functional.py

import torch
import torch.nn.functional as F
from torch._six import inf
from itertools import product
import warnings

__all__ = [
    'broadcast_tensors',
    'btrifact',
    'btrifact_with_info',
    'btrisolve',
    'btriunpack',
    'cartesian_prod',
    'chain_matmul',
    'einsum',
    'gesv',
    'isfinite',
    'isinf',
    'lu',
    'lu_unpack',
    'norm',
    'meshgrid',
    'pstrf',
    'potrf',
    'potri',
    'potrs',
    'split',
    'stft',
    'tensordot',
    'trtrs',
    'unique',
    'unique_consecutive',
]


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

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

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

    .. warning::

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

    Example::

        >>> x = torch.arange(3).view(1, 3)
        >>> y = torch.arange(2).view(2, 1)
        >>> a, b = torch.broadcast_tensors(x, y)
        >>> a.size()
        torch.Size([2, 3])
        >>> a
        tensor([[0, 1, 2],
                [0, 1, 2]])
    """
    return torch._C._VariableFunctions.broadcast_tensors(tensors)


def split(tensor, split_size_or_sections, dim=0):
    r"""Splits the tensor into chunks.

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

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

    Arguments:
        tensor (Tensor): tensor to split.
        split_size_or_sections (int) or (list(int)): size of a single chunk or
            list of sizes for each chunk
        dim (int): dimension along which to split the tensor.
    """
    # Overwriting reason:
    # This dispatches to two ATen functions depending on the type of
    # split_size_or_sections. The branching code is in tensor.py, which we
    # call here.
    return tensor.split(split_size_or_sections, dim)


def lu_unpack(LU_data, LU_pivots, unpack_data=True, unpack_pivots=True):
    r"""Unpacks the data and pivots from a LU factorization of a tensor.

    Returns a tuple of tensors as ``(the pivots, the L tensor, the U tensor)``.

    Arguments:
        LU_data (Tensor): the packed LU factorization data
        LU_pivots (Tensor): the packed LU factorization pivots
        unpack_data (bool): flag indicating if the data should be unpacked
        unpack_pivots (bool): flag indicating if the pivots should be unpacked

    Example::

        >>> A = torch.randn(2, 3, 3)
        >>> A_LU, pivots = A.lu()
        >>> P, A_L, A_U = torch.lu_unpack(A_LU, pivots)
        >>>
        >>> # can recover A from factorization
        >>> A_ = torch.bmm(P, torch.bmm(A_L, A_U))
    """

    sz = LU_data.size(-1)

    if unpack_data:
        U = LU_data.triu()
        L = LU_data.tril()
        L.diagonal(dim1=-2, dim2=-1).fill_(1)
    else:
        L = U = None

    if unpack_pivots:
        LU_pivots_zero_idx = LU_pivots - 1
        if LU_data.dim() > 2:
            P = torch.eye(sz, device=LU_data.device, dtype=LU_data.dtype).expand_as(LU_data).clone()
            for idx in product(*map(lambda x: list(range(x)), LU_data.shape[:-2])):
                final_order = list(range(sz))
                for k, j in enumerate(LU_pivots_zero_idx[idx]):
                    final_order[k], final_order[j] = final_order[j], final_order[k]
                P[idx] = P[idx].index_select(1, torch.as_tensor(final_order, device=LU_pivots.device))
        else:
            P = torch.eye(sz, device=LU_data.device, dtype=LU_data.dtype)
            final_order = list(range(sz))
            for k, j, in enumerate(LU_pivots_zero_idx):
                final_order[k], final_order[j] = final_order[j], final_order[k]
            P = P.index_select(1, torch.as_tensor(final_order, device=LU_pivots.device))
    else:
        P = None

    return P, L, U


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

This function provides a way of computing multilinear expressions (i.e. sums of products) using the
Einstein summation convention.

