linalg.py

#   Copyright (c) 2020 PaddlePaddle Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
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import numpy as np
from ..fluid.layer_helper import LayerHelper
from ..fluid.data_feeder import check_variable_and_dtype, check_type
from ..fluid.framework import in_dygraph_mode, _varbase_creator, Variable

from ..fluid.layers import transpose, cast  # noqa: F401
from paddle.common_ops_import import core
from paddle.common_ops_import import VarDesc
from paddle import _C_ops

__all__ = []


def matmul(x, y, transpose_x=False, transpose_y=False, name=None):
    """
    Applies matrix multiplication to two tensors. `matmul` follows
    the complete broadcast rules,
    and its behavior is consistent with `np.matmul`.

    Currently, the input tensors' number of dimensions can be any, `matmul` can be used to
    achieve the `dot`, `matmul` and `batchmatmul`.

    The actual behavior depends on the shapes of :math:`x`, :math:`y` and the
    flag values of :attr:`transpose_x`, :attr:`transpose_y`. Specifically:

    - If a transpose flag is specified, the last two dimensions of the tensor
      are transposed. If the tensor is ndim-1 of shape, the transpose is invalid. If the tensor
      is ndim-1 of shape :math:`[D]`, then for :math:`x` it is treated as :math:`[1, D]`, whereas
      for :math:`y` it is the opposite: It is treated as :math:`[D, 1]`.

    The multiplication behavior depends on the dimensions of `x` and `y`. Specifically:

    - If both tensors are 1-dimensional, the dot product result is obtained.

    - If both tensors are 2-dimensional, the matrix-matrix product is obtained.

    - If the `x` is 1-dimensional and the `y` is 2-dimensional,
      a `1` is prepended to its dimension in order to conduct the matrix multiply.
      After the matrix multiply, the prepended dimension is removed.

    - If the `x` is 2-dimensional and `y` is 1-dimensional,
      the matrix-vector product is obtained.

    - 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 obtained.
      If the first argument is 1-dimensional, a 1 is prepended to its dimension
      in order to conduct 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 (exclude the last two dimensions) dimensions are
      broadcasted according the broadcast rule.
      For example, if input is a (j, 1, n, m) tensor and the other is a (k, m, p) tensor,
      out will be a (j, k, n, p) tensor.

    Args:
        x (Tensor): The input tensor which is a Tensor.
        y (Tensor): The input tensor which is a Tensor.
        transpose_x (bool): Whether to transpose :math:`x` before multiplication.
        transpose_y (bool): Whether to transpose :math:`y` before multiplication.
        name(str|None): A name for this layer(optional). If set None, the layer
            will be named automatically.

    Returns:
        Tensor: The output Tensor.

    Examples:

    .. code-block:: python

        import paddle
        import numpy as np

        # vector * vector
        x_data = np.random.random([10]).astype(np.float32)
        y_data = np.random.random([10]).astype(np.float32)
        x = paddle.to_tensor(x_data)
        y = paddle.to_tensor(y_data)
        z = paddle.matmul(x, y)
        print(z.numpy().shape)
        # [1]

        # matrix * vector
        x_data = np.random.random([10, 5]).astype(np.float32)
        y_data = np.random.random([5]).astype(np.float32)
        x = paddle.to_tensor(x_data)
        y = paddle.to_tensor(y_data)
        z = paddle.matmul(x, y)
        print(z.numpy().shape)
        # [10]

        # batched matrix * broadcasted vector
        x_data = np.random.random([10, 5, 2]).astype(np.float32)
        y_data = np.random.random([2]).astype(np.float32)
        x = paddle.to_tensor(x_data)
        y = paddle.to_tensor(y_data)
        z = paddle.matmul(x, y)
        print(z.numpy().shape)
        # [10, 5]

