    tensor(3.1416)

    >>> torch.tensor([])  # Create an empty tensor (of size (0,))
    tensor([])
""".format(**factory_data_common_args))

add_docstr(torch.range,
           r"""
range(start=0, end, step=1, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a 1-D tensor of size :math:`\left\lfloor \frac{\text{end} - \text{start}}{\text{step}} \right\rfloor + 1`
with values from :attr:`start` to :attr:`end` with step :attr:`step`. Step is
the gap between two values in the tensor.

.. math::
    \text{out}_{i+1} = \text{out}_i + \text{step}.
""" + r"""
.. warning::
    This function is deprecated and will be removed in a future release because its behavior is inconsistent with
    Python's range builtin. Instead, use :func:`torch.arange`, which produces values in [start, end).

Args:
    start (float): the starting value for the set of points. Default: ``0``.
    end (float): the ending value for the set of points
    step (float): the gap between each pair of adjacent points. Default: ``1``.

Keyword args:
    {out}
    {dtype} If `dtype` is not given, infer the data type from the other input
        arguments. If any of `start`, `end`, or `stop` are floating-point, the
        `dtype` is inferred to be the default dtype, see
        :meth:`~torch.get_default_dtype`. Otherwise, the `dtype` is inferred to
        be `torch.int64`.
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.range(1, 4)
    tensor([ 1.,  2.,  3.,  4.])
    >>> torch.range(1, 4, 0.5)
    tensor([ 1.0000,  1.5000,  2.0000,  2.5000,  3.0000,  3.5000,  4.0000])
""".format(**factory_common_args))

add_docstr(torch.arange,
           r"""
arange(start=0, end, step=1, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a 1-D tensor of size :math:`\left\lceil \frac{\text{end} - \text{start}}{\text{step}} \right\rceil`
with values from the interval ``[start, end)`` taken with common difference
:attr:`step` beginning from `start`.

Note that non-integer :attr:`step` is subject to floating point rounding errors when
comparing against :attr:`end`; to avoid inconsistency, we advise adding a small epsilon to :attr:`end`
in such cases.

.. math::
    \text{out}_{{i+1}} = \text{out}_{i} + \text{step}
""" + r"""
Args:
    start (Number): the starting value for the set of points. Default: ``0``.
    end (Number): the ending value for the set of points
    step (Number): the gap between each pair of adjacent points. Default: ``1``.

Keyword args:
    {out}
    {dtype} If `dtype` is not given, infer the data type from the other input
        arguments. If any of `start`, `end`, or `stop` are floating-point, the
        `dtype` is inferred to be the default dtype, see
        :meth:`~torch.get_default_dtype`. Otherwise, the `dtype` is inferred to
        be `torch.int64`.
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.arange(5)
    tensor([ 0,  1,  2,  3,  4])
    >>> torch.arange(1, 4)
    tensor([ 1,  2,  3])
    >>> torch.arange(1, 2.5, 0.5)
    tensor([ 1.0000,  1.5000,  2.0000])
""".format(**factory_common_args))

add_docstr(torch.ravel,
           r"""
ravel(input) -> Tensor

Return a contiguous flattened tensor. A copy is made only if needed.

Args:
    {input}

Example::

    >>> t = torch.tensor([[[1, 2],
    ...                    [3, 4]],
    ...                   [[5, 6],
    ...                    [7, 8]]])
    >>> torch.ravel(t)
    tensor([1, 2, 3, 4, 5, 6, 7, 8])
""".format(**common_args))

add_docstr(torch.remainder,
           r"""
remainder(input, other, *, out=None) -> Tensor

Computes the element-wise remainder of division.

The dividend and divisor may contain both for integer and floating point
numbers. The remainder has the same sign as the divisor :attr:`other`.

When :attr:`other` is a tensor, the shapes of :attr:`input` and
:attr:`other` must be :ref:`broadcastable <broadcasting-semantics>`.

Args:
    input (Tensor): the dividend
    other (Tensor or float): the divisor that may be either a number or a
                               Tensor of the same shape as the dividend

Keyword args:
    {out}

Example::

    >>> torch.remainder(torch.tensor([-3., -2, -1, 1, 2, 3]), 2)
    tensor([ 1.,  0.,  1.,  1.,  0.,  1.])
    >>> torch.remainder(torch.tensor([1., 2, 3, 4, 5]), 1.5)
    tensor([ 1.0000,  0.5000,  0.0000,  1.0000,  0.5000])

.. seealso::

        :func:`torch.fmod`, which computes the element-wise remainder of
        division equivalently to the C library function ``fmod()``.
""".format(**common_args))

add_docstr(torch.renorm,
           r"""
renorm(input, p, dim, maxnorm, *, out=None) -> Tensor

Returns a tensor where each sub-tensor of :attr:`input` along dimension
:attr:`dim` is normalized such that the `p`-norm of the sub-tensor is lower
than the value :attr:`maxnorm`

.. note:: If the norm of a row is lower than `maxnorm`, the row is unchanged

Args:
    {input}
    p (float): the power for the norm computation
    dim (int): the dimension to slice over to get the sub-tensors
    maxnorm (float): the maximum norm to keep each sub-tensor under

Keyword args:
    {out}

Example::

    >>> x = torch.ones(3, 3)
    >>> x[1].fill_(2)
    tensor([ 2.,  2.,  2.])
    >>> x[2].fill_(3)
    tensor([ 3.,  3.,  3.])
    >>> x
    tensor([[ 1.,  1.,  1.],
            [ 2.,  2.,  2.],
            [ 3.,  3.,  3.]])
    >>> torch.renorm(x, 1, 0, 5)
    tensor([[ 1.0000,  1.0000,  1.0000],
            [ 1.6667,  1.6667,  1.6667],
            [ 1.6667,  1.6667,  1.6667]])
""".format(**common_args))

add_docstr(torch.reshape,
           r"""
reshape(input, shape) -> Tensor

Returns a tensor with the same data and number of elements as :attr:`input`,
but with the specified shape. When possible, the returned tensor will be a view
of :attr:`input`. Otherwise, it will be a copy. Contiguous inputs and inputs
with compatible strides can be reshaped without copying, but you should not
depend on the copying vs. viewing behavior.

See :meth:`torch.Tensor.view` on when it is possible to return a view.

A single dimension may be -1, in which case it's inferred from the remaining
dimensions and the number of elements in :attr:`input`.

Args:
    input (Tensor): the tensor to be reshaped
    shape (tuple of ints): the new shape

Example::

    >>> a = torch.arange(4.)
    >>> torch.reshape(a, (2, 2))
    tensor([[ 0.,  1.],
            [ 2.,  3.]])
    >>> b = torch.tensor([[0, 1], [2, 3]])
    >>> torch.reshape(b, (-1,))
    tensor([ 0,  1,  2,  3])
""")


add_docstr(torch.result_type,
           r"""
result_type(tensor1, tensor2) -> dtype

Returns the :class:`torch.dtype` that would result from performing an arithmetic
operation on the provided input tensors. See type promotion :ref:`documentation <type-promotion-doc>`
for more information on the type promotion logic.

Args:
    tensor1 (Tensor or Number): an input tensor or number
    tensor2 (Tensor or Number): an input tensor or number

Example::

    >>> torch.result_type(torch.tensor([1, 2], dtype=torch.int), 1.0)
    torch.float32
    >>> torch.result_type(torch.tensor([1, 2], dtype=torch.uint8), torch.tensor(1))
    torch.uint8
""")

add_docstr(torch.row_stack,
           r"""
row_stack(tensors, *, out=None) -> Tensor

Alias of :func:`torch.vstack`.
""".format(**common_args))

add_docstr(torch.round,
           r"""
round(input, *, out=None) -> Tensor

Returns a new tensor with each of the elements of :attr:`input` rounded
to the closest integer.

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.9920,  0.6077,  0.9734, -1.0362])
    >>> torch.round(a)
    tensor([ 1.,  1.,  1., -1.])
""".format(**common_args))

add_docstr(torch.rsqrt,
           r"""
rsqrt(input, *, out=None) -> Tensor

Returns a new tensor with the reciprocal of the square-root of each of
the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \frac{1}{\sqrt{\text{input}_{i}}}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.0370,  0.2970,  1.5420, -0.9105])
    >>> torch.rsqrt(a)
    tensor([    nan,  1.8351,  0.8053,     nan])
""".format(**common_args))

add_docstr(torch.set_flush_denormal,
           r"""
set_flush_denormal(mode) -> bool

Disables denormal floating numbers on CPU.

Returns ``True`` if your system supports flushing denormal numbers and it
successfully configures flush denormal mode.  :meth:`~torch.set_flush_denormal`
is only supported on x86 architectures supporting SSE3.

Args:
    mode (bool): Controls whether to enable flush denormal mode or not

Example::

    >>> torch.set_flush_denormal(True)
    True
    >>> torch.tensor([1e-323], dtype=torch.float64)
    tensor([ 0.], dtype=torch.float64)
    >>> torch.set_flush_denormal(False)
    True
    >>> torch.tensor([1e-323], dtype=torch.float64)
    tensor(9.88131e-324 *
           [ 1.0000], dtype=torch.float64)
""")

add_docstr(torch.set_num_threads, r"""
set_num_threads(int)

Sets the number of threads used for intraop parallelism on CPU.

.. warning::
    To ensure that the correct number of threads is used, set_num_threads
    must be called before running eager, JIT or autograd code.
""")

add_docstr(torch.set_num_interop_threads, r"""
set_num_interop_threads(int)

Sets the number of threads used for interop parallelism
(e.g. in JIT interpreter) on CPU.

.. warning::
    Can only be called once and before any inter-op parallel work
    is started (e.g. JIT execution).
""")

add_docstr(torch.sigmoid, r"""
sigmoid(input, *, out=None) -> Tensor

Returns a new tensor with the sigmoid of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \frac{1}{1 + e^{-\text{input}_{i}}}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.9213,  1.0887, -0.8858, -1.7683])
    >>> torch.sigmoid(a)
    tensor([ 0.7153,  0.7481,  0.2920,  0.1458])
""".format(**common_args))

add_docstr(torch.logit,
           r"""
logit(input, eps=None, *, out=None) -> Tensor

Returns a new tensor with the logit of the elements of :attr:`input`.
:attr:`input` is clamped to [eps, 1 - eps] when eps is not None.
When eps is None and :attr:`input` < 0 or :attr:`input` > 1, the function will yields NaN.

.. math::
    y_{i} = \ln(\frac{z_{i}}{1 - z_{i}}) \\
    z_{i} = \begin{cases}
        x_{i} & \text{if eps is None} \\
        \text{eps} & \text{if } x_{i} < \text{eps} \\
        x_{i} & \text{if } \text{eps} \leq x_{i} \leq 1 - \text{eps} \\
        1 - \text{eps} & \text{if } x_{i} > 1 - \text{eps}
    \end{cases}
""" + r"""
Args:
    {input}
    eps (float, optional): the epsilon for input clamp bound. Default: ``None``

Keyword args:
    {out}

Example::

    >>> a = torch.rand(5)
    >>> a
    tensor([0.2796, 0.9331, 0.6486, 0.1523, 0.6516])
    >>> torch.logit(a, eps=1e-6)
    tensor([-0.9466,  2.6352,  0.6131, -1.7169,  0.6261])
""".format(**common_args))

add_docstr(torch.sign,
           r"""
sign(input, *, out=None) -> Tensor

Returns a new tensor with the signs of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \operatorname{sgn}(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.tensor([0.7, -1.2, 0., 2.3])
    >>> a
    tensor([ 0.7000, -1.2000,  0.0000,  2.3000])
    >>> torch.sign(a)
    tensor([ 1., -1.,  0.,  1.])
""".format(**common_args))

add_docstr(torch.signbit,
           r"""
signbit(input, *, out=None) -> Tensor

Tests if each element of :attr:`input` has its sign bit set (is less than zero) or not.

Args:
  {input}

Keyword args:
  {out}

Example::

    >>> a = torch.tensor([0.7, -1.2, 0., 2.3])
    >>> torch.signbit(a)
    tensor([ False, True,  False,  False])
""".format(**common_args))

add_docstr(torch.sgn,
           r"""
sgn(input, *, out=None) -> Tensor

For complex tensors, this function returns a new tensor whose elemants have the same angle as that of the
elements of :attr:`input` and absolute value 1. For a non-complex tensor, this function
returns the signs of the elements of :attr:`input` (see :func:`torch.sign`).

:math:`\text{out}_{i} = 0`, if :math:`|{\text{{input}}_i}| == 0`
:math:`\text{out}_{i} = \frac{{\text{{input}}_i}}{|{\text{{input}}_i}|}`, otherwise

""" + r"""
Args:
    {input}

Keyword args:
  {out}

Example::

    >>> x=torch.tensor([3+4j, 7-24j, 0, 1+2j])
    >>> x.sgn()
    tensor([0.6000+0.8000j, 0.2800-0.9600j, 0.0000+0.0000j, 0.4472+0.8944j])
""".format(**common_args))

add_docstr(torch.sin,
           r"""
sin(input, *, out=None) -> Tensor

Returns a new tensor with the sine of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \sin(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-0.5461,  0.1347, -2.7266, -0.2746])
    >>> torch.sin(a)
    tensor([-0.5194,  0.1343, -0.4032, -0.2711])
""".format(**common_args))

add_docstr(torch.sinh,
           r"""
sinh(input, *, out=None) -> Tensor

Returns a new tensor with the hyperbolic sine of the elements of
:attr:`input`.

.. math::
    \text{out}_{i} = \sinh(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.5380, -0.8632, -0.1265,  0.9399])
    >>> torch.sinh(a)
    tensor([ 0.5644, -0.9744, -0.1268,  1.0845])
""".format(**common_args))

add_docstr(torch.sort,
           r"""
sort(input, dim=-1, descending=False, *, out=None) -> (Tensor, LongTensor)

Sorts the elements of the :attr:`input` tensor along a given dimension
in ascending order by value.

If :attr:`dim` is not given, the last dimension of the `input` is chosen.

If :attr:`descending` is ``True`` then the elements are sorted in descending
order by value.

A namedtuple of (values, indices) is returned, where the `values` are the
sorted values and `indices` are the indices of the elements in the original
`input` tensor.