Args:
    equation (string): The equation is given in terms of lower case letters (indices) to be associated
           with each dimension of the operands and result. The left hand side lists the operands
           dimensions, separated by commas. There should be one index letter per tensor dimension.
           The right hand side follows after `->` and gives the indices for the output.
           If the `->` and right hand side are omitted, it implicitly defined as the alphabetically
           sorted list of all indices appearing exactly once in the left hand side.
           The indices not apprearing in the output are summed over after multiplying the operands
           entries.
           If an index appears several times for the same operand, a diagonal is taken.
           Ellipses `...` represent a fixed number of dimensions. If the right hand side is inferred,
           the ellipsis dimensions are at the beginning of the output.
    operands (list of Tensors): The operands to compute the Einstein sum of.

Examples::

    >>> x = torch.randn(5)
    >>> y = torch.randn(4)
    >>> torch.einsum('i,j->ij', x, y)  # outer product
    tensor([[-0.0570, -0.0286, -0.0231,  0.0197],
            [ 1.2616,  0.6335,  0.5113, -0.4351],
            [ 1.4452,  0.7257,  0.5857, -0.4984],
            [-0.4647, -0.2333, -0.1883,  0.1603],
            [-1.1130, -0.5588, -0.4510,  0.3838]])


    >>> A = torch.randn(3,5,4)
    >>> l = torch.randn(2,5)
    >>> r = torch.randn(2,4)
    >>> torch.einsum('bn,anm,bm->ba', l, A, r) # compare torch.nn.functional.bilinear
    tensor([[-0.3430, -5.2405,  0.4494],
            [ 0.3311,  5.5201, -3.0356]])


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

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

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

    >>> A = torch.randn(3, 3)
    >>> torch.einsum('ii->i', A) # diagonal
    tensor([-0.7825,  0.8291, -0.1936])

    >>> A = torch.randn(4, 3, 3)
    >>> torch.einsum('...ii->...i', A) # batch diagonal
    tensor([[-1.0864,  0.7292,  0.0569],
            [-0.9725, -1.0270,  0.6493],
            [ 0.5832, -1.1716, -1.5084],
            [ 0.4041, -1.1690,  0.8570]])

    >>> A = torch.randn(2, 3, 4, 5)
    >>> torch.einsum('...ij->...ji', A).shape # batch permute
    torch.Size([2, 3, 5, 4])
"""
    if len(operands) == 1 and isinstance(operands[0], (list, tuple)):
        # the old interface of passing the operands as one list argument
        operands = operands[0]
    return torch._C._VariableFunctions.einsum(equation, operands)


def isfinite(tensor):
    r"""Returns a new tensor with boolean elements representing if each element is `Finite` or not.

    Arguments:
        tensor (Tensor): A tensor to check

    Returns:
        Tensor: A ``torch.ByteTensor`` containing a 1 at each location of finite elements and 0 otherwise

    Example::

        >>> torch.isfinite(torch.tensor([1, float('inf'), 2, float('-inf'), float('nan')]))
        tensor([ 1,  0,  1,  0,  0], dtype=torch.uint8)
    """
    if not isinstance(tensor, torch.Tensor):
        raise TypeError("The argument is not a tensor: {}".format(repr(tensor)))

    # Support int input, nan and inf are concepts in floating point numbers.
    # Numpy uses type 'Object' when the int overflows long, but we don't
    # have a similar concept. It's safe to assume any created LongTensor doesn't
    # overflow and it's finite.
    if not tensor.is_floating_point():
        return torch.ones_like(tensor, dtype=torch.uint8)
    return (tensor == tensor) & (tensor.abs() != inf)


def isinf(tensor):
    r"""Returns a new tensor with boolean elements representing if each element is `+/-INF` or not.