        # batched matrix * batched matrix
        x_data = np.random.random([10, 5, 2]).astype(np.float32)
        y_data = np.random.random([10, 2, 5]).astype(np.float32)
        x = paddle.to_tensor(x_data)
        y = paddle.to_tensor(y_data)
        z = paddle.matmul(x, y)
        print(z.numpy().shape)
        # [10, 5, 5]

        # batched matrix * broadcasted matrix
        x_data = np.random.random([10, 1, 5, 2]).astype(np.float32)
        y_data = np.random.random([1, 3, 2, 5]).astype(np.float32)
        x = paddle.to_tensor(x_data)
        y = paddle.to_tensor(y_data)
        z = paddle.matmul(x, y)
        print(z.numpy().shape)
        # [10, 3, 5, 5]

    """
    op_type = 'matmul_v2'
    if in_dygraph_mode():
        op = getattr(_C_ops, op_type)
        return op(x, y, 'trans_x', transpose_x, 'trans_y', transpose_y)

    attrs = {
        'trans_x': transpose_x,
        'trans_y': transpose_y,
    }

    def __check_input(x, y):
        var_names = {'x': x, 'y': y}
        for name, val in var_names.items():
            check_variable_and_dtype(
                val, name, ['float16', 'float32', 'float64'], 'matmul')

    __check_input(x, y)

    helper = LayerHelper('matmul_v2', **locals())
    out = helper.create_variable_for_type_inference(dtype=x.dtype)
    helper.append_op(
        type='matmul_v2',
        inputs={'X': x,
                'Y': y},
        outputs={'Out': out},
        attrs=attrs)
    return out


def norm(x, p='fro', axis=None, keepdim=False, name=None):
    """

    Returns the matrix norm (Frobenius) or vector norm (the 1-norm, the Euclidean
    or 2-norm, and in general the p-norm for p > 0) of a given tensor.

    .. note::
        This norm API is different from `numpy.linalg.norm`.
        This api supports high-order input tensors (rank >= 3), and certain axis need to be pointed out to calculate the norm.
        But `numpy.linalg.norm` only supports 1-D vector or 2-D matrix as input tensor.
        For p-order matrix norm, this api actually treats matrix as a flattened vector to calculate the vector norm, NOT REAL MATRIX NORM.

    Args:
        x (Tensor): The input tensor could be N-D tensor, and the input data
            type could be float32 or float64.
        p (float|string, optional): Order of the norm. Supported values are `fro`, `0`, `1`, `2`,
            `inf`, `-inf` and any positive real number yielding the corresponding p-norm. Not supported: ord < 0 and nuclear norm.
            Default value is `fro`.
        axis (int|list|tuple, optional): The axis on which to apply norm operation. If axis is int
            or list(int)/tuple(int)  with only one element, the vector norm is computed over the axis.
            If `axis < 0`, the dimension to norm operation is rank(input) + axis.
            If axis is a list(int)/tuple(int) with two elements, the matrix norm is computed over the axis.
            Defalut value is `None`.
        keepdim (bool, optional): Whether to reserve the reduced dimension in the
            output Tensor. The result tensor will have fewer dimension
            than the :attr:`input` unless :attr:`keepdim` is true, default
            value is False.
        name (str, optional): The default value is None. Normally there is no need for
            user to set this property. For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tensor: results of norm operation on the specified axis of input tensor,
        it's data type is the same as input's Tensor.

    Examples:
        .. code-block:: python

            import paddle
            import numpy as np
            shape=[2, 3, 4]
            np_input = np.arange(24).astype('float32') - 12
            np_input = np_input.reshape(shape)
            x = paddle.to_tensor(np_input)
            #[[[-12. -11. -10.  -9.] [ -8.  -7.  -6.  -5.] [ -4.  -3.  -2.  -1.]]
            # [[  0.   1.   2.   3.] [  4.   5.   6.   7.] [  8.   9.  10.  11.]]]