Args:
    {input}
    dim (int, optional): the dimension to sort along
    descending (bool, optional): controls the sorting order (ascending or descending)

Keyword args:
    out (tuple, optional): the output tuple of (`Tensor`, `LongTensor`) that can
        be optionally given to be used as output buffers

Example::

    >>> x = torch.randn(3, 4)
    >>> sorted, indices = torch.sort(x)
    >>> sorted
    tensor([[-0.2162,  0.0608,  0.6719,  2.3332],
            [-0.5793,  0.0061,  0.6058,  0.9497],
            [-0.5071,  0.3343,  0.9553,  1.0960]])
    >>> indices
    tensor([[ 1,  0,  2,  3],
            [ 3,  1,  0,  2],
            [ 0,  3,  1,  2]])

    >>> sorted, indices = torch.sort(x, 0)
    >>> sorted
    tensor([[-0.5071, -0.2162,  0.6719, -0.5793],
            [ 0.0608,  0.0061,  0.9497,  0.3343],
            [ 0.6058,  0.9553,  1.0960,  2.3332]])
    >>> indices
    tensor([[ 2,  0,  0,  1],
            [ 0,  1,  1,  2],
            [ 1,  2,  2,  0]])
""".format(**common_args))

add_docstr(torch.argsort,
           r"""
argsort(input, dim=-1, descending=False) -> LongTensor

Returns the indices that sort a tensor along a given dimension in ascending
order by value.

This is the second value returned by :meth:`torch.sort`.  See its documentation
for the exact semantics of this method.

Args:
    {input}
    dim (int, optional): the dimension to sort along
    descending (bool, optional): controls the sorting order (ascending or descending)

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.0785,  1.5267, -0.8521,  0.4065],
            [ 0.1598,  0.0788, -0.0745, -1.2700],
            [ 1.2208,  1.0722, -0.7064,  1.2564],
            [ 0.0669, -0.2318, -0.8229, -0.9280]])


    >>> torch.argsort(a, dim=1)
    tensor([[2, 0, 3, 1],
            [3, 2, 1, 0],
            [2, 1, 0, 3],
            [3, 2, 1, 0]])
""".format(**common_args))

add_docstr(torch.sparse_coo_tensor,
           r"""
sparse_coo_tensor(indices, values, size=None, *, dtype=None, device=None, requires_grad=False) -> Tensor

Constructs a :ref:`sparse tensor in COO(rdinate) format
<sparse-coo-docs>` with specified values at the given
:attr:`indices`.

.. note::

   This function returns an :ref:`uncoalesced tensor <sparse-uncoalesced-coo-docs>`.

Args:
    indices (array_like): Initial data for the tensor. Can be a list, tuple,
        NumPy ``ndarray``, scalar, and other types. Will be cast to a :class:`torch.LongTensor`
        internally. The indices are the coordinates of the non-zero values in the matrix, and thus
        should be two-dimensional where the first dimension is the number of tensor dimensions and
        the second dimension is the number of non-zero values.
    values (array_like): Initial values for the tensor. Can be a list, tuple,
        NumPy ``ndarray``, scalar, and other types.
    size (list, tuple, or :class:`torch.Size`, optional): Size of the sparse tensor. If not
        provided the size will be inferred as the minimum size big enough to hold all non-zero
        elements.

Keyword args:
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        Default: if None, infers data type from :attr:`values`.
    device (:class:`torch.device`, optional): the desired device of returned tensor.
        Default: if None, uses the current device for the default tensor type
        (see :func:`torch.set_default_tensor_type`). :attr:`device` will be the CPU
        for CPU tensor types and the current CUDA device for CUDA tensor types.
    {requires_grad}


Example::

    >>> i = torch.tensor([[0, 1, 1],
                          [2, 0, 2]])
    >>> v = torch.tensor([3, 4, 5], dtype=torch.float32)
    >>> torch.sparse_coo_tensor(i, v, [2, 4])
    tensor(indices=tensor([[0, 1, 1],
                           [2, 0, 2]]),
           values=tensor([3., 4., 5.]),
           size=(2, 4), nnz=3, layout=torch.sparse_coo)

    >>> torch.sparse_coo_tensor(i, v)  # Shape inference
    tensor(indices=tensor([[0, 1, 1],
                           [2, 0, 2]]),
           values=tensor([3., 4., 5.]),
           size=(2, 3), nnz=3, layout=torch.sparse_coo)

    >>> torch.sparse_coo_tensor(i, v, [2, 4],
                                dtype=torch.float64,
                                device=torch.device('cuda:0'))
    tensor(indices=tensor([[0, 1, 1],
                           [2, 0, 2]]),
           values=tensor([3., 4., 5.]),
           device='cuda:0', size=(2, 4), nnz=3, dtype=torch.float64,
           layout=torch.sparse_coo)

    # Create an empty sparse tensor with the following invariants:
    #   1. sparse_dim + dense_dim = len(SparseTensor.shape)
    #   2. SparseTensor._indices().shape = (sparse_dim, nnz)
    #   3. SparseTensor._values().shape = (nnz, SparseTensor.shape[sparse_dim:])
    #
    # For instance, to create an empty sparse tensor with nnz = 0, dense_dim = 0 and
    # sparse_dim = 1 (hence indices is a 2D tensor of shape = (1, 0))
    >>> S = torch.sparse_coo_tensor(torch.empty([1, 0]), [], [1])
    tensor(indices=tensor([], size=(1, 0)),
           values=tensor([], size=(0,)),
           size=(1,), nnz=0, layout=torch.sparse_coo)

    # and to create an empty sparse tensor with nnz = 0, dense_dim = 1 and
    # sparse_dim = 1
    >>> S = torch.sparse_coo_tensor(torch.empty([1, 0]), torch.empty([0, 2]), [1, 2])
    tensor(indices=tensor([], size=(1, 0)),
           values=tensor([], size=(0, 2)),
           size=(1, 2), nnz=0, layout=torch.sparse_coo)

.. _torch.sparse: https://pytorch.org/docs/stable/sparse.html
""".format(**factory_common_args))

add_docstr(torch.sqrt,
           r"""
sqrt(input, *, out=None) -> Tensor

Returns a new tensor with the square-root of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \sqrt{\text{input}_{i}}
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-2.0755,  1.0226,  0.0831,  0.4806])
    >>> torch.sqrt(a)
    tensor([    nan,  1.0112,  0.2883,  0.6933])
""".format(**common_args))

add_docstr(torch.square,
           r"""
square(input, *, out=None) -> Tensor

Returns a new tensor with the square of the elements of :attr:`input`.

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-2.0755,  1.0226,  0.0831,  0.4806])
    >>> torch.square(a)
    tensor([ 4.3077,  1.0457,  0.0069,  0.2310])
""".format(**common_args))

add_docstr(torch.squeeze,
           r"""
squeeze(input, dim=None, *, out=None) -> Tensor

Returns a tensor with all the dimensions of :attr:`input` of size `1` removed.

For example, if `input` is of shape:
:math:`(A \times 1 \times B \times C \times 1 \times D)` then the `out` tensor
will be of shape: :math:`(A \times B \times C \times D)`.

When :attr:`dim` is given, a squeeze operation is done only in the given
dimension. If `input` is of shape: :math:`(A \times 1 \times B)`,
``squeeze(input, 0)`` leaves the tensor unchanged, but ``squeeze(input, 1)``
will squeeze the tensor to the shape :math:`(A \times B)`.

.. note:: The returned tensor shares the storage with the input tensor,
          so changing the contents of one will change the contents of the other.

.. warning:: If the tensor has a batch dimension of size 1, then `squeeze(input)`
          will also remove the batch dimension, which can lead to unexpected
          errors.

Args:
    {input}
    dim (int, optional): if given, the input will be squeezed only in
           this dimension

Keyword args:
    {out}

Example::

    >>> x = torch.zeros(2, 1, 2, 1, 2)
    >>> x.size()
    torch.Size([2, 1, 2, 1, 2])
    >>> y = torch.squeeze(x)
    >>> y.size()
    torch.Size([2, 2, 2])
    >>> y = torch.squeeze(x, 0)
    >>> y.size()
    torch.Size([2, 1, 2, 1, 2])
    >>> y = torch.squeeze(x, 1)
    >>> y.size()
    torch.Size([2, 2, 1, 2])
""".format(**common_args))

add_docstr(torch.std, r"""
std(input, unbiased=True) -> Tensor

Returns the standard-deviation of all elements in the :attr:`input` tensor.

If :attr:`unbiased` is ``False``, then the standard-deviation will be calculated
via the biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    unbiased (bool): whether to use the unbiased estimation or not

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[-0.8166, -1.3802, -0.3560]])
    >>> torch.std(a)
    tensor(0.5130)

.. function:: std(input, dim, unbiased=True, keepdim=False, *, out=None) -> Tensor

Returns the standard-deviation of each row of the :attr:`input` tensor in the
dimension :attr:`dim`. If :attr:`dim` is a list of dimensions,
reduce over all of them.

{keepdim_details}

If :attr:`unbiased` is ``False``, then the standard-deviation will be calculated
via the biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    {dim}
    unbiased (bool): whether to use the unbiased estimation or not
    {keepdim}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.2035,  1.2959,  1.8101, -0.4644],
            [ 1.5027, -0.3270,  0.5905,  0.6538],
            [-1.5745,  1.3330, -0.5596, -0.6548],
            [ 0.1264, -0.5080,  1.6420,  0.1992]])
    >>> torch.std(a, dim=1)
    tensor([ 1.0311,  0.7477,  1.2204,  0.9087])
""".format(**multi_dim_common))

add_docstr(torch.std_mean,
           r"""
std_mean(input, unbiased=True) -> (Tensor, Tensor)

Returns the standard-deviation and mean of all elements in the :attr:`input` tensor.

If :attr:`unbiased` is ``False``, then the standard-deviation will be calculated
via the biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    unbiased (bool): whether to use the unbiased estimation or not

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[0.3364, 0.3591, 0.9462]])
    >>> torch.std_mean(a)
    (tensor(0.3457), tensor(0.5472))

.. function:: std_mean(input, dim, unbiased=True, keepdim=False) -> (Tensor, Tensor)

Returns the standard-deviation and mean of each row of the :attr:`input` tensor in the
dimension :attr:`dim`. If :attr:`dim` is a list of dimensions,
reduce over all of them.

{keepdim_details}

If :attr:`unbiased` is ``False``, then the standard-deviation will be calculated
via the biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    {dim}
    unbiased (bool): whether to use the unbiased estimation or not
    {keepdim}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.5648, -0.5984, -1.2676, -1.4471],
            [ 0.9267,  1.0612,  1.1050, -0.6014],
            [ 0.0154,  1.9301,  0.0125, -1.0904],
            [-1.9711, -0.7748, -1.3840,  0.5067]])
    >>> torch.std_mean(a, 1)
    (tensor([0.9110, 0.8197, 1.2552, 1.0608]), tensor([-0.6871,  0.6229,  0.2169, -0.9058]))
""".format(**multi_dim_common))

add_docstr(torch.sub, r"""
sub(input, other, *, alpha=1, out=None) -> Tensor

Subtracts :attr:`other`, scaled by :attr:`alpha`, from :attr:`input`.

.. math::
    \text{{out}}_i = \text{{input}}_i - \text{{alpha}} \times \text{{other}}_i
""" + r"""

Supports :ref:`broadcasting to a common shape <broadcasting-semantics>`,
:ref:`type promotion <type-promotion-doc>`, and integer, float, and complex inputs.

Args:
    {input}
    other (Tensor or Scalar): the tensor or scalar to subtract from :attr:`input`

Keyword args:
    alpha (Scalar): the scalar multiplier for :attr:`other`
    {out}

Example::

    >>> a = torch.tensor((1, 2))
    >>> b = torch.tensor((0, 1))
    >>> torch.sub(a, b, alpha=2)
    tensor([1, 0])
""".format(**common_args))

add_docstr(torch.subtract, r"""
subtract(input, other, *, alpha=1, out=None) -> Tensor

Alias for :func:`torch.sub`.
""")

add_docstr(torch.sum,
           r"""
sum(input, *, dtype=None) -> Tensor

Returns the sum of all elements in the :attr:`input` tensor.

Args:
    {input}

Keyword args:
    {dtype}

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[ 0.1133, -0.9567,  0.2958]])
    >>> torch.sum(a)
    tensor(-0.5475)

.. function:: sum(input, dim, keepdim=False, *, dtype=None) -> Tensor

Returns the sum of each row of the :attr:`input` tensor in the given
dimension :attr:`dim`. If :attr:`dim` is a list of dimensions,
reduce over all of them.

{keepdim_details}

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    {dtype}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[ 0.0569, -0.2475,  0.0737, -0.3429],
            [-0.2993,  0.9138,  0.9337, -1.6864],
            [ 0.1132,  0.7892, -0.1003,  0.5688],
            [ 0.3637, -0.9906, -0.4752, -1.5197]])
    >>> torch.sum(a, 1)
    tensor([-0.4598, -0.1381,  1.3708, -2.6217])
    >>> b = torch.arange(4 * 5 * 6).view(4, 5, 6)
    >>> torch.sum(b, (2, 1))
    tensor([  435.,  1335.,  2235.,  3135.])
""".format(**multi_dim_common))

add_docstr(torch.nansum,
           r"""
nansum(input, *, dtype=None) -> Tensor

Returns the sum of all elements, treating Not a Numbers (NaNs) as zero.

Args:
    {input}

Keyword args:
    {dtype}

Example::

    >>> a = torch.tensor([1., 2., float('nan'), 4.])
    >>> torch.nansum(a)
    tensor(7.)

.. function:: nansum(input, dim, keepdim=False, *, dtype=None) -> Tensor

Returns the sum of each row of the :attr:`input` tensor in the given
dimension :attr:`dim`, treating Not a Numbers (NaNs) as zero.
If :attr:`dim` is a list of dimensions, reduce over all of them.

{keepdim_details}

Args:
    {input}
    {dim}
    {keepdim}

Keyword args:
    {dtype}

Example::

    >>> torch.nansum(torch.tensor([1., float("nan")]))
    1.0
    >>> a = torch.tensor([[1, 2], [3., float("nan")]])
    >>> torch.nansum(a)
    tensor(6.)
    >>> torch.nansum(a, dim=0)
    tensor([4., 2.])
    >>> torch.nansum(a, dim=1)
    tensor([3., 3.])
""".format(**multi_dim_common))

add_docstr(torch.svd,
           r"""
svd(input, some=True, compute_uv=True, *, out=None) -> (Tensor, Tensor, Tensor)

This function returns a namedtuple ``(U, S, V)`` which is the singular value
decomposition of a input real matrix or batches of real matrices :attr:`input` such that
:math:`input = U \times diag(S) \times V^T`.

If :attr:`some` is ``True`` (default), the method returns the reduced singular value decomposition
i.e., if the last two dimensions of :attr:`input` are ``m`` and ``n``, then the returned
`U` and `V` matrices will contain only :math:`min(n, m)` orthonormal columns.