    Arguments:
        tensor (Tensor): A tensor to check

    Returns:
        Tensor: A ``torch.ByteTensor`` containing a 1 at each location of `+/-INF` elements and 0 otherwise

    Example::

        >>> torch.isinf(torch.tensor([1, float('inf'), 2, float('-inf'), float('nan')]))
        tensor([ 0,  1,  0,  1,  0], dtype=torch.uint8)
    """
    if not isinstance(tensor, torch.Tensor):
        raise TypeError("The argument is not a tensor: {}".format(repr(tensor)))
    if tensor.dtype in [torch.uint8, torch.int8, torch.int16, torch.int32, torch.int64]:
        return torch.zeros_like(tensor, dtype=torch.uint8)
    return tensor.abs() == inf


def meshgrid(*tensors, **kwargs):
    r"""Take :math:`N` tensors, each of which can be either scalar or 1-dimensional
vector, and create :math:`N` N-dimensional grids, where the :math:`i` :sup:`th` grid is defined by
expanding the :math:`i` :sup:`th` input over dimensions defined by other inputs.


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

    Returns:
        seq (sequence of Tensors): If the input has :math:`k` tensors of size
        :math:`(N_1,), (N_2,), \ldots , (N_k,)`, then the output would also have :math:`k` tensors,
        where all tensors are of size :math:`(N_1, N_2, \ldots , N_k)`.

    Example::

        >>> x = torch.tensor([1, 2, 3])
        >>> y = torch.tensor([4, 5, 6])
        >>> grid_x, grid_y = torch.meshgrid(x, y)
        >>> grid_x
        tensor([[1, 1, 1],
                [2, 2, 2],
                [3, 3, 3]])
        >>> grid_y
        tensor([[4, 5, 6],
                [4, 5, 6],
                [4, 5, 6]])
    """
    if kwargs:
        raise TypeError("meshgrid() got an unexpected keyword argument '%s'" % (list(kwargs)[0],))
    if len(tensors) == 1 and isinstance(tensors[0], (list, tuple)):
        # the old interface of passing the operands as one list argument
        tensors = tensors[0]
    return torch._C._VariableFunctions.meshgrid(tensors)


def stft(input, n_fft, hop_length=None, win_length=None, window=None,
         center=True, pad_mode='reflect', normalized=False, onesided=True):
    r"""Short-time Fourier transform (STFT).

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

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

    where :math:`m` is the index of the sliding window, and :math:`\omega` is
    the frequency that :math:`0 \leq \omega < \text{n\_fft}`. When
    :attr:`onesided` is the default value ``True``,

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

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

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

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

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

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

    * If :attr:`onesided` is ``True`` (default), only values for :math:`\omega`
      in :math:`\left[0, 1, 2, \dots, \left\lfloor \frac{\text{n\_fft}}{2} \right\rfloor + 1\right]`
      are returned because the real-to-complex Fourier transform satisfies the
      conjugate symmetry, i.e., :math:`X[m, \omega] = X[m, \text{n\_fft} - \omega]^*`.

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

    Returns the real and the imaginary parts together as one tensor of size
    :math:`(* \times N \times T \times 2)`, where :math:`*` is the optional
    batch size of :attr:`input`, :math:`N` is the number of frequencies where
    STFT is applied, :math:`T` is the total number of frames used, and each pair
    in the last dimension represents a complex number as the real part and the
    imaginary part.

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

    Arguments:
        input (Tensor): the input tensor
        n_fft (int): size of Fourier transform
        hop_length (int, optional): the distance between neighboring sliding window
            frames. Default: ``None`` (treated as equal to ``floor(n_fft / 4)``)
        win_length (int, optional): the size of window frame and STFT filter.
            Default: ``None``  (treated as equal to :attr:`n_fft`)
        window (Tensor, optional): the optional window function.
            Default: ``None`` (treated as window of all :math:`1` s)
        center (bool, optional): whether to pad :attr:`input` on both sides so
            that the :math:`t`-th frame is centered at time :math:`t \times \text{hop\_length}`.
            Default: ``True``
        pad_mode (string, optional): controls the padding method used when
            :attr:`center` is ``True``. Default: ``"reflect"``
        normalized (bool, optional): controls whether to return the normalized STFT results
             Default: ``False``
        onesided (bool, optional): controls whether to return half of results to
            avoid redundancy Default: ``True``

    Returns:
        Tensor: A tensor containing the STFT result with shape described above