            # compute frobenius norm along last two dimensions.
            out_fro = paddle.norm(x, p='fro', axis=[0,1])
            # out_fro.numpy() [17.435596 16.911535 16.7332   16.911535]

            # compute 2-order vector norm along last dimension.
            out_pnorm = paddle.norm(x, p=2, axis=-1)
            #out_pnorm.numpy(): [[21.118711  13.190906   5.477226]
            #                    [ 3.7416575 11.224972  19.131126]]

            # compute 2-order  norm along [0,1] dimension.
            out_pnorm = paddle.norm(x, p=2, axis=[0,1])
            #out_pnorm.numpy(): [17.435596 16.911535 16.7332   16.911535]

            # compute inf-order  norm
            out_pnorm = paddle.norm(x, p=np.inf)
            #out_pnorm.numpy()  = [12.]
            out_pnorm = paddle.norm(x, p=np.inf, axis=0)
            #out_pnorm.numpy(): [[12. 11. 10. 9.] [8. 7. 6. 7.] [8. 9. 10. 11.]]

            # compute -inf-order  norm
            out_pnorm = paddle.norm(x, p=-np.inf)
            #out_pnorm.numpy(): [0.]
            out_pnorm = paddle.norm(x, p=-np.inf, axis=0)
            #out_pnorm.numpy(): [[0. 1. 2. 3.] [4. 5. 6. 5.] [4. 3. 2. 1.]]
    """

    def frobenius_norm(input, dim=None, keepdim=False, name=None):
        """
        The frobenius norm OP is to calculate the frobenius norm of certain two dimensions of Tensor `input`.
        Args:
          input (Variable): Tensor, data type float32, float64.
          dim (list, optional): None for last two dimensions.
          keepdim (bool, optional): Whether keep the dimensions as the `input`, Default False.
        """
        if dim is not None and not (isinstance(dim, list) and len(dim) == 2):
            raise ValueError(
                "The dim of frobenius norm op should be None or two elements list!"
            )
        if in_dygraph_mode():
            if dim is None:
                return _C_ops.frobenius_norm(input, 'keep_dim', keepdim,
                                             'reduce_all', True)
            return _C_ops.frobenius_norm(input, 'dim', dim, 'keep_dim', keepdim,
                                         'reduce_all', False)
        attrs = {'dim': dim, 'keep_dim': keepdim, 'reduce_all': False}
        if dim is None:
            attrs['reduce_all'] = True
        check_variable_and_dtype(input, 'input', ['float32', 'float64'],
                                 'frobenius_norm')

        helper = LayerHelper('frobenius_norm', **locals())
        out = helper.create_variable_for_type_inference(
            dtype=helper.input_dtype())

        helper.append_op(
            type='frobenius_norm',
            inputs={'X': input},
            outputs={'Out': out},
            attrs=attrs)
        return out

    def vector_norm(input,
                    porder=None,
                    axis=None,
                    keepdim=False,
                    asvector=False,
                    name=None):
        """
        Calculate the p-order vector norm for certain  dimension of Tensor `input`.
        Args:
          input (Variable): Tensor, data type float32, float64.
          porder (float, optional): None for porder=2.0.
          axis (int, optional): None for last dimension.
          keepdim (bool, optional): Whether keep the dimensions as the `input`, Default False.
        """
        if in_dygraph_mode():
            if axis is None: axis = -1
            return _C_ops.p_norm(input, 'porder', porder, 'axis', axis,
                                 'keepdim', keepdim, 'asvector', asvector)
        if porder is not None:
            check_type(porder, 'porder', (float, int), 'p_norm')
        if axis is not None:
            check_type(axis, 'axis', (int), 'p_norm')
        check_variable_and_dtype(input, 'input', ['float32', 'float64'],
                                 'p_norm')

        attrs = {
            'axis': axis if axis is not None else -1,
            'porder': float(porder) if porder is not None else 2.0,
            'keepdim': keepdim,
            'asvector': asvector,
            'epsilon': 1e-12,
        }
        helper = LayerHelper('p_norm', **locals())
        out = helper.create_variable_for_type_inference(
            dtype=helper.input_dtype())