If :attr:`compute_uv` is ``False``, the returned `U` and `V` matrices will be zero matrices
of shape :math:`(m \times m)` and :math:`(n \times n)` respectively. :attr:`some` will be ignored here.

.. note:: The singular values are returned in descending order. If :attr:`input` is a batch of matrices,
          then the singular values of each matrix in the batch is returned in descending order.

.. note:: The implementation of SVD on CPU uses the LAPACK routine `?gesdd` (a divide-and-conquer
          algorithm) instead of `?gesvd` for speed. Analogously, the SVD on GPU uses the MAGMA routine
          `gesdd` as well.

.. note:: Irrespective of the original strides, the returned matrix `U`
          will be transposed, i.e. with strides :code:`U.contiguous().transpose(-2, -1).stride()`

.. note:: Extra care needs to be taken when backward through `U` and `V`
          outputs. Such operation is really only stable when :attr:`input` is
          full rank with all distinct singular values. Otherwise, ``NaN`` can
          appear as the gradients are not properly defined. Also, notice that
          double backward will usually do an additional backward through `U` and
          `V` even if the original backward is only on `S`.

.. note:: When :attr:`some` = ``False``, the gradients on :code:`U[..., :, min(m, n):]`
          and :code:`V[..., :, min(m, n):]` will be ignored in backward as those vectors
          can be arbitrary bases of the subspaces.

.. note:: When :attr:`compute_uv` = ``False``, backward cannot be performed since `U` and `V`
          from the forward pass is required for the backward operation.

Args:
    input (Tensor): the input tensor of size :math:`(*, m, n)` where `*` is zero or more
                    batch dimensions consisting of :math:`m \times n` matrices.
    some (bool, optional): controls the shape of returned `U` and `V`
    compute_uv (bool, optional): option whether to compute `U` and `V` or not

Keyword args:
    out (tuple, optional): the output tuple of tensors

Example::

    >>> a = torch.randn(5, 3)
    >>> a
    tensor([[ 0.2364, -0.7752,  0.6372],
            [ 1.7201,  0.7394, -0.0504],
            [-0.3371, -1.0584,  0.5296],
            [ 0.3550, -0.4022,  1.5569],
            [ 0.2445, -0.0158,  1.1414]])
    >>> u, s, v = torch.svd(a)
    >>> u
    tensor([[ 0.4027,  0.0287,  0.5434],
            [-0.1946,  0.8833,  0.3679],
            [ 0.4296, -0.2890,  0.5261],
            [ 0.6604,  0.2717, -0.2618],
            [ 0.4234,  0.2481, -0.4733]])
    >>> s
    tensor([2.3289, 2.0315, 0.7806])
    >>> v
    tensor([[-0.0199,  0.8766,  0.4809],
            [-0.5080,  0.4054, -0.7600],
            [ 0.8611,  0.2594, -0.4373]])
    >>> torch.dist(a, torch.mm(torch.mm(u, torch.diag(s)), v.t()))
    tensor(8.6531e-07)
    >>> a_big = torch.randn(7, 5, 3)
    >>> u, s, v = torch.svd(a_big)
    >>> torch.dist(a_big, torch.matmul(torch.matmul(u, torch.diag_embed(s)), v.transpose(-2, -1)))
    tensor(2.6503e-06)
""")

add_docstr(torch.symeig,
           r"""
symeig(input, eigenvectors=False, upper=True, *, out=None) -> (Tensor, Tensor)

This function returns eigenvalues and eigenvectors
of a real symmetric matrix :attr:`input` or a batch of real symmetric matrices,
represented by a namedtuple (eigenvalues, eigenvectors).

This function calculates all eigenvalues (and vectors) of :attr:`input`
such that :math:`\text{input} = V \text{diag}(e) V^T`.

The boolean argument :attr:`eigenvectors` defines computation of
both eigenvectors and eigenvalues or eigenvalues only.

If it is ``False``, only eigenvalues are computed. If it is ``True``,
both eigenvalues and eigenvectors are computed.

Since the input matrix :attr:`input` is supposed to be symmetric,
only the upper triangular portion is used by default.

If :attr:`upper` is ``False``, then lower triangular portion is used.

.. note:: The eigenvalues are returned in ascending order. If :attr:`input` is a batch of matrices,
          then the eigenvalues of each matrix in the batch is returned in ascending order.

.. note:: Irrespective of the original strides, the returned matrix `V` will
          be transposed, i.e. with strides `V.contiguous().transpose(-1, -2).stride()`.

.. note:: Extra care needs to be taken when backward through outputs. Such
          operation is really only stable when all eigenvalues are distinct.
          Otherwise, ``NaN`` can appear as the gradients are not properly defined.

Args:
    input (Tensor): the input tensor of size :math:`(*, n, n)` where `*` is zero or more
                    batch dimensions consisting of symmetric matrices.
    eigenvectors(boolean, optional): controls whether eigenvectors have to be computed
    upper(boolean, optional): controls whether to consider upper-triangular or lower-triangular region

Keyword args:
    out (tuple, optional): the output tuple of (Tensor, Tensor)

Returns:
    (Tensor, Tensor): A namedtuple (eigenvalues, eigenvectors) containing

        - **eigenvalues** (*Tensor*): Shape :math:`(*, m)`. The eigenvalues in ascending order.
        - **eigenvectors** (*Tensor*): Shape :math:`(*, m, m)`.
          If ``eigenvectors=False``, it's an empty tensor.
          Otherwise, this tensor contains the orthonormal eigenvectors of the ``input``.

Examples::


    >>> a = torch.randn(5, 5)
    >>> a = a + a.t()  # To make a symmetric
    >>> a
    tensor([[-5.7827,  4.4559, -0.2344, -1.7123, -1.8330],
            [ 4.4559,  1.4250, -2.8636, -3.2100, -0.1798],
            [-0.2344, -2.8636,  1.7112, -5.5785,  7.1988],
            [-1.7123, -3.2100, -5.5785, -2.6227,  3.1036],
            [-1.8330, -0.1798,  7.1988,  3.1036, -5.1453]])
    >>> e, v = torch.symeig(a, eigenvectors=True)
    >>> e
    tensor([-13.7012,  -7.7497,  -2.3163,   5.2477,   8.1050])
    >>> v
    tensor([[ 0.1643,  0.9034, -0.0291,  0.3508,  0.1817],
            [-0.2417, -0.3071, -0.5081,  0.6534,  0.4026],
            [-0.5176,  0.1223, -0.0220,  0.3295, -0.7798],
            [-0.4850,  0.2695, -0.5773, -0.5840,  0.1337],
            [ 0.6415, -0.0447, -0.6381, -0.0193, -0.4230]])
    >>> a_big = torch.randn(5, 2, 2)
    >>> a_big = a_big + a_big.transpose(-2, -1)  # To make a_big symmetric
    >>> e, v = a_big.symeig(eigenvectors=True)
    >>> torch.allclose(torch.matmul(v, torch.matmul(e.diag_embed(), v.transpose(-2, -1))), a_big)
    True
""")

add_docstr(torch.t,
           r"""
t(input) -> Tensor

Expects :attr:`input` to be <= 2-D tensor and transposes dimensions 0
and 1.

0-D and 1-D tensors are returned as is. When input is a 2-D tensor this
is equivalent to ``transpose(input, 0, 1)``.

Args:
    {input}

Example::

    >>> x = torch.randn(())
    >>> x
    tensor(0.1995)
    >>> torch.t(x)
    tensor(0.1995)
    >>> x = torch.randn(3)
    >>> x
    tensor([ 2.4320, -0.4608,  0.7702])
    >>> torch.t(x)
    tensor([ 2.4320, -0.4608,  0.7702])
    >>> x = torch.randn(2, 3)
    >>> x
    tensor([[ 0.4875,  0.9158, -0.5872],
            [ 0.3938, -0.6929,  0.6932]])
    >>> torch.t(x)
    tensor([[ 0.4875,  0.3938],
            [ 0.9158, -0.6929],
            [-0.5872,  0.6932]])
""".format(**common_args))

add_docstr(torch.flip,
           r"""
flip(input, dims) -> Tensor

Reverse the order of a n-D tensor along given axis in dims.

Args:
    {input}
    dims (a list or tuple): axis to flip on

Example::

    >>> x = torch.arange(8).view(2, 2, 2)
    >>> x
    tensor([[[ 0,  1],
             [ 2,  3]],

            [[ 4,  5],
             [ 6,  7]]])
    >>> torch.flip(x, [0, 1])
    tensor([[[ 6,  7],
             [ 4,  5]],

            [[ 2,  3],
             [ 0,  1]]])
""".format(**common_args))

add_docstr(torch.fliplr,
           r"""
fliplr(input) -> Tensor

Flip array in the left/right direction, returning a new tensor.

Flip the entries in each row in the left/right direction.
Columns are preserved, but appear in a different order than before.

Note:
    Equivalent to input[:,::-1]. Requires the array to be at least 2-D.

Args:
    input (Tensor): Must be at least 2-dimensional.

Example::

    >>> x = torch.arange(4).view(2, 2)
    >>> x
    tensor([[0, 1],
            [2, 3]])
    >>> torch.fliplr(x)
    tensor([[1, 0],
            [3, 2]])
""".format(**common_args))

add_docstr(torch.flipud,
           r"""
flipud(input) -> Tensor

Flip array in the up/down direction, returning a new tensor.

Flip the entries in each column in the up/down direction.
Rows are preserved, but appear in a different order than before.

Note:
    Equivalent to input[::-1,...]. Requires the array to be at least 1-D.

Args:
    input (Tensor): Must be at least 1-dimensional.

Example::

    >>> x = torch.arange(4).view(2, 2)
    >>> x
    tensor([[0, 1],
            [2, 3]])
    >>> torch.flipud(x)
    tensor([[2, 3],
            [0, 1]])
""".format(**common_args))

add_docstr(torch.roll,
           r"""
roll(input, shifts, dims=None) -> Tensor

Roll the tensor along the given dimension(s). Elements that are shifted beyond the
last position are re-introduced at the first position. If a dimension is not
specified, the tensor will be flattened before rolling and then restored
to the original shape.

Args:
    {input}
    shifts (int or tuple of ints): The number of places by which the elements
        of the tensor are shifted. If shifts is a tuple, dims must be a tuple of
        the same size, and each dimension will be rolled by the corresponding
        value
    dims (int or tuple of ints): Axis along which to roll

Example::

    >>> x = torch.tensor([1, 2, 3, 4, 5, 6, 7, 8]).view(4, 2)
    >>> x
    tensor([[1, 2],
            [3, 4],
            [5, 6],
            [7, 8]])
    >>> torch.roll(x, 1, 0)
    tensor([[7, 8],
            [1, 2],
            [3, 4],
            [5, 6]])
    >>> torch.roll(x, -1, 0)
    tensor([[3, 4],
            [5, 6],
            [7, 8],
            [1, 2]])
    >>> torch.roll(x, shifts=(2, 1), dims=(0, 1))
    tensor([[6, 5],
            [8, 7],
            [2, 1],
            [4, 3]])
""".format(**common_args))

add_docstr(torch.rot90,
           r"""
rot90(input, k, dims) -> Tensor

Rotate a n-D tensor by 90 degrees in the plane specified by dims axis.
Rotation direction is from the first towards the second axis if k > 0, and from the second towards the first for k < 0.

Args:
    {input}
    k (int): number of times to rotate
    dims (a list or tuple): axis to rotate

Example::

    >>> x = torch.arange(4).view(2, 2)
    >>> x
    tensor([[0, 1],
            [2, 3]])
    >>> torch.rot90(x, 1, [0, 1])
    tensor([[1, 3],
            [0, 2]])

    >>> x = torch.arange(8).view(2, 2, 2)
    >>> x
    tensor([[[0, 1],
             [2, 3]],

            [[4, 5],
             [6, 7]]])
    >>> torch.rot90(x, 1, [1, 2])
    tensor([[[1, 3],
             [0, 2]],

            [[5, 7],
             [4, 6]]])
""".format(**common_args))

add_docstr(torch.take,
           r"""
take(input, index) -> Tensor

Returns a new tensor with the elements of :attr:`input` at the given indices.
The input tensor is treated as if it were viewed as a 1-D tensor. The result
takes the same shape as the indices.

Args:
    {input}
    indices (LongTensor): the indices into tensor

Example::

    >>> src = torch.tensor([[4, 3, 5],
                            [6, 7, 8]])
    >>> torch.take(src, torch.tensor([0, 2, 5]))
    tensor([ 4,  5,  8])
""".format(**common_args))

add_docstr(torch.tan,
           r"""
tan(input, *, out=None) -> Tensor

Returns a new tensor with the tangent of the elements of :attr:`input`.

.. math::
    \text{out}_{i} = \tan(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([-1.2027, -1.7687,  0.4412, -1.3856])
    >>> torch.tan(a)
    tensor([-2.5930,  4.9859,  0.4722, -5.3366])
""".format(**common_args))

add_docstr(torch.tanh,
           r"""
tanh(input, *, out=None) -> Tensor

Returns a new tensor with the hyperbolic tangent of the elements
of :attr:`input`.

.. math::
    \text{out}_{i} = \tanh(\text{input}_{i})
""" + r"""
Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 0.8986, -0.7279,  1.1745,  0.2611])
    >>> torch.tanh(a)
    tensor([ 0.7156, -0.6218,  0.8257,  0.2553])
""".format(**common_args))

add_docstr(torch.topk,
           r"""
topk(input, k, dim=None, largest=True, sorted=True, *, out=None) -> (Tensor, LongTensor)

Returns the :attr:`k` largest elements of the given :attr:`input` tensor along
a given dimension.

If :attr:`dim` is not given, the last dimension of the `input` is chosen.

If :attr:`largest` is ``False`` then the `k` smallest elements are returned.

A namedtuple of `(values, indices)` is returned, where the `indices` are the indices
of the elements in the original `input` tensor.

The boolean option :attr:`sorted` if ``True``, will make sure that the returned
`k` elements are themselves sorted

Args:
    {input}
    k (int): the k in "top-k"
    dim (int, optional): the dimension to sort along
    largest (bool, optional): controls whether to return largest or
           smallest elements
    sorted (bool, optional): controls whether to return the elements
           in sorted order

Keyword args:
    out (tuple, optional): the output tuple of (Tensor, LongTensor) that can be
        optionally given to be used as output buffers

Example::

    >>> x = torch.arange(1., 6.)
    >>> x
    tensor([ 1.,  2.,  3.,  4.,  5.])
    >>> torch.topk(x, 3)
    torch.return_types.topk(values=tensor([5., 4., 3.]), indices=tensor([4, 3, 2]))
""".format(**common_args))

add_docstr(torch.trace,
           r"""
trace(input) -> Tensor

Returns the sum of the elements of the diagonal of the input 2-D matrix.