    """
    # TODO: after having proper ways to map Python strings to ATen Enum, move
    #       this and F.pad to ATen.
    if center:
        signal_dim = input.dim()
        extended_shape = [1] * (3 - signal_dim) + list(input.size())
        pad = int(n_fft // 2)
        input = F.pad(input.view(extended_shape), (pad, pad), pad_mode)
        input = input.view(input.shape[-signal_dim:])
    return torch._C._VariableFunctions.stft(input, n_fft, hop_length, win_length, window, normalized, onesided)


del torch.unique_dim


def unique(input, sorted=True, return_inverse=False, return_counts=False, dim=None):
    r"""Returns the unique elements of the input tensor.

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

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

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

    Example::

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

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

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

    """
    if dim is not None:
        output, inverse_indices, counts = torch._C._VariableFunctions.unique_dim(
            input,
            dim,
            sorted=sorted,
            return_inverse=return_inverse,
            return_counts=return_counts,
        )
    else:
        output, inverse_indices, counts = torch._unique2(
            input,
            sorted=sorted,
            return_inverse=return_inverse,
            return_counts=return_counts,
        )
    if return_inverse and return_counts:
        return output, inverse_indices, counts
    elif return_inverse:
        return output, inverse_indices
    elif return_counts:
        return output, counts
    else:
        return output


def unique_consecutive(input, return_inverse=False, return_counts=False, dim=None):
    r"""Eliminates all but the first element from every consecutive group of equivalent elements.

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

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

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

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

    Example::

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

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

        >>> output, counts = torch.unique_consecutive(x, return_counts=True)
        >>> output
        tensor([1, 2, 3, 1, 2])
        >>> counts
        tensor([2, 2, 1, 2, 1])
    """
    output, inverse_indices, counts = torch._C._VariableFunctions.unique_consecutive(
        input, return_inverse=return_inverse, return_counts=return_counts, dim=dim)
    if return_inverse and return_counts:
        return output, inverse_indices, counts
    if return_inverse:
        return output, inverse_indices
    if return_counts:
        return output, counts
    return output


def tensordot(a, b, dims=2):
    r"""Returns a contraction of a and b over multiple dimensions.

    :attr:`tensordot` implements a generalizes the matrix product.

    Args:
      a (Tensor): Left tensor to contract
      b (Tensor): Right tensor to contract
      dims (int or tuple of two lists of integers): number of dimensions to
         contract or explicit lists of dimensions for :attr:`a` and
         :attr:`b` respectively

    When called with an integer argument :attr:`dims` = :math:`d`, and the number of
    dimensions of :attr:`a` and :attr:`b` is :math:`m` and :math:`n`, respectively,
    it computes

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

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

    Examples::

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

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

    """
    if isinstance(dims, (list, tuple)) or \
       (isinstance(dims, torch.Tensor) and dims.numel() > 1):
        dims_a, dims_b = dims
    else:
        if isinstance(dims, torch.Tensor):
            dims = dims.item()
        dims_a = list(range(-dims, 0))
        dims_b = list(range(dims))
    return torch._C._VariableFunctions.tensordot(a, b, dims_a, dims_b)


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

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

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

    Example::

        >>> a = [1, 2, 3]
        >>> b = [4, 5]
        >>> list(itertools.product(a, b))
        [(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)]
        >>> tensor_a = torch.tensor(a)
        >>> tensor_b = torch.tensor(b)
        >>> torch.cartesian_prod(tensor_a, tensor_b)
        tensor([[1, 4],
                [1, 5],
                [2, 4],
                [2, 5],
                [3, 4],
                [3, 5]])
    """
    return torch._C._VariableFunctions.cartesian_prod(tensors)


def norm(input, p="fro", dim=None, keepdim=False, out=None, dtype=None):
    r"""Returns the matrix norm or vector norm of a given tensor.