        helper.append_op(
            type='p_norm',
            inputs={'X': input},
            outputs={'Out': out},
            attrs=attrs)
        return out

    def inf_norm(input,
                 porder=None,
                 axis=axis,
                 keepdim=False,
                 asvector=False,
                 name=None):
        helper = LayerHelper('frobenius_norm', **locals())
        out = helper.create_variable_for_type_inference(
            dtype=helper.input_dtype())
        helper.append_op(type='abs', inputs={'X': input}, outputs={'Out': out})
        reduce_out = helper.create_variable_for_type_inference(
            dtype=helper.input_dtype())

        reduce_all = True if axis == None or axis == [] or asvector == True else False
        axis = axis if axis != None and axis != [] else [0]

        reduce_type = 'reduce_max' if porder == np.float(
            'inf') else 'reduce_min'
        helper.append_op(
            type=reduce_type,
            inputs={'X': out},
            outputs={'Out': reduce_out},
            attrs={'dim': axis,
                   'keep_dim': keepdim,
                   'reduce_all': reduce_all})

        return reduce_out

    def p_matrix_norm(input, porder=1., axis=axis, keepdim=False, name=None):
        """
        NOTE:
            This function actually treats the matrix as flattened vector to calculate vector norm instead of matrix norm.
        """
        block = LayerHelper('norm', **locals())
        out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        abs_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        block.append_op(
            type='abs', inputs={'X': input}, outputs={'Out': abs_out})
        pow_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())

        block.append_op(
            type='pow',
            inputs={'X': abs_out},
            outputs={'Out': pow_out},
            attrs={'factor': porder})
        sum_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        block.append_op(
            type='reduce_sum',
            inputs={'X': pow_out},
            outputs={'Out': sum_out},
            attrs={
                'dim': axis,
                'keep_dim': keepdim,
                'reduce_all': True if axis is None else False
            })
        porder
        block.append_op(
            type='pow',
            inputs={'X': sum_out},
            outputs={'Out': out},
            attrs={'factor': float(1. / porder)})
        return out

    if axis is None and p is not None:
        if isinstance(p, str):
            if p == "fro":
                return frobenius_norm(x, dim=axis, keepdim=keepdim, name=name)
            else:
                raise ValueError(
                    "only valid string values are 'fro', found {}".format(p))
        elif isinstance(p, (int, float)):
            return vector_norm(
                x,
                porder=p,
                axis=axis,
                keepdim=keepdim,
                asvector=True,
                name=name)
        else:
            raise ValueError("only valid p type is string or float, found {}".
                             format(type(p)))

    if isinstance(axis, tuple):
        axis = list(axis)
    if isinstance(axis, list) and len(axis) == 1:
        axis = axis[0]

    #calculate vector norm, where axis is int or list with only one integer
    if isinstance(axis, int):
        if isinstance(p, str):
            if p == "fro":
                return vector_norm(
                    x,
                    porder=2,
                    axis=axis,
                    keepdim=keepdim,
                    asvector=False,
                    name=name)

            else:
                raise ValueError(
                    "only valid string values are 'fro', found {}".format(p))
        elif isinstance(p, (int, float)):
            return vector_norm(
                x,
                axis=axis,
                porder=p,
                keepdim=keepdim,
                asvector=False,
                name=name)
        else:
            raise ValueError(
                "unspport p for p-order vector norm. except float, found {}".
                format(p))
    #calculate matrix norm, where axis is list with two integers
    elif isinstance(axis, list) and len(axis) == 2:
        if p == "fro":
            return frobenius_norm(x, dim=axis, keepdim=keepdim, name=name)
        elif p == np.inf or p == -np.inf:
            return inf_norm(x, porder=p, axis=axis, keepdim=keepdim, name=name)
        elif p == 0:
            raise ValueError(
                "just suport axis type int or list (length of list <=1) if p = 0, found {}".
                format(axis))
        else:
            return p_matrix_norm(
                x, porder=p, axis=axis, keepdim=keepdim, name=name)
    else:
        raise ValueError(
            "except axis type int or list (length of list <=2), found {}".
            format(axis))


def dist(x, y, p=2):
    r"""