Example::

    >>> x = torch.arange(1., 10.).view(3, 3)
    >>> x
    tensor([[ 1.,  2.,  3.],
            [ 4.,  5.,  6.],
            [ 7.,  8.,  9.]])
    >>> torch.trace(x)
    tensor(15.)
""")

add_docstr(torch.transpose,
           r"""
transpose(input, dim0, dim1) -> Tensor

Returns a tensor that is a transposed version of :attr:`input`.
The given dimensions :attr:`dim0` and :attr:`dim1` are swapped.

The resulting :attr:`out` tensor shares it's underlying storage with the
:attr:`input` tensor, so changing the content of one would change the content
of the other.

Args:
    {input}
    dim0 (int): the first dimension to be transposed
    dim1 (int): the second dimension to be transposed

Example::

    >>> x = torch.randn(2, 3)
    >>> x
    tensor([[ 1.0028, -0.9893,  0.5809],
            [-0.1669,  0.7299,  0.4942]])
    >>> torch.transpose(x, 0, 1)
    tensor([[ 1.0028, -0.1669],
            [-0.9893,  0.7299],
            [ 0.5809,  0.4942]])
""".format(**common_args))

add_docstr(torch.triangular_solve,
           r"""
triangular_solve(input, A, upper=True, transpose=False, unitriangular=False) -> (Tensor, Tensor)

Solves a system of equations with a triangular coefficient matrix :math:`A`
and multiple right-hand sides :math:`b`.

In particular, solves :math:`AX = b` and assumes :math:`A` is upper-triangular
with the default keyword arguments.

`torch.triangular_solve(b, A)` can take in 2D inputs `b, A` or inputs that are
batches of 2D matrices. If the inputs are batches, then returns
batched outputs `X`

Args:
    input (Tensor): multiple right-hand sides of size :math:`(*, m, k)` where
                :math:`*` is zero of more batch dimensions (:math:`b`)
    A (Tensor): the input triangular coefficient matrix of size :math:`(*, m, m)`
                where :math:`*` is zero or more batch dimensions
    upper (bool, optional): whether to solve the upper-triangular system
        of equations (default) or the lower-triangular system of equations. Default: ``True``.
    transpose (bool, optional): whether :math:`A` should be transposed before
        being sent into the solver. Default: ``False``.
    unitriangular (bool, optional): whether :math:`A` is unit triangular.
        If True, the diagonal elements of :math:`A` are assumed to be
        1 and not referenced from :math:`A`. Default: ``False``.

Returns:
    A namedtuple `(solution, cloned_coefficient)` where `cloned_coefficient`
    is a clone of :math:`A` and `solution` is the solution :math:`X` to :math:`AX = b`
    (or whatever variant of the system of equations, depending on the keyword arguments.)

Examples::

    >>> A = torch.randn(2, 2).triu()
    >>> A
    tensor([[ 1.1527, -1.0753],
            [ 0.0000,  0.7986]])
    >>> b = torch.randn(2, 3)
    >>> b
    tensor([[-0.0210,  2.3513, -1.5492],
            [ 1.5429,  0.7403, -1.0243]])
    >>> torch.triangular_solve(b, A)
    torch.return_types.triangular_solve(
    solution=tensor([[ 1.7841,  2.9046, -2.5405],
            [ 1.9320,  0.9270, -1.2826]]),
    cloned_coefficient=tensor([[ 1.1527, -1.0753],
            [ 0.0000,  0.7986]]))
""")

add_docstr(torch.tril,
           r"""
tril(input, diagonal=0, *, out=None) -> Tensor

Returns the lower triangular part of the matrix (2-D tensor) or batch of matrices
:attr:`input`, the other elements of the result tensor :attr:`out` are set to 0.

The lower triangular part of the matrix is defined as the elements on and
below the diagonal.

The argument :attr:`diagonal` controls which diagonal to consider. If
:attr:`diagonal` = 0, all elements on and below the main diagonal are
retained. A positive value includes just as many diagonals above the main
diagonal, and similarly a negative value excludes just as many diagonals below
the main diagonal. The main diagonal are the set of indices
:math:`\lbrace (i, i) \rbrace` for :math:`i \in [0, \min\{d_{1}, d_{2}\} - 1]` where
:math:`d_{1}, d_{2}` are the dimensions of the matrix.
""" + r"""
Args:
    {input}
    diagonal (int, optional): the diagonal to consider

Keyword args:
    {out}

Example::

    >>> a = torch.randn(3, 3)
    >>> a
    tensor([[-1.0813, -0.8619,  0.7105],
            [ 0.0935,  0.1380,  2.2112],
            [-0.3409, -0.9828,  0.0289]])
    >>> torch.tril(a)
    tensor([[-1.0813,  0.0000,  0.0000],
            [ 0.0935,  0.1380,  0.0000],
            [-0.3409, -0.9828,  0.0289]])

    >>> b = torch.randn(4, 6)
    >>> b
    tensor([[ 1.2219,  0.5653, -0.2521, -0.2345,  1.2544,  0.3461],
            [ 0.4785, -0.4477,  0.6049,  0.6368,  0.8775,  0.7145],
            [ 1.1502,  3.2716, -1.1243, -0.5413,  0.3615,  0.6864],
            [-0.0614, -0.7344, -1.3164, -0.7648, -1.4024,  0.0978]])
    >>> torch.tril(b, diagonal=1)
    tensor([[ 1.2219,  0.5653,  0.0000,  0.0000,  0.0000,  0.0000],
            [ 0.4785, -0.4477,  0.6049,  0.0000,  0.0000,  0.0000],
            [ 1.1502,  3.2716, -1.1243, -0.5413,  0.0000,  0.0000],
            [-0.0614, -0.7344, -1.3164, -0.7648, -1.4024,  0.0000]])
    >>> torch.tril(b, diagonal=-1)
    tensor([[ 0.0000,  0.0000,  0.0000,  0.0000,  0.0000,  0.0000],
            [ 0.4785,  0.0000,  0.0000,  0.0000,  0.0000,  0.0000],
            [ 1.1502,  3.2716,  0.0000,  0.0000,  0.0000,  0.0000],
            [-0.0614, -0.7344, -1.3164,  0.0000,  0.0000,  0.0000]])
""".format(**common_args))

# docstr is split in two parts to avoid format mis-captureing :math: braces '{}'
# as common args.
add_docstr(torch.tril_indices,
           r"""
tril_indices(row, col, offset=0, *, dtype=torch.long, device='cpu', layout=torch.strided) -> Tensor

Returns the indices of the lower triangular part of a :attr:`row`-by-
:attr:`col` matrix in a 2-by-N Tensor, where the first row contains row
coordinates of all indices and the second row contains column coordinates.
Indices are ordered based on rows and then columns.

The lower triangular part of the matrix is defined as the elements on and
below the diagonal.

The argument :attr:`offset` controls which diagonal to consider. If
:attr:`offset` = 0, all elements on and below the main diagonal are
retained. A positive value includes just as many diagonals above the main
diagonal, and similarly a negative value excludes just as many diagonals below
the main diagonal. The main diagonal are the set of indices
:math:`\lbrace (i, i) \rbrace` for :math:`i \in [0, \min\{d_{1}, d_{2}\} - 1]`
where :math:`d_{1}, d_{2}` are the dimensions of the matrix.

.. note::
    When running on CUDA, ``row * col`` must be less than :math:`2^{59}` to
    prevent overflow during calculation.
""" + r"""
Args:
    row (``int``): number of rows in the 2-D matrix.
    col (``int``): number of columns in the 2-D matrix.
    offset (``int``): diagonal offset from the main diagonal.
        Default: if not provided, 0.

Keyword args:
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        Default: if ``None``, ``torch.long``.
    {device}
    layout (:class:`torch.layout`, optional): currently only support ``torch.strided``.

Example::
    >>> a = torch.tril_indices(3, 3)
    >>> a
    tensor([[0, 1, 1, 2, 2, 2],
            [0, 0, 1, 0, 1, 2]])

    >>> a = torch.tril_indices(4, 3, -1)
    >>> a
    tensor([[1, 2, 2, 3, 3, 3],
            [0, 0, 1, 0, 1, 2]])

    >>> a = torch.tril_indices(4, 3, 1)
    >>> a
    tensor([[0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3],
            [0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2]])
""".format(**factory_common_args))

add_docstr(torch.triu,
           r"""
triu(input, diagonal=0, *, out=None) -> Tensor

Returns the upper triangular part of a matrix (2-D tensor) or batch of matrices
:attr:`input`, the other elements of the result tensor :attr:`out` are set to 0.

The upper triangular part of the matrix is defined as the elements on and
above the diagonal.

The argument :attr:`diagonal` controls which diagonal to consider. If
:attr:`diagonal` = 0, all elements on and above the main diagonal are
retained. A positive value excludes just as many diagonals above the main
diagonal, and similarly a negative value includes just as many diagonals below
the main diagonal. The main diagonal are the set of indices
:math:`\lbrace (i, i) \rbrace` for :math:`i \in [0, \min\{d_{1}, d_{2}\} - 1]` where
:math:`d_{1}, d_{2}` are the dimensions of the matrix.
""" + r"""
Args:
    {input}
    diagonal (int, optional): the diagonal to consider

Keyword args:
    {out}

Example::

    >>> a = torch.randn(3, 3)
    >>> a
    tensor([[ 0.2309,  0.5207,  2.0049],
            [ 0.2072, -1.0680,  0.6602],
            [ 0.3480, -0.5211, -0.4573]])
    >>> torch.triu(a)
    tensor([[ 0.2309,  0.5207,  2.0049],
            [ 0.0000, -1.0680,  0.6602],
            [ 0.0000,  0.0000, -0.4573]])
    >>> torch.triu(a, diagonal=1)
    tensor([[ 0.0000,  0.5207,  2.0049],
            [ 0.0000,  0.0000,  0.6602],
            [ 0.0000,  0.0000,  0.0000]])
    >>> torch.triu(a, diagonal=-1)
    tensor([[ 0.2309,  0.5207,  2.0049],
            [ 0.2072, -1.0680,  0.6602],
            [ 0.0000, -0.5211, -0.4573]])

    >>> b = torch.randn(4, 6)
    >>> b
    tensor([[ 0.5876, -0.0794, -1.8373,  0.6654,  0.2604,  1.5235],
            [-0.2447,  0.9556, -1.2919,  1.3378, -0.1768, -1.0857],
            [ 0.4333,  0.3146,  0.6576, -1.0432,  0.9348, -0.4410],
            [-0.9888,  1.0679, -1.3337, -1.6556,  0.4798,  0.2830]])
    >>> torch.triu(b, diagonal=1)
    tensor([[ 0.0000, -0.0794, -1.8373,  0.6654,  0.2604,  1.5235],
            [ 0.0000,  0.0000, -1.2919,  1.3378, -0.1768, -1.0857],
            [ 0.0000,  0.0000,  0.0000, -1.0432,  0.9348, -0.4410],
            [ 0.0000,  0.0000,  0.0000,  0.0000,  0.4798,  0.2830]])
    >>> torch.triu(b, diagonal=-1)
    tensor([[ 0.5876, -0.0794, -1.8373,  0.6654,  0.2604,  1.5235],
            [-0.2447,  0.9556, -1.2919,  1.3378, -0.1768, -1.0857],
            [ 0.0000,  0.3146,  0.6576, -1.0432,  0.9348, -0.4410],
            [ 0.0000,  0.0000, -1.3337, -1.6556,  0.4798,  0.2830]])
""".format(**common_args))

# docstr is split in two parts to avoid format mis-capturing :math: braces '{}'
# as common args.
add_docstr(torch.triu_indices,
           r"""
triu_indices(row, col, offset=0, *, dtype=torch.long, device='cpu', layout=torch.strided) -> Tensor

Returns the indices of the upper triangular part of a :attr:`row` by
:attr:`col` matrix in a 2-by-N Tensor, where the first row contains row
coordinates of all indices and the second row contains column coordinates.
Indices are ordered based on rows and then columns.

The upper triangular part of the matrix is defined as the elements on and
above the diagonal.

The argument :attr:`offset` controls which diagonal to consider. If
:attr:`offset` = 0, all elements on and above the main diagonal are
retained. A positive value excludes just as many diagonals above the main
diagonal, and similarly a negative value includes just as many diagonals below
the main diagonal. The main diagonal are the set of indices
:math:`\lbrace (i, i) \rbrace` for :math:`i \in [0, \min\{d_{1}, d_{2}\} - 1]`
where :math:`d_{1}, d_{2}` are the dimensions of the matrix.

.. note::
    When running on CUDA, ``row * col`` must be less than :math:`2^{59}` to
    prevent overflow during calculation.
""" + r"""
Args:
    row (``int``): number of rows in the 2-D matrix.
    col (``int``): number of columns in the 2-D matrix.
    offset (``int``): diagonal offset from the main diagonal.
        Default: if not provided, 0.

Keyword args:
    dtype (:class:`torch.dtype`, optional): the desired data type of returned tensor.
        Default: if ``None``, ``torch.long``.
    {device}
    layout (:class:`torch.layout`, optional): currently only support ``torch.strided``.

Example::
    >>> a = torch.triu_indices(3, 3)
    >>> a
    tensor([[0, 0, 0, 1, 1, 2],
            [0, 1, 2, 1, 2, 2]])

    >>> a = torch.triu_indices(4, 3, -1)
    >>> a
    tensor([[0, 0, 0, 1, 1, 1, 2, 2, 3],
            [0, 1, 2, 0, 1, 2, 1, 2, 2]])

    >>> a = torch.triu_indices(4, 3, 1)
    >>> a
    tensor([[0, 0, 1],
            [1, 2, 2]])
""".format(**factory_common_args))

add_docstr(torch.true_divide, r"""
true_divide(dividend, divisor, *, out) -> Tensor

Alias for :func:`torch.div`.
""".format(**common_args))

add_docstr(torch.trunc,
           r"""
trunc(input, *, out=None) -> Tensor

Returns a new tensor with the truncated integer values of
the elements of :attr:`input`.

Args:
    {input}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4)
    >>> a
    tensor([ 3.4742,  0.5466, -0.8008, -0.9079])
    >>> torch.trunc(a)
    tensor([ 3.,  0., -0., -0.])
""".format(**common_args))

add_docstr(torch.fix,
           r"""
fix(input, *, out=None) -> Tensor

Alias for :func:`torch.trunc`
""".format(**common_args))

add_docstr(torch.unsqueeze,
           r"""
unsqueeze(input, dim) -> Tensor

Returns a new tensor with a dimension of size one inserted at the
specified position.