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

            =====  ============================  ==========================
            ord    matrix norm                   vector norm
            =====  ============================  ==========================
            None   Frobenius norm                2-norm
            'fro'  Frobenius norm                --
            'nuc'  nuclear norm                  --
            Other  as vec norm when dim is None  sum(abs(x)**ord)**(1./ord)
            =====  ============================  ==========================

        dim (int, 2-tuple of ints, 2-list of ints, optional): If it is an int,
            vector norm will be calculated, if it is 2-tuple of ints, matrix norm
            will be calculated. If the value is None, matrix norm will be calculated
            when the input tensor only has two dimensions, vector norm will be
            calculated when the input tensor only has one dimension. If the input
            tensor has more than two dimensions, the vector norm will be applied to
            last dimension.
        keepdim (bool, optional): whether the output tensors have :attr:`dim`
            retained or not. Ignored if :attr:`dim` = ``None`` and
            :attr:`out` = ``None``. Default: ``False``
        out (Tensor, optional): the output tensor. Ignored if
            :attr:`dim` = ``None`` and :attr:`out` = ``None``.
        dtype (:class:`torch.dtype`, optional): the desired data type of
            returned tensor. If specified, the input tensor is casted to
            :attr:'dtype' while performing the operation. Default: None.


    Example::

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

    # catch default case
    if dim is None and out is None and dtype is None:
        if p == "fro":
            return torch._C._VariableFunctions.frobenius_norm(input)
        elif p != "nuc":
            return torch._C._VariableFunctions.norm(input, p)

    if p == "fro":
        if dtype is not None:
            raise ValueError("dtype argument is not supported in frobenius norm")
        if dim is None:
            dim = tuple(range(ndim))
        if out is None:
            return torch._C._VariableFunctions.frobenius_norm(input, dim, keepdim=keepdim)
        return torch._C._VariableFunctions.frobenius_norm(input, dim, keepdim=keepdim, out=out)
    elif p == "nuc":
        if dtype is not None:
            raise ValueError("dtype argument is not supported in nuclear norm")
        if out is None:
            torch._C._VariableFunctions.nuclear_norm(input, keepdim=keepdim)
        return torch._C._VariableFunctions.nuclear_norm(input, keepdim=keepdim, out=out)
    else:
        if dim is None:
            dim = tuple(range(ndim))
        if out is None and dtype is None:
            return torch._C._VariableFunctions.norm(input, p, dim, keepdim=keepdim)
        elif out is None:
            return torch._C._VariableFunctions.norm(input, p, dim, keepdim=keepdim, dtype=dtype)
        elif dtype is None:
            return torch._C._VariableFunctions.norm(input, p, dim, keepdim=keepdim, out=out)
    return torch._C._VariableFunctions.norm(input, p, dim, keepdim=keepdim, dtype=dtype, out=out)


def chain_matmul(*matrices):
    r"""Returns the matrix product of the :math:`N` 2-D tensors. This product is efficiently computed
    using the matrix chain order algorithm which selects the order in which incurs the lowest cost in terms
    of arithmetic operations (`[CLRS]`_). Note that since this is a function to compute the product, :math:`N`
    needs to be greater than or equal to 2; if equal to 2 then a trivial matrix-matrix product is returned.
    If :math:`N` is 1, then this is a no-op - the original matrix is returned as is.


    Args:
        matrices (Tensors...): a sequence of 2 or more 2-D tensors whose product is to be determined.


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

    Example::

        >>> a = torch.randn(3, 4)
        >>> b = torch.randn(4, 5)
        >>> c = torch.randn(5, 6)
        >>> d = torch.randn(6, 7)
        >>> torch.chain_matmul(a, b, c, d)
        tensor([[ -2.3375,  -3.9790,  -4.1119,  -6.6577,   9.5609, -11.5095,  -3.2614],
                [ 21.4038,   3.3378,  -8.4982,  -5.2457, -10.2561,  -2.4684,   2.7163],
                [ -0.9647,  -5.8917,  -2.3213,  -5.2284,  12.8615, -12.2816,  -2.5095]])

    .. _`[CLRS]`: https://mitpress.mit.edu/books/introduction-algorithms-third-edition
    """
    return torch._C._VariableFunctions.chain_matmul(matrices)


def pstrf(a, upper=True, out=None):
    r"""Computes the pivoted Cholesky decomposition of a symmetric positive-definite
    matrix :attr:`a`. returns a namedtuple (u, pivot) of matrice.