    This OP returns the p-norm of (x - y). It is not a norm in a strict sense, only as a measure
    of distance. The shapes of x and y must be broadcastable. The definition is as follows, for
    details, please refer to the `numpy's broadcasting <https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html>`_:

    - Each input has at least one dimension.
    - Match the two input dimensions from back to front, the dimension sizes must either be equal, one of them is 1, or one of them does not exist.

    Where, z = x - y, the shapes of x and y are broadcastable, then the shape of z can be
    obtained as follows:

    1. If the number of dimensions of x and y are not equal, prepend 1 to the dimensions of the
    tensor with fewer dimensions.

    For example, The shape of x is [8, 1, 6, 1], the shape of y is [7, 1, 5], prepend 1 to the
    dimension of y.

    x (4-D Tensor):  8 x 1 x 6 x 1

    y (4-D Tensor):  1 x 7 x 1 x 5

    2. Determine the size of each dimension of the output z: choose the maximum value from the
    two input dimensions.

    z (4-D Tensor):  8 x 7 x 6 x 5

    If the number of dimensions of the two inputs are the same, the size of the output can be
    directly determined in step 2. When p takes different values, the norm formula is as follows:

    When p = 0, defining $0^0=0$, the zero-norm of z is simply the number of non-zero elements of z.

    .. math::

        ||z||_{0}=\lim_{p \\rightarrow 0}\sum_{i=1}^{m}|z_i|^{p}

    When p = inf, the inf-norm of z is the maximum element of z.

    .. math::

        ||z||_\infty=\max_i |z_i|

    When p = -inf, the negative-inf-norm of z is the minimum element of z.

    .. math::

        ||z||_{-\infty}=\min_i |z_i|

    Otherwise, the p-norm of z follows the formula,

    .. math::

        ||z||_{p}=(\sum_{i=1}^{m}|z_i|^p)^{\\frac{1}{p}}

    Args:
        x (Tensor): 1-D to 6-D Tensor, its data type is float32 or float64.
        y (Tensor): 1-D to 6-D Tensor, its data type is float32 or float64.
        p (float, optional): The norm to be computed, its data type is float32 or float64. Default: 2.

    Returns:
        Tensor: Tensor that is the p-norm of (x - y).

    Examples:
        .. code-block:: python

            import paddle
            import numpy as np

            x = paddle.to_tensor(np.array([[3, 3],[3, 3]]), "float32")
            y = paddle.to_tensor(np.array([[3, 3],[3, 1]]), "float32")
            out = paddle.dist(x, y, 0)
            print(out) # out = [1.]

            out = paddle.dist(x, y, 2)
            print(out) # out = [2.]

            out = paddle.dist(x, y, float("inf"))
            print(out) # out = [2.]

            out = paddle.dist(x, y, float("-inf"))
            print(out) # out = [0.]
    """
    check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'dist')
    check_variable_and_dtype(y, 'dtype', ['float32', 'float64'], 'dist')
    check_type(p, 'p', (float, int), 'dist')
    helper = LayerHelper("dist", **locals())
    out = helper.create_variable_for_type_inference(x.dtype)

    inputs = {"X": [x], "Y": [y]}
    outputs = {'Out': [out]}
    attrs = {"p": float(p)}
    helper.append_op(
        type='dist', inputs=inputs, outputs={'Out': out}, attrs=attrs)
    return out


def cond(x, p=None, name=None):
    """

    Computes the condition number of a matrix or batches of matrices with respect to a matrix norm ``p``.