The returned tensor shares the same underlying data with this tensor.

A :attr:`dim` value within the range ``[-input.dim() - 1, input.dim() + 1)``
can be used. Negative :attr:`dim` will correspond to :meth:`unsqueeze`
applied at :attr:`dim` = ``dim + input.dim() + 1``.

Args:
    {input}
    dim (int): the index at which to insert the singleton dimension

Example::

    >>> x = torch.tensor([1, 2, 3, 4])
    >>> torch.unsqueeze(x, 0)
    tensor([[ 1,  2,  3,  4]])
    >>> torch.unsqueeze(x, 1)
    tensor([[ 1],
            [ 2],
            [ 3],
            [ 4]])
""".format(**common_args))

add_docstr(torch.var, r"""
var(input, unbiased=True) -> Tensor

Returns the variance of all elements in the :attr:`input` tensor.

If :attr:`unbiased` is ``False``, then the variance will be calculated via the
biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    unbiased (bool): whether to use the unbiased estimation or not

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[-0.3425, -1.2636, -0.4864]])
    >>> torch.var(a)
    tensor(0.2455)


.. function:: var(input, dim, unbiased=True, keepdim=False, *, out=None) -> Tensor

Returns the variance of each row of the :attr:`input` tensor in the given
dimension :attr:`dim`.

{keepdim_details}

If :attr:`unbiased` is ``False``, then the variance will be calculated via the
biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    {dim}
    unbiased (bool): whether to use the unbiased estimation or not
    {keepdim}

Keyword args:
    {out}

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[-0.3567,  1.7385, -1.3042,  0.7423],
            [ 1.3436, -0.1015, -0.9834, -0.8438],
            [ 0.6056,  0.1089, -0.3112, -1.4085],
            [-0.7700,  0.6074, -0.1469,  0.7777]])
    >>> torch.var(a, 1)
    tensor([ 1.7444,  1.1363,  0.7356,  0.5112])
""".format(**multi_dim_common))

add_docstr(torch.var_mean,
           r"""
var_mean(input, unbiased=True) -> (Tensor, Tensor)

Returns the variance and mean of all elements in the :attr:`input` tensor.

If :attr:`unbiased` is ``False``, then the variance will be calculated via the
biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    unbiased (bool): whether to use the unbiased estimation or not

Example::

    >>> a = torch.randn(1, 3)
    >>> a
    tensor([[0.0146, 0.4258, 0.2211]])
    >>> torch.var_mean(a)
    (tensor(0.0423), tensor(0.2205))

.. function:: var_mean(input, dim, keepdim=False, unbiased=True) -> (Tensor, Tensor)

Returns the variance and mean of each row of the :attr:`input` tensor in the given
dimension :attr:`dim`.

{keepdim_details}

If :attr:`unbiased` is ``False``, then the variance will be calculated via the
biased estimator. Otherwise, Bessel's correction will be used.

Args:
    {input}
    {dim}
    {keepdim}
    unbiased (bool): whether to use the unbiased estimation or not

Example::

    >>> a = torch.randn(4, 4)
    >>> a
    tensor([[-1.5650,  2.0415, -0.1024, -0.5790],
            [ 0.2325, -2.6145, -1.6428, -0.3537],
            [-0.2159, -1.1069,  1.2882, -1.3265],
            [-0.6706, -1.5893,  0.6827,  1.6727]])
    >>> torch.var_mean(a, 1)
    (tensor([2.3174, 1.6403, 1.4092, 2.0791]), tensor([-0.0512, -1.0946, -0.3403,  0.0239]))
""".format(**multi_dim_common))

add_docstr(torch.zeros,
           r"""
zeros(*size, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Returns a tensor filled with the scalar value `0`, with the shape defined
by the variable argument :attr:`size`.

Args:
    size (int...): a sequence of integers defining the shape of the output tensor.
        Can be a variable number of arguments or a collection like a list or tuple.

Keyword args:
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.zeros(2, 3)
    tensor([[ 0.,  0.,  0.],
            [ 0.,  0.,  0.]])

    >>> torch.zeros(5)
    tensor([ 0.,  0.,  0.,  0.,  0.])
""".format(**factory_common_args))

add_docstr(torch.zeros_like,
           r"""
zeros_like(input, *, dtype=None, layout=None, device=None, requires_grad=False, memory_format=torch.preserve_format) -> Tensor

Returns a tensor filled with the scalar value `0`, with the same size as
:attr:`input`. ``torch.zeros_like(input)`` is equivalent to
``torch.zeros(input.size(), dtype=input.dtype, layout=input.layout, device=input.device)``.

.. warning::
    As of 0.4, this function does not support an :attr:`out` keyword. As an alternative,
    the old ``torch.zeros_like(input, out=output)`` is equivalent to
    ``torch.zeros(input.size(), out=output)``.

Args:
    {input}

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}

Example::

    >>> input = torch.empty(2, 3)
    >>> torch.zeros_like(input)
    tensor([[ 0.,  0.,  0.],
            [ 0.,  0.,  0.]])
""".format(**factory_like_common_args))

add_docstr(torch.empty,
           r"""
empty(*size, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False, pin_memory=False) -> Tensor

Returns a tensor filled with uninitialized data. The shape of the tensor is
defined by the variable argument :attr:`size`.

Args:
    size (int...): a sequence of integers defining the shape of the output tensor.
        Can be a variable number of arguments or a collection like a list or tuple.

Keyword args:
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {pin_memory}
    {memory_format}

Example::

    >>> torch.empty(2, 3)
    tensor(1.00000e-08 *
           [[ 6.3984,  0.0000,  0.0000],
            [ 0.0000,  0.0000,  0.0000]])

""".format(**factory_common_args))

add_docstr(torch.empty_like,
           r"""
empty_like(input, *, dtype=None, layout=None, device=None, requires_grad=False, memory_format=torch.preserve_format) -> Tensor

Returns an uninitialized tensor with the same size as :attr:`input`.
``torch.empty_like(input)`` is equivalent to
``torch.empty(input.size(), dtype=input.dtype, layout=input.layout, device=input.device)``.

Args:
    {input}

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}

Example::

    >>> torch.empty((2,3), dtype=torch.int64)
    tensor([[ 9.4064e+13,  2.8000e+01,  9.3493e+13],
            [ 7.5751e+18,  7.1428e+18,  7.5955e+18]])
""".format(**factory_like_common_args))

add_docstr(torch.empty_strided,
           r"""
empty_strided(size, stride, *, dtype=None, layout=None, device=None, requires_grad=False, pin_memory=False) -> Tensor

Returns a tensor filled with uninitialized data. The shape and strides of the tensor is
defined by the variable argument :attr:`size` and :attr:`stride` respectively.
``torch.empty_strided(size, stride)`` is equivalent to
``torch.empty(size).as_strided(size, stride)``.

.. warning::
    More than one element of the created 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.

Args:
    size (tuple of ints): the shape of the output tensor
    stride (tuple of ints): the strides of the output tensor

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {pin_memory}

Example::

    >>> a = torch.empty_strided((2, 3), (1, 2))
    >>> a
    tensor([[8.9683e-44, 4.4842e-44, 5.1239e+07],
            [0.0000e+00, 0.0000e+00, 3.0705e-41]])
    >>> a.stride()
    (1, 2)
    >>> a.size()
    torch.Size([2, 3])
""".format(**factory_common_args))

add_docstr(torch.full, r"""
full(size, fill_value, *, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False) -> Tensor

Creates a tensor of size :attr:`size` filled with :attr:`fill_value`. The
tensor's dtype is inferred from :attr:`fill_value`.

Args:
    size (int...): a list, tuple, or :class:`torch.Size` of integers defining the
        shape of the output tensor.
    fill_value (Scalar): the value to fill the output tensor with.

Keyword args:
    {out}
    {dtype}
    {layout}
    {device}
    {requires_grad}

Example::

    >>> torch.full((2, 3), 3.141592)
    tensor([[ 3.1416,  3.1416,  3.1416],
            [ 3.1416,  3.1416,  3.1416]])
""".format(**factory_common_args))

add_docstr(torch.full_like,
           """
full_like(input, fill_value, \\*, out=None, dtype=None, layout=torch.strided, device=None, requires_grad=False, \
memory_format=torch.preserve_format) -> Tensor

Returns a tensor with the same size as :attr:`input` filled with :attr:`fill_value`.
``torch.full_like(input, fill_value)`` is equivalent to
``torch.full(input.size(), fill_value, dtype=input.dtype, layout=input.layout, device=input.device)``.

Args:
    {input}
    fill_value: the number to fill the output tensor with.

Keyword args:
    {dtype}
    {layout}
    {device}
    {requires_grad}
    {memory_format}
""".format(**factory_like_common_args))

add_docstr(torch.det,
           r"""
det(input) -> Tensor

Calculates determinant of a square matrix or batches of square matrices.

.. note::
    Backward through :meth:`det` internally uses SVD results when :attr:`input` is
    not invertible. In this case, double backward through :meth:`det` will be
    unstable in when :attr:`input` doesn't have distinct singular values. See
    :meth:`~torch.svd` for details.

Arguments:
    input (Tensor): the input tensor of size ``(*, n, n)`` where ``*`` is zero or more
                batch dimensions.

Example::

    >>> A = torch.randn(3, 3)
    >>> torch.det(A)
    tensor(3.7641)

    >>> A = torch.randn(3, 2, 2)
    >>> A
    tensor([[[ 0.9254, -0.6213],
             [-0.5787,  1.6843]],

            [[ 0.3242, -0.9665],
             [ 0.4539, -0.0887]],

            [[ 1.1336, -0.4025],
             [-0.7089,  0.9032]]])
    >>> A.det()
    tensor([1.1990, 0.4099, 0.7386])
""")

add_docstr(torch.where,
           r"""
where(condition, x, y) -> Tensor

Return a tensor of elements selected from either :attr:`x` or :attr:`y`, depending on :attr:`condition`.

The operation is defined as:

.. math::
    \text{out}_i = \begin{cases}
        \text{x}_i & \text{if } \text{condition}_i \\
        \text{y}_i & \text{otherwise} \\
    \end{cases}

.. note::
    The tensors :attr:`condition`, :attr:`x`, :attr:`y` must be :ref:`broadcastable <broadcasting-semantics>`.

.. note::
    Currently valid scalar and tensor combination are
    1. Scalar of floating dtype and torch.double
    2. Scalar of integral dtype and torch.long
    3. Scalar of complex dtype and torch.complex128

Arguments:
    condition (BoolTensor): When True (nonzero), yield x, otherwise yield y
    x (Tensor or Scalar): value (if :attr:x is a scalar) or values selected at indices
                          where :attr:`condition` is ``True``
    y (Tensor or Scalar): value (if :attr:x is a scalar) or values selected at indices
                          where :attr:`condition` is ``False``

Returns:
    Tensor: A tensor of shape equal to the broadcasted shape of :attr:`condition`, :attr:`x`, :attr:`y`

Example::

    >>> x = torch.randn(3, 2)
    >>> y = torch.ones(3, 2)
    >>> x
    tensor([[-0.4620,  0.3139],
            [ 0.3898, -0.7197],
            [ 0.0478, -0.1657]])
    >>> torch.where(x > 0, x, y)
    tensor([[ 1.0000,  0.3139],
            [ 0.3898,  1.0000],
            [ 0.0478,  1.0000]])
    >>> x = torch.randn(2, 2, dtype=torch.double)
    >>> x
    tensor([[ 1.0779,  0.0383],
            [-0.8785, -1.1089]], dtype=torch.float64)
    >>> torch.where(x > 0, x, 0.)
    tensor([[1.0779, 0.0383],
            [0.0000, 0.0000]], dtype=torch.float64)

.. function:: where(condition) -> tuple of LongTensor

``torch.where(condition)`` is identical to
``torch.nonzero(condition, as_tuple=True)``.

.. note::
    See also :func:`torch.nonzero`.
""")

add_docstr(torch.logdet,
           r"""
logdet(input) -> Tensor

Calculates log determinant of a square matrix or batches of square matrices.

.. note::
    Result is ``-inf`` if :attr:`input` has zero log determinant, and is ``nan`` if
    :attr:`input` has negative determinant.

.. note::
    Backward through :meth:`logdet` internally uses SVD results when :attr:`input`
    is not invertible. In this case, double backward through :meth:`logdet` will
    be unstable in when :attr:`input` doesn't have distinct singular values. See
    :meth:`~torch.svd` for details.

Arguments:
    input (Tensor): the input tensor of size ``(*, n, n)`` where ``*`` is zero or more
                batch dimensions.

Example::

    >>> A = torch.randn(3, 3)
    >>> torch.det(A)
    tensor(0.2611)
    >>> torch.logdet(A)
    tensor(-1.3430)
    >>> A
    tensor([[[ 0.9254, -0.6213],
             [-0.5787,  1.6843]],

            [[ 0.3242, -0.9665],
             [ 0.4539, -0.0887]],

            [[ 1.1336, -0.4025],
             [-0.7089,  0.9032]]])
    >>> A.det()
    tensor([1.1990, 0.4099, 0.7386])
    >>> A.det().log()
    tensor([ 0.1815, -0.8917, -0.3031])
""")

add_docstr(torch.slogdet,
           r"""
slogdet(input) -> (Tensor, Tensor)

Calculates the sign and log absolute value of the determinant(s) of a square matrix or batches of square matrices.

.. note::
    If ``input`` has zero determinant, this returns ``(0, -inf)``.

.. note::
    Backward through :meth:`slogdet` internally uses SVD results when :attr:`input`
    is not invertible. In this case, double backward through :meth:`slogdet`
    will be unstable in when :attr:`input` doesn't have distinct singular values.
    See :meth:`~torch.svd` for details.

Arguments:
    input (Tensor): the input tensor of size ``(*, n, n)`` where ``*`` is zero or more
                batch dimensions.

Returns:
    A namedtuple (sign, logabsdet) containing the sign of the determinant, and the log
    value of the absolute determinant.

Example::

    >>> A = torch.randn(3, 3)
    >>> A
    tensor([[ 0.0032, -0.2239, -1.1219],
            [-0.6690,  0.1161,  0.4053],
            [-1.6218, -0.9273, -0.0082]])
    >>> torch.det(A)
    tensor(-0.7576)
    >>> torch.logdet(A)
    tensor(nan)
    >>> torch.slogdet(A)
    torch.return_types.slogdet(sign=tensor(-1.), logabsdet=tensor(-0.2776))
""")

add_docstr(torch.pinverse,
           r"""
pinverse(input, rcond=1e-15) -> Tensor

Calculates the pseudo-inverse (also known as the Moore-Penrose inverse) of a 2D tensor.
Please look at `Moore-Penrose inverse`_ for more details

.. note::
    This method is implemented using the Singular Value Decomposition.