    If :attr:`upper` is ``True`` or not provided, `u` is upper triangular
    such that :math:`a = p^T u^T u p`, with `p` the permutation given by `pivot`.

    If :attr:`upper` is ``False``, `u` is lower triangular such that
    :math:`a = p^T u u^T p`.

    .. warning::
        :func:`torch.pstrf` is deprecated in favour of :func:`torch.cholesky` and will
        be removed in the next release.

    Args:
        a (Tensor): the input 2-D tensor
        upper (bool, optional): whether to return a upper (default) or lower triangular matrix
        out (tuple, optional): namedtuple of `u` and `pivot` tensors

    Example::

        >>> a = torch.randn(3, 3)
        >>> a = torch.mm(a, a.t()) # make symmetric positive definite
        >>> a
        tensor([[ 3.5405, -0.4577,  0.8342],
                [-0.4577,  1.8244, -0.1996],
                [ 0.8342, -0.1996,  3.7493]])
        >>> u,piv = torch.pstrf(a)
        >>> u
        tensor([[ 1.9363,  0.4308, -0.1031],
                [ 0.0000,  1.8316, -0.2256],
                [ 0.0000,  0.0000,  1.3277]])
        >>> piv
        tensor([ 2,  0,  1], dtype=torch.int32)
        >>> p = torch.eye(3).index_select(0,piv.long()).index_select(0,piv.long()).t() # make pivot permutation
        >>> torch.mm(torch.mm(p.t(),torch.mm(u.t(),u)),p) # reconstruct
        tensor([[ 3.5405, -0.4577,  0.8342],
                [-0.4577,  1.8244, -0.1996],
                [ 0.8342, -0.1996,  3.7493]])
    """
    warnings.warn("torch.pstrf is deprecated in favour of torch.cholesky and will be removed "
                  "in the next release.", stacklevel=2)
    return torch._C._VariableFunctions.pstrf(a, upper=upper, out=out)


def potrf(a, upper=True, out=None):
    r"""Computes the Cholesky decomposition of a symmetric positive-definite
    matrix :math:`A`.

    For more information regarding :func:`torch.potrf`, please check :func:`torch.cholesky`.

    .. warning::
        :func:`torch.potrf` is deprecated in favour of :func:`torch.cholesky` and will be removed
        in the next release. Please use :func:`torch.cholesky` instead and note that the :attr:`upper`
        argument in :func:`torch.cholesky` defaults to ``False``.
    """
    warnings.warn("torch.potrf is deprecated in favour of torch.cholesky and will be removed in the next "
                  "release. Please use torch.cholesky instead and note that the :attr:`upper` argument in"
                  " torch.cholesky defaults to ``False``.", stacklevel=2)
    return torch.cholesky(a, upper=upper, out=out)


def potri(a, upper=True, out=None):
    r"""Computes the inverse of a symmetric positive-definite matrix :math:`A` using its
    Cholesky factor.

    For more information regarding :func:`torch.potri`, please check :func:`torch.cholesky_inverse`.

    .. warning::
        :func:`torch.potri` is deprecated in favour of :func:`torch.cholesky_inverse` and will be removed
        in the next release. Please use :func:`torch.cholesky_inverse` instead and note that the :attr:`upper`
        argument in :func:`torch.cholesky_inverse` defaults to ``False``.
    """
    warnings.warn("torch.potri is deprecated in favour of torch.cholesky_inverse and will be removed in "
                  "the next release. Please use torch.cholesky_inverse instead and note that the :attr:`upper` "
                  "argument in torch.cholesky_inverse defaults to ``False``.", stacklevel=2)
    return torch.cholesky_inverse(a, upper=upper, out=out)


def potrs(b, u, upper=True, out=None):
    r"""Solves a linear system of equations with a positive semidefinite
    matrix to be inverted given its Cholesky factor matrix :attr:`u`.

    For more information regarding :func:`torch.potrs`, please check :func:`torch.cholesky_solve`.

    .. warning::
        :func:`torch.potrs` is deprecated in favour of :func:`torch.cholesky_solve` and will be
        removed in the next release. Please use :func:`torch.cholesky_solve` instead and note that
        the :attr:`upper` argument in :func:`torch.cholesky_solve` defaults to ``False``.
    """
    warnings.warn("torch.potrs is deprecated in favour of torch.cholesky_solve and will be removed "
                  "in the next release. Please use torch.cholesky instead and note that the "
                  ":attr:`upper` argument in torch.cholesky_solve defaults to ``False``.", stacklevel=2)
    return torch.cholesky_solve(b, u, upper=upper, out=out)


def gesv(b, A, out=None):
    r"""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 tuple `X, LU`.

    For more information regarding :func:`torch.gesv`, please check :func:`torch.solve`.

    .. warning::
        :func:`torch.gesv` is deprecated in favour of :func:`torch.solve` and will be removed in the
        next release. Please use :func:`torch.solve` instead.
    """
    warnings.warn("torch.gesv is deprecated in favour of torch.solve and will be removed in the "
                  "next release. Please use torch.solve instead.", stacklevel=2)
    return torch.solve(b, A, out=out)


def trtrs(b, A, upper=True, transpose=False, unitriangular=False, out=None):
    r"""Solves a system of equations with a triangular coefficient matrix :math:`A`
    and multiple right-hand sides :attr:`b`.

    In particular, solves :math:`AX = b` and assumes :math:`A` is upper-triangular
    with the default keyword arguments.

    For more information regarding :func:`torch.trtrs`, please check :func:`torch.triangular_solve`.

    .. warning::
        :func:`torch.trtrs` is deprecated in favour of :func:`torch.triangular_solve` and will be
        removed in the next release. Please use :func:`torch.triangular_solve` instead.
    """
    warnings.warn("torch.trtrs is deprecated in favour of torch.triangular_solve and will be "
                  "removed in the next release. Please use torch.triangular_solve instead.", stacklevel=2)
    return torch.triangular_solve(b, A, upper=upper, transpose=transpose, unitriangular=unitriangular, out=out)


def btrifact(A, pivot=True, out=None):
    r"""Returns a tuple containing the LU factorization and pivots of :attr:`A`.
    Pivoting is done if :attr:`pivot` is set.

    For more information regarding :func:`torch.btrifact`, please check :func:`torch.lu`.

    .. warning::
        :func:`torch.btrifact` is deprecated in favour of :func:`torch.lu` and will be
        removed in the next release. Please use :func:`torch.lu` instead.
    """
    warnings.warn("torch.btrifact is deprecated in favour of torch.lu and will be "
                  "removed in the next release. Please use torch.lu instead.", stacklevel=2)
    return lu(A, pivot=pivot, get_infos=False, out=out)


def btrifact_with_info(A, pivot=True, out=None):
    r"""Performs LU factorization and returns additional status information along with the LU
    factorization and pivots.

    For more information regarding :func:`torch.btrifact_with_info`, please check :func:`torch.lu`.

    .. warning::
        :func:`torch.btrifact_with_info` is deprecated in favour of :func:`torch.lu` and will
        be removed in the next release. Please use :func:`torch.lu` with the :attr:`get_infos`
        argument set to ``True`` instead.
    """
    warnings.warn("torch.btrifact_with_info is deprecated in favour of torch.lu and will be "
                  "removed in the next release. Please use torch.lu with the get_infos argument "
                  "set to True instead.",
                  stacklevel=2)
    return lu(A, pivot=pivot, get_infos=True, out=out)


def btriunpack(LU_data, LU_pivots, unpack_data=True, unpack_pivots=True):
    r"""Unpacks the data and pivots from a LU factorization of a tensor.