.. note::
    The pseudo-inverse is not necessarily a continuous function in the elements of the matrix `[1]`_.
    Therefore, derivatives are not always existent, and exist for a constant rank only `[2]`_.
    However, this method is backprop-able due to the implementation by using SVD results, and
    could be unstable. Double-backward will also be unstable due to the usage of SVD internally.
    See :meth:`~torch.svd` for more details.

.. note::
    Supports real and complex inputs.
    Batched version for complex inputs is only supported on the CPU.

Arguments:
    input (Tensor): The input tensor of size :math:`(*, m, n)` where :math:`*` is zero or more batch dimensions
    rcond (float): A floating point value to determine the cutoff for small singular values.
                   Default: 1e-15

Returns:
    The pseudo-inverse of :attr:`input` of dimensions :math:`(*, n, m)`

Example::

    >>> input = torch.randn(3, 5)
    >>> input
    tensor([[ 0.5495,  0.0979, -1.4092, -0.1128,  0.4132],
            [-1.1143, -0.3662,  0.3042,  1.6374, -0.9294],
            [-0.3269, -0.5745, -0.0382, -0.5922, -0.6759]])
    >>> torch.pinverse(input)
    tensor([[ 0.0600, -0.1933, -0.2090],
            [-0.0903, -0.0817, -0.4752],
            [-0.7124, -0.1631, -0.2272],
            [ 0.1356,  0.3933, -0.5023],
            [-0.0308, -0.1725, -0.5216]])
    >>> # Batched pinverse example
    >>> a = torch.randn(2,6,3)
    >>> b = torch.pinverse(a)
    >>> torch.matmul(b, a)
    tensor([[[ 1.0000e+00,  1.6391e-07, -1.1548e-07],
            [ 8.3121e-08,  1.0000e+00, -2.7567e-07],
            [ 3.5390e-08,  1.4901e-08,  1.0000e+00]],

            [[ 1.0000e+00, -8.9407e-08,  2.9802e-08],
            [-2.2352e-07,  1.0000e+00,  1.1921e-07],
            [ 0.0000e+00,  8.9407e-08,  1.0000e+00]]])

.. _Moore-Penrose inverse: https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse

.. _[1]: https://epubs.siam.org/doi/10.1137/0117004

.. _[2]: https://www.jstor.org/stable/2156365
""")

add_docstr(torch.fft, r"""
fft(input, signal_ndim, normalized=False) -> Tensor

Complex-to-complex Discrete Fourier Transform.

.. warning::
    The function :func:`torch.fft` is deprecated and will be removed in
    PyTorch 1.8. Use the new :ref:`torch.fft <torch-fft-module>` module
    functions, instead, by importing :ref:`torch.fft <torch-fft-module>` and
    calling :func:`torch.fft.fft` or :func:`torch.fft.fftn`.

This method computes the complex-to-complex discrete Fourier transform.
Ignoring the batch dimensions, it computes the following expression:

.. math::
    X[\omega_1, \dots, \omega_d] =
        \sum_{n_1=0}^{N_1-1} \dots \sum_{n_d=0}^{N_d-1} x[n_1, \dots, n_d]
         e^{-j\ 2 \pi \sum_{i=0}^d \frac{\omega_i n_i}{N_i}},

where :math:`d` = :attr:`signal_ndim` is number of dimensions for the
signal, and :math:`N_i` is the size of signal dimension :math:`i`.

This method supports 1D, 2D and 3D complex-to-complex transforms, indicated
by :attr:`signal_ndim`. :attr:`input` must be a tensor with last dimension
of size 2, representing the real and imaginary components of complex
numbers, and should have at least ``signal_ndim + 1`` dimensions with optionally
arbitrary number of leading batch dimensions. If :attr:`normalized` is set to
``True``, this normalizes the result by dividing it with
:math:`\sqrt{\prod_{i=1}^K N_i}` so that the operator is unitary.

Returns the real and the imaginary parts together as one tensor of the same
shape of :attr:`input`.

The inverse of this function is :func:`~torch.ifft`.

.. note::
    For CUDA tensors, an LRU cache is used for cuFFT plans to speed up
    repeatedly running FFT methods on tensors of same geometry with same
    configuration. See :ref:`cufft-plan-cache` for more details on how to
    monitor and control the cache.

.. warning::
    If the torch.fft module is imported then "torch.fft" will refer to the
    module and not this function. Use :meth:`torch.Tensor.fft` instead.

.. warning::
    Due to limited dynamic range of half datatype, performing this operation in half
    precision may cause the first element of result to overflow for certain inputs.

.. warning::
    For CPU tensors, this method is currently only available with MKL. Use
    :func:`torch.backends.mkl.is_available` to check if MKL is installed.

Arguments:
    input (Tensor): the input tensor of at least :attr:`signal_ndim` ``+ 1``
        dimensions
    signal_ndim (int): the number of dimensions in each signal.
        :attr:`signal_ndim` can only be 1, 2 or 3
    normalized (bool, optional): controls whether to return normalized results.
        Default: ``False``

Returns:
    Tensor: A tensor containing the complex-to-complex Fourier transform result

Example::

    >>> # unbatched 2D FFT
    >>> x = torch.randn(4, 3, 2)
    >>> torch.fft(x, 2)
    tensor([[[-0.0876,  1.7835],
             [-2.0399, -2.9754],
             [ 4.4773, -5.0119]],

            [[-1.5716,  2.7631],
             [-3.8846,  5.2652],
             [ 0.2046, -0.7088]],

            [[ 1.9938, -0.5901],
             [ 6.5637,  6.4556],
             [ 2.9865,  4.9318]],

            [[ 7.0193,  1.1742],
             [-1.3717, -2.1084],
             [ 2.0289,  2.9357]]])
    >>> # batched 1D FFT
    >>> torch.fft(x, 1)
    tensor([[[ 1.8385,  1.2827],
             [-0.1831,  1.6593],
             [ 2.4243,  0.5367]],

            [[-0.9176, -1.5543],
             [-3.9943, -2.9860],
             [ 1.2838, -2.9420]],

            [[-0.8854, -0.6860],
             [ 2.4450,  0.0808],
             [ 1.3076, -0.5768]],

            [[-0.1231,  2.7411],
             [-0.3075, -1.7295],
             [-0.5384, -2.0299]]])
    >>> # arbitrary number of batch dimensions, 2D FFT
    >>> x = torch.randn(3, 3, 5, 5, 2)
    >>> y = torch.fft(x, 2)
    >>> y.shape
    torch.Size([3, 3, 5, 5, 2])

""")

add_docstr(torch.ifft, r"""
ifft(input, signal_ndim, normalized=False) -> Tensor

Complex-to-complex Inverse Discrete Fourier Transform.

.. warning::
    The function :func:`torch.ifft` is deprecated and will be removed in a
    future PyTorch release. Use the new :ref:`torch.fft <torch-fft-module>`
    module functions, instead, by importing :ref:`torch.fft <torch-fft-module>`
    and calling :func:`torch.fft.ifft` or :func:`torch.fft.ifftn`.

This method computes the complex-to-complex inverse discrete Fourier
transform. Ignoring the batch dimensions, it computes the following
expression:

.. math::
    X[\omega_1, \dots, \omega_d] =
        \frac{1}{\prod_{i=1}^d N_i} \sum_{n_1=0}^{N_1-1} \dots \sum_{n_d=0}^{N_d-1} x[n_1, \dots, n_d]
         e^{\ j\ 2 \pi \sum_{i=0}^d \frac{\omega_i n_i}{N_i}},

where :math:`d` = :attr:`signal_ndim` is number of dimensions for the
signal, and :math:`N_i` is the size of signal dimension :math:`i`.

The argument specifications are almost identical with :func:`~torch.fft`.
However, if :attr:`normalized` is set to ``True``, this instead returns the
results multiplied by :math:`\sqrt{\prod_{i=1}^d N_i}`, to become a unitary
operator. Therefore, to invert a :func:`~torch.fft`, the :attr:`normalized`
argument should be set identically for :func:`~torch.fft`.

Returns the real and the imaginary parts together as one tensor of the same
shape of :attr:`input`.

The inverse of this function is :func:`~torch.fft`.

.. note::
    For CUDA tensors, an LRU cache is used for cuFFT plans to speed up
    repeatedly running FFT methods on tensors of same geometry with same
    configuration. See :ref:`cufft-plan-cache` for more details on how to
    monitor and control the cache.

.. warning::
    Due to limited dynamic range of half datatype, performing this operation in half
    precision may cause the first element of result to overflow for certain inputs.

.. warning::
    For CPU tensors, this method is currently only available with MKL. Use
    :func:`torch.backends.mkl.is_available` to check if MKL is installed.

Arguments:
    input (Tensor): the input tensor of at least :attr:`signal_ndim` ``+ 1``
        dimensions
    signal_ndim (int): the number of dimensions in each signal.
        :attr:`signal_ndim` can only be 1, 2 or 3
    normalized (bool, optional): controls whether to return normalized results.
        Default: ``False``

Returns:
    Tensor: A tensor containing the complex-to-complex inverse Fourier transform result

Example::

    >>> x = torch.randn(3, 3, 2)
    >>> x
    tensor([[[ 1.2766,  1.3680],
             [-0.8337,  2.0251],
             [ 0.9465, -1.4390]],

            [[-0.1890,  1.6010],
             [ 1.1034, -1.9230],
             [-0.9482,  1.0775]],

            [[-0.7708, -0.8176],
             [-0.1843, -0.2287],
             [-1.9034, -0.2196]]])
    >>> y = torch.fft(x, 2)
    >>> torch.ifft(y, 2)  # recover x
    tensor([[[ 1.2766,  1.3680],
             [-0.8337,  2.0251],
             [ 0.9465, -1.4390]],

            [[-0.1890,  1.6010],
             [ 1.1034, -1.9230],
             [-0.9482,  1.0775]],

            [[-0.7708, -0.8176],
             [-0.1843, -0.2287],
             [-1.9034, -0.2196]]])

""")

add_docstr(torch.rfft, r"""
rfft(input, signal_ndim, normalized=False, onesided=True) -> Tensor

Real-to-complex Discrete Fourier Transform.

.. warning::
    The function :func:`torch.rfft` is deprecated and will be removed in a
    future PyTorch release. Use the new :ref:`torch.fft <torch-fft-module>`
    module functions, instead, by importing :ref:`torch.fft <torch-fft-module>`
    and calling :func:`torch.fft.rfft` for one-sided output, or
    :func:`torch.fft.fft` for two-sided output.

This method computes the real-to-complex discrete Fourier transform. It is
mathematically equivalent with :func:`~torch.fft` with differences only in
formats of the input and output.

This method supports 1D, 2D and 3D real-to-complex transforms, indicated
by :attr:`signal_ndim`. :attr:`input` must be a tensor with at least
``signal_ndim`` dimensions with optionally arbitrary number of leading batch
dimensions. If :attr:`normalized` is set to ``True``, this normalizes the result
by dividing it with :math:`\sqrt{\prod_{i=1}^K N_i}` so that the operator is
unitary, where :math:`N_i` is the size of signal dimension :math:`i`.

The real-to-complex Fourier transform results follow conjugate symmetry:

.. math::
    X[\omega_1, \dots, \omega_d] = X^*[N_1 - \omega_1, \dots, N_d - \omega_d],

where the index arithmetic is computed modulus the size of the corresponding
dimension, :math:`\ ^*` is the conjugate operator, and
:math:`d` = :attr:`signal_ndim`. :attr:`onesided` flag controls whether to avoid
redundancy in the output results. If set to ``True`` (default), the output will
not be full complex result of shape :math:`(*, 2)`, where :math:`*` is the shape
of :attr:`input`, but instead the last dimension will be halfed as of size
:math:`\lfloor \frac{N_d}{2} \rfloor + 1`.

The inverse of this function is :func:`~torch.irfft`.

.. note::
    For CUDA tensors, an LRU cache is used for cuFFT plans to speed up
    repeatedly running FFT methods on tensors of same geometry with same
    configuration. See :ref:`cufft-plan-cache` for more details on how to
    monitor and control the cache.

.. warning::
    Due to limited dynamic range of half datatype, performing this operation in half
    precision may cause the first element of result to overflow for certain inputs.

.. warning::
    For CPU tensors, this method is currently only available with MKL. Use
    :func:`torch.backends.mkl.is_available` to check if MKL is installed.

Arguments:
    input (Tensor): the input tensor of at least :attr:`signal_ndim` dimensions
    signal_ndim (int): the number of dimensions in each signal.
        :attr:`signal_ndim` can only be 1, 2 or 3
    normalized (bool, optional): controls whether to return normalized results.
        Default: ``False``
    onesided (bool, optional): controls whether to return half of results to
        avoid redundancy. Default: ``True``

Returns:
    Tensor: A tensor containing the real-to-complex Fourier transform result

Example::

    >>> x = torch.randn(5, 5)
    >>> torch.rfft(x, 2).shape
    torch.Size([5, 3, 2])
    >>> torch.rfft(x, 2, onesided=False).shape
    torch.Size([5, 5, 2])

""")


add_docstr(torch.irfft, r"""
irfft(input, signal_ndim, normalized=False, onesided=True, signal_sizes=None) -> Tensor

Complex-to-real Inverse Discrete Fourier Transform.

.. warning::
    The function :func:`torch.irfft` is deprecated and will be removed in a
    future PyTorch release. Use the new :ref:`torch.fft <torch-fft-module>`
    module functions, instead, by importing :ref:`torch.fft <torch-fft-module>`
    and calling :func:`torch.fft.irfft` for one-sided input, or
    :func:`torch.fft.ifft` for two-sided input.

This method computes the complex-to-real inverse discrete Fourier transform.
It is mathematically equivalent with :func:`ifft` with differences only in
formats of the input and output.

The argument specifications are almost identical with :func:`~torch.ifft`.
Similar to :func:`~torch.ifft`, if :attr:`normalized` is set to ``True``,
this normalizes the result by multiplying it with
:math:`\sqrt{\prod_{i=1}^K N_i}` so that the operator is unitary, where
:math:`N_i` is the size of signal dimension :math:`i`.

.. note::
    Due to the conjugate symmetry, :attr:`input` do not need to contain the full
    complex frequency values. Roughly half of the values will be sufficient, as
    is the case when :attr:`input` is given by :func:`~torch.rfft` with
    ``rfft(signal, onesided=True)``. In such case, set the :attr:`onesided`
    argument of this method to ``True``. Moreover, the original signal shape
    information can sometimes be lost, optionally set :attr:`signal_sizes` to be
    the size of the original signal (without the batch dimensions if in batched
    mode) to recover it with correct shape.