    For more information regarding :func:`torch.btriunpack`, please check :func:`torch.lu_unpack`.

    .. warning::
        :func:`torch.btriunpack` is deprecated in favour of :func:`torch.lu_unpack` and will be
        removed in the next release. Please use :func:`torch.lu_unpack` instead.
    """
    warnings.warn("torch.btriunpack is deprecated in favour of torch.lu_unpack and will be "
                  "removed in the next release. Please use torch.lu_unpack instead.", stacklevel=2)
    return lu_unpack(LU_data=LU_data, LU_pivots=LU_pivots,
                     unpack_data=unpack_data, unpack_pivots=unpack_pivots)


def btrisolve(b, LU_data, LU_pivots, out=None):
    r"""Solves the system of equations :math:`Ax = b` using the partially pivoted LU
    factorization of :math:`A` given by :attr:`LU_data` and :attr:`LU_pivots`.

    For more information regarding :func:`torch.btrisolve`, please check
    :func:`torch.lu_solve`.

    .. warning::
        :func:`torch.btrisolve` is deprecated in favour of :func:`torch.lu_solve` and will be
        removed in the next release. Please use :func:`torch.lu_solve` instead.
    """
    warnings.warn("torch.btrisolve is deprecated in favour of torch.lu_solve and will be "
                  "removed in the next release. Please use torch.lu_solve instead.", stacklevel=2)
    return torch.lu_solve(b, LU_data=LU_data, LU_pivots=LU_pivots, out=out)


def lu(A, pivot=True, get_infos=False, out=None):
    r"""Computes the LU factorization of a square matrix or batches of square matrices
    :attr:`A`. Returns a tuple containing the LU factorization and pivots of :attr:`A`.
    Pivoting is done if :attr:`pivot` is set to ``True``.

    .. note::
        The pivots returned by the function are 1-indexed. If :attr:`pivot` is ``False``,
        then the returned pivots is a tensor filled with zeros of the appropriate size.

    .. note::
        LU factorization with :attr:`pivot` = ``False`` is not available for CPU, and attempting
        to do so will throw an error. However, LU factorization with :attr:`pivot` = ``False`` is
        available for CUDA.

    .. note::
        This function does not check if the factorization was successful or not if
        :attr:`get_infos` is ``True`` since the status of the factorization is present in the
        third element of the return tuple.

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

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

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

            - **pivots** (*IntTensor*): the pivots of size :math:`(*, m)`

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

    Example::

        >>> A = torch.randn(2, 3, 3)
        >>> A_LU, pivots = torch.lu(A)
        >>> A_LU
        tensor([[[ 1.3506,  2.5558, -0.0816],
                 [ 0.1684,  1.1551,  0.1940],
                 [ 0.1193,  0.6189, -0.5497]],

                [[ 0.4526,  1.2526, -0.3285],
                 [-0.7988,  0.7175, -0.9701],
                 [ 0.2634, -0.9255, -0.3459]]])
        >>> pivots
        tensor([[ 3,  3,  3],
                [ 3,  3,  3]], dtype=torch.int32)
        >>> A_LU, pivots, info = torch.lu(A, get_infos=True)
        >>> if info.nonzero().size(0) == 0:
        ...   print('LU factorization succeeded for all samples!')
        LU factorization succeeded for all samples!
    """
    # If get_infos is True, then we don't need to check for errors and vice versa
    result = torch._lu_with_info(A, pivot=pivot, check_errors=(not get_infos))
    if out is not None:
        if not isinstance(out, (tuple, list)):
            raise TypeError("argument 'out' must be tuple of Tensors, not {}"
                            .format(type(out).__name__))
        if len(out) - int(get_infos) != 2:
            raise TypeError("expected tuple of {} elements but got {}"
                            .format(2 + int(get_infos), len(out)))
        return (out[i].resize_as_(result[i]).copy_(result[i]) for i in range(len(out)))
    if get_infos:
        return result  # A_LU, pivots, infos
    else:
        return result[0], result[1]  # A_LU, pivots