    Therefore, to invert an :func:`~torch.rfft`, the :attr:`normalized` and
    :attr:`onesided` arguments should be set identically for :func:`~torch.irfft`,
    and preferably a :attr:`signal_sizes` is given to avoid size mismatch. See the
    example below for a case of size mismatch.

    See :func:`~torch.rfft` for details on conjugate symmetry.

The inverse of this function is :func:`~torch.rfft`.

.. warning::
    Generally speaking, input to this function should contain values
    following conjugate symmetry. Note that even if :attr:`onesided` is
    ``True``, often symmetry on some part is still needed. When this
    requirement is not satisfied, the behavior of :func:`~torch.irfft` is
    undefined. Since :func:`torch.autograd.gradcheck` estimates numerical
    Jacobian with point perturbations, :func:`~torch.irfft` will almost
    certainly fail the check.

.. note::
    For CUDA tensors, an LRU cache is used for cuFFT plans to speed up
    repeatedly running FFT methods on tensors of same geometry with same
    configuration. See :ref:`cufft-plan-cache` for more details on how to
    monitor and control the cache.

.. warning::
    Due to limited dynamic range of half datatype, performing this operation in half
    precision may cause the first element of result to overflow for certain inputs.

.. warning::
    For CPU tensors, this method is currently only available with MKL. Use
    :func:`torch.backends.mkl.is_available` to check if MKL is installed.

Arguments:
    input (Tensor): the input tensor of at least :attr:`signal_ndim` ``+ 1``
        dimensions
    signal_ndim (int): the number of dimensions in each signal.
        :attr:`signal_ndim` can only be 1, 2 or 3
    normalized (bool, optional): controls whether to return normalized results.
        Default: ``False``
    onesided (bool, optional): controls whether :attr:`input` was halfed to avoid
        redundancy, e.g., by :func:`rfft`. Default: ``True``
    signal_sizes (list or :class:`torch.Size`, optional): the size of the original
        signal (without batch dimension). Default: ``None``

Returns:
    Tensor: A tensor containing the complex-to-real inverse Fourier transform result

Example::

    >>> x = torch.randn(4, 4)
    >>> torch.rfft(x, 2, onesided=True).shape
    torch.Size([4, 3, 2])
    >>>
    >>> # notice that with onesided=True, output size does not determine the original signal size
    >>> x = torch.randn(4, 5)

    >>> torch.rfft(x, 2, onesided=True).shape
    torch.Size([4, 3, 2])
    >>>
    >>> # now we use the original shape to recover x
    >>> x
    tensor([[-0.8992,  0.6117, -1.6091, -0.4155, -0.8346],
            [-2.1596, -0.0853,  0.7232,  0.1941, -0.0789],
            [-2.0329,  1.1031,  0.6869, -0.5042,  0.9895],
            [-0.1884,  0.2858, -1.5831,  0.9917, -0.8356]])
    >>> y = torch.rfft(x, 2, onesided=True)
    >>> torch.irfft(y, 2, onesided=True, signal_sizes=x.shape)  # recover x
    tensor([[-0.8992,  0.6117, -1.6091, -0.4155, -0.8346],
            [-2.1596, -0.0853,  0.7232,  0.1941, -0.0789],
            [-2.0329,  1.1031,  0.6869, -0.5042,  0.9895],
            [-0.1884,  0.2858, -1.5831,  0.9917, -0.8356]])

""")


add_docstr(torch.hann_window,
           """
hann_window(window_length, periodic=True, *, dtype=None, \
layout=torch.strided, device=None, requires_grad=False) -> Tensor
""" + r"""
Hann window function.

.. math::
    w[n] = \frac{1}{2}\ \left[1 - \cos \left( \frac{2 \pi n}{N - 1} \right)\right] =
            \sin^2 \left( \frac{\pi n}{N - 1} \right),

where :math:`N` is the full window size.

The input :attr:`window_length` is a positive integer controlling the
returned window size. :attr:`periodic` flag determines whether the returned
window trims off the last duplicate value from the symmetric window and is
ready to be used as a periodic window with functions like
:meth:`torch.stft`. Therefore, if :attr:`periodic` is true, the :math:`N` in
above formula is in fact :math:`\text{window\_length} + 1`. Also, we always have
``torch.hann_window(L, periodic=True)`` equal to
``torch.hann_window(L + 1, periodic=False)[:-1])``.

.. note::
    If :attr:`window_length` :math:`=1`, the returned window contains a single value 1.
""" + r"""
Arguments:
    window_length (int): the size of returned window
    periodic (bool, optional): If True, returns a window to be used as periodic
        function. If False, return a symmetric window.

Keyword args:
    {dtype} Only floating point types are supported.
    layout (:class:`torch.layout`, optional): the desired layout of returned window tensor. Only
          ``torch.strided`` (dense layout) is supported.
    {device}
    {requires_grad}

Returns:
    Tensor: A 1-D tensor of size :math:`(\text{{window\_length}},)` containing the window

""".format(**factory_common_args))


add_docstr(torch.hamming_window,
           """
hamming_window(window_length, periodic=True, alpha=0.54, beta=0.46, *, dtype=None, \
layout=torch.strided, device=None, requires_grad=False) -> Tensor
""" + r"""
Hamming window function.

.. math::
    w[n] = \alpha - \beta\ \cos \left( \frac{2 \pi n}{N - 1} \right),

where :math:`N` is the full window size.

The input :attr:`window_length` is a positive integer controlling the
returned window size. :attr:`periodic` flag determines whether the returned
window trims off the last duplicate value from the symmetric window and is
ready to be used as a periodic window with functions like
:meth:`torch.stft`. Therefore, if :attr:`periodic` is true, the :math:`N` in
above formula is in fact :math:`\text{window\_length} + 1`. Also, we always have
``torch.hamming_window(L, periodic=True)`` equal to
``torch.hamming_window(L + 1, periodic=False)[:-1])``.

.. note::
    If :attr:`window_length` :math:`=1`, the returned window contains a single value 1.

.. note::
    This is a generalized version of :meth:`torch.hann_window`.
""" + r"""
Arguments:
    window_length (int): the size of returned window
    periodic (bool, optional): If True, returns a window to be used as periodic
        function. If False, return a symmetric window.
    alpha (float, optional): The coefficient :math:`\alpha` in the equation above
    beta (float, optional): The coefficient :math:`\beta` in the equation above

Keyword args:
    {dtype} Only floating point types are supported.
    layout (:class:`torch.layout`, optional): the desired layout of returned window tensor. Only
          ``torch.strided`` (dense layout) is supported.
    {device}
    {requires_grad}

Returns:
    Tensor: A 1-D tensor of size :math:`(\text{{window\_length}},)` containing the window

""".format(**factory_common_args))


add_docstr(torch.bartlett_window,
           """
bartlett_window(window_length, periodic=True, *, dtype=None, \
layout=torch.strided, device=None, requires_grad=False) -> Tensor
""" + r"""
Bartlett window function.

.. math::
    w[n] = 1 - \left| \frac{2n}{N-1} - 1 \right| = \begin{cases}
        \frac{2n}{N - 1} & \text{if } 0 \leq n \leq \frac{N - 1}{2} \\
        2 - \frac{2n}{N - 1} & \text{if } \frac{N - 1}{2} < n < N \\
    \end{cases},

where :math:`N` is the full window size.

The input :attr:`window_length` is a positive integer controlling the
returned window size. :attr:`periodic` flag determines whether the returned
window trims off the last duplicate value from the symmetric window and is
ready to be used as a periodic window with functions like
:meth:`torch.stft`. Therefore, if :attr:`periodic` is true, the :math:`N` in
above formula is in fact :math:`\text{window\_length} + 1`. Also, we always have
``torch.bartlett_window(L, periodic=True)`` equal to
``torch.bartlett_window(L + 1, periodic=False)[:-1])``.

.. note::
    If :attr:`window_length` :math:`=1`, the returned window contains a single value 1.
""" + r"""
Arguments:
    window_length (int): the size of returned window
    periodic (bool, optional): If True, returns a window to be used as periodic
        function. If False, return a symmetric window.

Keyword args:
    {dtype} Only floating point types are supported.
    layout (:class:`torch.layout`, optional): the desired layout of returned window tensor. Only
          ``torch.strided`` (dense layout) is supported.
    {device}
    {requires_grad}

Returns:
    Tensor: A 1-D tensor of size :math:`(\text{{window\_length}},)` containing the window

""".format(**factory_common_args))


add_docstr(torch.blackman_window,
           """
blackman_window(window_length, periodic=True, *, dtype=None, \
layout=torch.strided, device=None, requires_grad=False) -> Tensor
""" + r"""
Blackman window function.

.. math::
    w[n] = 0.42 - 0.5 \cos \left( \frac{2 \pi n}{N - 1} \right) + 0.08 \cos \left( \frac{4 \pi n}{N - 1} \right)

where :math:`N` is the full window size.

The input :attr:`window_length` is a positive integer controlling the
returned window size. :attr:`periodic` flag determines whether the returned
window trims off the last duplicate value from the symmetric window and is
ready to be used as a periodic window with functions like
:meth:`torch.stft`. Therefore, if :attr:`periodic` is true, the :math:`N` in
above formula is in fact :math:`\text{window\_length} + 1`. Also, we always have
``torch.blackman_window(L, periodic=True)`` equal to
``torch.blackman_window(L + 1, periodic=False)[:-1])``.

.. note::
    If :attr:`window_length` :math:`=1`, the returned window contains a single value 1.
""" + r"""
Arguments:
    window_length (int): the size of returned window
    periodic (bool, optional): If True, returns a window to be used as periodic
        function. If False, return a symmetric window.

Keyword args:
    {dtype} Only floating point types are supported.
    layout (:class:`torch.layout`, optional): the desired layout of returned window tensor. Only
          ``torch.strided`` (dense layout) is supported.
    {device}
    {requires_grad}

Returns:
    Tensor: A 1-D tensor of size :math:`(\text{{window\_length}},)` containing the window

""".format(**factory_common_args))


add_docstr(torch.kaiser_window, """
kaiser_window(window_length, periodic=True, beta=12.0, *, dtype=None, \
layout=torch.strided, device=None, requires_grad=False) -> Tensor
""" + r"""
Computes the Kaiser window with window length :attr:`window_length` and shape parameter :attr:`beta`.

Let I_0 be the zeroth order modified Bessel function of the first kind (see :func:`torch.i0`) and
``N = L - 1`` if :attr:`periodic` is False and ``L`` if :attr:`periodic` is True,
where ``L`` is the :attr:`window_length`. This function computes:

.. math::
    out_i = I_0 \left( \beta \sqrt{1 - \left( {\frac{i - N/2}{N/2}} \right) ^2 } \right) / I_0( \beta )

Calling ``torch.kaiser_window(L, B, periodic=True)`` is equivalent to calling
``torch.kaiser_window(L + 1, B, periodic=False)[:-1])``.
The :attr:`periodic` argument is intended as a helpful shorthand
to produce a periodic window as input to functions like :func:`torch.stft`.

.. note::
    If :attr:`window_length` is one, then the returned window is a single element tensor containing a one.

""" + r"""
Args:
    window_length (int): length of the window.
    periodic (bool, optional): If True, returns a periodic window suitable for use in spectral analysis.
        If False, returns a symmetric window suitable for use in filter design.
    beta (float, optional): shape parameter for the window.

Keyword args:
    {dtype}
    layout (:class:`torch.layout`, optional): the desired layout of returned window tensor. Only
          ``torch.strided`` (dense layout) is supported.
    {device}
    {requires_grad}

""".format(**factory_common_args))


add_docstr(torch.vander,
           """
vander(x, N=None, increasing=False) -> Tensor
""" + r"""
Generates a Vandermonde matrix.

The columns of the output matrix are elementwise powers of the input vector :math:`x^{{(N-1)}}, x^{{(N-2)}}, ..., x^0`.
If increasing is True, the order of the columns is reversed :math:`x^0, x^1, ..., x^{{(N-1)}}`. Such a
matrix with a geometric progression in each row is named for Alexandre-Theophile Vandermonde.

Arguments:
    x (Tensor): 1-D input tensor.
    N (int, optional): Number of columns in the output. If N is not specified,
        a square array is returned :math:`(N = len(x))`.
    increasing (bool, optional): Order of the powers of the columns. If True,
        the powers increase from left to right, if False (the default) they are reversed.

Returns:
    Tensor: Vandermonde matrix. If increasing is False, the first column is :math:`x^{{(N-1)}}`,
    the second :math:`x^{{(N-2)}}` and so forth. If increasing is True, the columns
    are :math:`x^0, x^1, ..., x^{{(N-1)}}`.

Example::

    >>> x = torch.tensor([1, 2, 3, 5])
    >>> torch.vander(x)
    tensor([[  1,   1,   1,   1],
            [  8,   4,   2,   1],
            [ 27,   9,   3,   1],
            [125,  25,   5,   1]])
    >>> torch.vander(x, N=3)
    tensor([[ 1,  1,  1],
            [ 4,  2,  1],
            [ 9,  3,  1],
            [25,  5,  1]])
    >>> torch.vander(x, N=3, increasing=True)
    tensor([[ 1,  1,  1],
            [ 1,  2,  4],
            [ 1,  3,  9],
            [ 1,  5, 25]])

""".format(**factory_common_args))


add_docstr(torch.unbind,
           r"""
unbind(input, dim=0) -> seq

Removes a tensor dimension.

Returns a tuple of all slices along a given dimension, already without it.

Arguments:
    input (Tensor): the tensor to unbind
    dim (int): dimension to remove

Example::

    >>> torch.unbind(torch.tensor([[1, 2, 3],
    >>>                            [4, 5, 6],
    >>>                            [7, 8, 9]]))
    (tensor([1, 2, 3]), tensor([4, 5, 6]), tensor([7, 8, 9]))
""")


add_docstr(torch.combinations,
           r"""
combinations(input, r=2, with_replacement=False) -> seq

Compute combinations of length :math:`r` of the given tensor. The behavior is similar to
python's `itertools.combinations` when `with_replacement` is set to `False`, and
`itertools.combinations_with_replacement` when `with_replacement` is set to `True`.

Arguments:
    input (Tensor): 1D vector.
    r (int, optional): number of elements to combine
    with_replacement (boolean, optional): whether to allow duplication in combination

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

Example::

    >>> a = [1, 2, 3]
    >>> list(itertools.combinations(a, r=2))
    [(1, 2), (1, 3), (2, 3)]
    >>> list(itertools.combinations(a, r=3))
    [(1, 2, 3)]
    >>> list(itertools.combinations_with_replacement(a, r=2))
    [(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)]
    >>> tensor_a = torch.tensor(a)
    >>> torch.combinations(tensor_a)
    tensor([[1, 2],
            [1, 3],
            [2, 3]])
    >>> torch.combinations(tensor_a, r=3)
    tensor([[1, 2, 3]])
    >>> torch.combinations(tensor_a, with_replacement=True)
    tensor([[1, 1],
            [1, 2],
            [1, 3],
            [2, 2],
            [2, 3],
            [3, 3]])
""")

add_docstr(torch.trapz,
           r"""
trapz(y, x, *, dim=-1) -> Tensor

Estimate :math:`\int y\,dx` along `dim`, using the trapezoid rule.

Arguments:
    y (Tensor): The values of the function to integrate
    x (Tensor): The points at which the function `y` is sampled.
        If `x` is not in ascending order, intervals on which it is decreasing
        contribute negatively to the estimated integral (i.e., the convention
        :math:`\int_a^b f = -\int_b^a f` is followed).
    dim (int): The dimension along which to integrate.
        By default, use the last dimension.

Returns:
    A Tensor with the same shape as the input, except with `dim` removed.
    Each element of the returned tensor represents the estimated integral
    :math:`\int y\,dx` along `dim`.

Example::

    >>> y = torch.randn((2, 3))
    >>> y
    tensor([[-2.1156,  0.6857, -0.2700],
            [-1.2145,  0.5540,  2.0431]])
    >>> x = torch.tensor([[1, 3, 4], [1, 2, 3]])
    >>> torch.trapz(y, x)
    tensor([-1.2220,  0.9683])

.. function:: trapz(y, *, dx=1, dim=-1) -> Tensor

As above, but the sample points are spaced uniformly at a distance of `dx`.

Arguments:
    y (Tensor): The values of the function to integrate
    dx (float): The distance between points at which `y` is sampled.
    dim (int): The dimension along which to integrate.
        By default, use the last dimension.

Returns:
    A Tensor with the same shape as the input, except with `dim` removed.
    Each element of the returned tensor represents the estimated integral
    :math:`\int y\,dx` along `dim`.
""")

add_docstr(torch.repeat_interleave,
           r"""
repeat_interleave(input, repeats, dim=None) -> Tensor

Repeat elements of a tensor.

.. warning::

    This is different from :meth:`torch.Tensor.repeat` but similar to ``numpy.repeat``.

Args:
    {input}
    repeats (Tensor or int): The number of repetitions for each element.
        repeats is broadcasted to fit the shape of the given axis.
    dim (int, optional): The dimension along which to repeat values.
        By default, use the flattened input array, and return a flat output
        array.

Returns:
    Tensor: Repeated tensor which has the same shape as input, except along the
     given axis.

Example::

    >>> x = torch.tensor([1, 2, 3])
    >>> x.repeat_interleave(2)
    tensor([1, 1, 2, 2, 3, 3])
    >>> y = torch.tensor([[1, 2], [3, 4]])
    >>> torch.repeat_interleave(y, 2)
    tensor([1, 1, 2, 2, 3, 3, 4, 4])
    >>> torch.repeat_interleave(y, 3, dim=1)
    tensor([[1, 1, 1, 2, 2, 2],
            [3, 3, 3, 4, 4, 4]])
    >>> torch.repeat_interleave(y, torch.tensor([1, 2]), dim=0)
    tensor([[1, 2],
            [3, 4],
            [3, 4]])

.. function:: repeat_interleave(repeats) -> Tensor

If the `repeats` is `tensor([n1, n2, n3, ...])`, then the output will be
`tensor([0, 0, ..., 1, 1, ..., 2, 2, ..., ...])` where `0` appears `n1` times,
`1` appears `n2` times, `2` appears `n3` times, etc.
""".format(**common_args))


add_docstr(torch.quantize_per_tensor,
           r"""
quantize_per_tensor(input, scale, zero_point, dtype) -> Tensor

Converts a float tensor to a quantized tensor with given scale and zero point.

Arguments:
    input (Tensor): float tensor to quantize
    scale (float): scale to apply in quantization formula
    zero_point (int): offset in integer value that maps to float zero
    dtype (:class:`torch.dtype`): the desired data type of returned tensor.
        Has to be one of the quantized dtypes: ``torch.quint8``, ``torch.qint8``, ``torch.qint32``

Returns:
    Tensor: A newly quantized tensor

Example::

    >>> torch.quantize_per_tensor(torch.tensor([-1.0, 0.0, 1.0, 2.0]), 0.1, 10, torch.quint8)
    tensor([-1.,  0.,  1.,  2.], size=(4,), dtype=torch.quint8,
           quantization_scheme=torch.per_tensor_affine, scale=0.1, zero_point=10)
    >>> torch.quantize_per_tensor(torch.tensor([-1.0, 0.0, 1.0, 2.0]), 0.1, 10, torch.quint8).int_repr()
    tensor([ 0, 10, 20, 30], dtype=torch.uint8)
""")

add_docstr(torch.quantize_per_channel,
           r"""
quantize_per_channel(input, scales, zero_points, axis, dtype) -> Tensor

Converts a float tensor to a per-channel quantized tensor with given scales and zero points.

Arguments:
    input (Tensor): float tensor to quantize
    scales (Tensor): float 1D tensor of scales to use, size should match ``input.size(axis)``
    zero_points (int): integer 1D tensor of offset to use, size should match ``input.size(axis)``
    axis (int): dimension on which apply per-channel quantization
    dtype (:class:`torch.dtype`): the desired data type of returned tensor.
        Has to be one of the quantized dtypes: ``torch.quint8``, ``torch.qint8``, ``torch.qint32``

Returns:
    Tensor: A newly quantized tensor

Example::

    >>> x = torch.tensor([[-1.0, 0.0], [1.0, 2.0]])
    >>> torch.quantize_per_channel(x, torch.tensor([0.1, 0.01]), torch.tensor([10, 0]), 0, torch.quint8)
    tensor([[-1.,  0.],
            [ 1.,  2.]], size=(2, 2), dtype=torch.quint8,
           quantization_scheme=torch.per_channel_affine,
           scale=tensor([0.1000, 0.0100], dtype=torch.float64),
           zero_point=tensor([10,  0]), axis=0)
    >>> torch.quantize_per_channel(x, torch.tensor([0.1, 0.01]), torch.tensor([10, 0]), 0, torch.quint8).int_repr()
    tensor([[  0,  10],
            [100, 200]], dtype=torch.uint8)
""")

add_docstr(torch.Generator,
           r"""
Generator(device='cpu') -> Generator

Creates and returns a generator object that manages the state of the algorithm which
produces pseudo random numbers. Used as a keyword argument in many :ref:`inplace-random-sampling`
functions.

Arguments:
    device (:class:`torch.device`, optional): the desired device for the generator.

Returns:
    Generator: An torch.Generator object.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cuda = torch.Generator(device='cuda')
""")


add_docstr(torch.Generator.set_state,
           r"""
Generator.set_state(new_state) -> void

Sets the Generator state.

Arguments:
    new_state (torch.ByteTensor): The desired state.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cpu_other = torch.Generator()
    >>> g_cpu.set_state(g_cpu_other.get_state())
""")


add_docstr(torch.Generator.get_state,
           r"""
Generator.get_state() -> Tensor

Returns the Generator state as a ``torch.ByteTensor``.

Returns:
    Tensor: A ``torch.ByteTensor`` which contains all the necessary bits
    to restore a Generator to a specific point in time.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cpu.get_state()
""")


add_docstr(torch.Generator.manual_seed,
           r"""
Generator.manual_seed(seed) -> Generator

Sets the seed for generating random numbers. Returns a `torch.Generator` object.
It is recommended to set a large seed, i.e. a number that has a good balance of 0
and 1 bits. Avoid having many 0 bits in the seed.

Arguments:
    seed (int): The desired seed. Value must be within the inclusive range
        `[-0x8000_0000_0000_0000, 0xffff_ffff_ffff_ffff]`. Otherwise, a RuntimeError
        is raised. Negative inputs are remapped to positive values with the formula
        `0xffff_ffff_ffff_ffff + seed`.

Returns:
    Generator: An torch.Generator object.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cpu.manual_seed(2147483647)
""")


add_docstr(torch.Generator.initial_seed,
           r"""
Generator.initial_seed() -> int

Returns the initial seed for generating random numbers.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cpu.initial_seed()
    2147483647
""")


add_docstr(torch.Generator.seed,
           r"""
Generator.seed() -> int

Gets a non-deterministic random number from std::random_device or the current
time and uses it to seed a Generator.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cpu.seed()
    1516516984916
""")


add_docstr(torch.Generator.device,
           r"""
Generator.device -> device

Gets the current device of the generator.

Example::

    >>> g_cpu = torch.Generator()
    >>> g_cpu.device
    device(type='cpu')
""")

add_docstr(torch.searchsorted,
           r"""
searchsorted(sorted_sequence, values, *, out_int32=False, right=False, out=None) -> Tensor

Find the indices from the *innermost* dimension of :attr:`sorted_sequence` such that, if the
corresponding values in :attr:`values` were inserted before the indices, the order of the
corresponding *innermost* dimension within :attr:`sorted_sequence` would be preserved.
Return a new tensor with the same size as :attr:`values`. If :attr:`right` is False (default),
then the left boundary of :attr:`sorted_sequence` is closed. More formally, the returned index
satisfies the following rules:

.. list-table::
   :widths: 12 10 78
   :header-rows: 1

   * - :attr:`sorted_sequence`
     - :attr:`right`
     - *returned index satisfies*
   * - 1-D
     - False
     - ``sorted_sequence[i-1] < values[m][n]...[l][x] <= sorted_sequence[i]``
   * - 1-D
     - True
     - ``sorted_sequence[i-1] <= values[m][n]...[l][x] < sorted_sequence[i]``
   * - N-D
     - False
     - ``sorted_sequence[m][n]...[l][i-1] < values[m][n]...[l][x] <= sorted_sequence[m][n]...[l][i]``
   * - N-D
     - True
     - ``sorted_sequence[m][n]...[l][i-1] <= values[m][n]...[l][x] < sorted_sequence[m][n]...[l][i]``

Args:
    sorted_sequence (Tensor): N-D or 1-D tensor, containing monotonically increasing sequence on the *innermost*
                              dimension.
    values (Tensor or Scalar): N-D tensor or a Scalar containing the search value(s).

Keyword args:
    out_int32 (bool, optional): indicate the output data type. torch.int32 if True, torch.int64 otherwise.
                                Default value is False, i.e. default output data type is torch.int64.
    right (bool, optional): if False, return the first suitable location that is found. If True, return the
                            last such index. If no suitable index found, return 0 for non-numerical value
                            (eg. nan, inf) or the size of *innermost* dimension within :attr:`sorted_sequence`
                            (one pass the last index of the *innermost* dimension). In other words, if False,
                            gets the lower bound index for each value in :attr:`values` on the corresponding
                            *innermost* dimension of the :attr:`sorted_sequence`. If True, gets the upper
                            bound index instead. Default value is False.
    out (Tensor, optional): the output tensor, must be the same size as :attr:`values` if provided.

.. note:: If your use case is always 1-D sorted sequence, :func:`torch.bucketize` is preferred,
          because it has fewer dimension checks resulting in slightly better performance.


Example::

    >>> sorted_sequence = torch.tensor([[1, 3, 5, 7, 9], [2, 4, 6, 8, 10]])
    >>> sorted_sequence
    tensor([[ 1,  3,  5,  7,  9],
            [ 2,  4,  6,  8, 10]])
    >>> values = torch.tensor([[3, 6, 9], [3, 6, 9]])
    >>> values
    tensor([[3, 6, 9],
            [3, 6, 9]])
    >>> torch.searchsorted(sorted_sequence, values)
    tensor([[1, 3, 4],
            [1, 2, 4]])
    >>> torch.searchsorted(sorted_sequence, values, right=True)
    tensor([[2, 3, 5],
            [1, 3, 4]])

    >>> sorted_sequence_1d = torch.tensor([1, 3, 5, 7, 9])
    >>> sorted_sequence_1d
    tensor([1, 3, 5, 7, 9])
    >>> torch.searchsorted(sorted_sequence_1d, values)
    tensor([[1, 3, 4],
            [1, 3, 4]])
""")

add_docstr(torch.bucketize,
           r"""
bucketize(input, boundaries, *, out_int32=False, right=False, out=None) -> Tensor

Returns the indices of the buckets to which each value in the :attr:`input` belongs, where the
boundaries of the buckets are set by :attr:`boundaries`. Return a new tensor with the same size
as :attr:`input`. If :attr:`right` is False (default), then the left boundary is closed. More
formally, the returned index satisfies the following rules:

.. list-table::
   :widths: 15 85
   :header-rows: 1

   * - :attr:`right`
     - *returned index satisfies*
   * - False
     - ``boundaries[i-1] < input[m][n]...[l][x] <= boundaries[i]``
   * - True
     - ``boundaries[i-1] <= input[m][n]...[l][x] < boundaries[i]``

Args:
    input (Tensor or Scalar): N-D tensor or a Scalar containing the search value(s).
    boundaries (Tensor): 1-D tensor, must contain a monotonically increasing sequence.

Keyword args:
    out_int32 (bool, optional): indicate the output data type. torch.int32 if True, torch.int64 otherwise.
                                Default value is False, i.e. default output data type is torch.int64.
    right (bool, optional): if False, return the first suitable location that is found. If True, return the
                            last such index. If no suitable index found, return 0 for non-numerical value
                            (eg. nan, inf) or the size of :attr:`boundaries` (one pass the last index).
                            In other words, if False, gets the lower bound index for each value in :attr:`input`
                            from :attr:`boundaries`. If True, gets the upper bound index instead.
                            Default value is False.
    out (Tensor, optional): the output tensor, must be the same size as :attr:`input` if provided.


Example::

    >>> boundaries = torch.tensor([1, 3, 5, 7, 9])
    >>> boundaries
    tensor([1, 3, 5, 7, 9])
    >>> v = torch.tensor([[3, 6, 9], [3, 6, 9]])
    >>> v
    tensor([[3, 6, 9],
            [3, 6, 9]])
    >>> torch.bucketize(v, boundaries)
    tensor([[1, 3, 4],
            [1, 3, 4]])
    >>> torch.bucketize(v, boundaries, right=True)
    tensor([[2, 3, 5],
            [2, 3, 5]])
""")