    Args:
        x (Tensor): The input tensor could be tensor of shape ``(*, m, n)`` where ``*`` is zero or more batch dimensions 
            for ``p`` in ``(2, -2)``, or of shape ``(*, n, n)`` where every matrix is invertible for any supported ``p``. 
            And the input data type could be ``float32`` or ``float64``.
        p (float|string, optional): Order of the norm. Supported values are `fro`, `nuc`, `1`, `-1`, `2`, `-2`,
            `inf`, `-inf`. Default value is `None`, meaning that the order of the norm is `2`.
        name (str, optional): The default value is `None`. Normally there is no need for
            user to set this property. For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tensor: computing results of condition number, its data type is the same as input Tensor ``x``.

    Examples:
        .. code-block:: python

            import paddle
            import numpy as np

            x = paddle.to_tensor([[1., 0, -1], [0, 1, 0], [1, 0, 1]])

            # compute conditional number when p is None
            out = paddle.linalg.cond(x)
            # out.numpy() [1.4142135]

            # compute conditional number when order of the norm is 'fro'
            out_fro = paddle.linalg.cond(x, p='fro')
            # out_fro.numpy() [3.1622777]

            # compute conditional number when order of the norm is 'nuc'
            out_nuc = paddle.linalg.cond(x, p='nuc')
            # out_nuc.numpy() [9.2426405]

            # compute conditional number when order of the norm is 1
            out_1 = paddle.linalg.cond(x, p=1)
            # out_1.numpy() [2.]

            # compute conditional number when order of the norm is -1
            out_minus_1 = paddle.linalg.cond(x, p=-1)
            # out_minus_1.numpy() [1.]

            # compute conditional number when order of the norm is 2
            out_2 = paddle.linalg.cond(x, p=2)
            # out_2.numpy() [1.4142135]

            # compute conditional number when order of the norm is -1
            out_minus_2 = paddle.linalg.cond(x, p=-2)
            # out_minus_2.numpy() [0.70710677]

            # compute conditional number when order of the norm is inf
            out_inf = paddle.linalg.cond(x, p=np.inf)
            # out_inf.numpy() [2.]

            # compute conditional number when order of the norm is -inf
            out_minus_inf = paddle.linalg.cond(x, p=-np.inf)
            # out_minus_inf.numpy() [1.]

            a = paddle.to_tensor(np.random.randn(2, 4, 4).astype('float32'))
            # a.numpy() 
            # [[[ 0.14063153 -0.996288    0.7996131  -0.02571543]
            #   [-0.16303636  1.5534962  -0.49919784 -0.04402903]
            #   [-1.1341571  -0.6022629   0.5445269   0.29154757]
            #   [-0.16816919 -0.30972657  1.7521842  -0.5402487 ]]
            #  [[-0.58081484  0.12402827  0.7229862  -0.55046535]
            #   [-0.15178485 -1.1604939   0.75810957  0.30971205]
            #   [-0.9669573   1.0940945  -0.27363303 -0.35416734]
            #   [-1.216529    2.0018666  -0.7773689  -0.17556527]]]
            a_cond_fro = paddle.linalg.cond(a, p='fro')
            # a_cond_fro.numpy()  [31.572273 28.120834]

            b = paddle.to_tensor(np.random.randn(2, 3, 4).astype('float64'))
            # b.numpy()
            # [[[ 1.61707487  0.46829144  0.38130416  0.82546736]
            #   [-1.72710298  0.08866375 -0.62518804  0.16128892]
            #   [-0.02822879 -1.67764516  0.11141444  0.3220113 ]]
            #  [[ 0.22524372  0.62474921 -0.85503233 -1.03960523]
            #   [-0.76620689  0.56673047  0.85064753 -0.45158196]
            #   [ 1.47595418  2.23646462  1.5701758   0.10497519]]]
            b_cond_2 = paddle.linalg.cond(b, p=2)
            # b_cond_2.numpy()  [3.30064451 2.51976252]

    """

    def mat_norm(input, porder=1., axis=None):
        """
        NOTE:
            Calculate the matrix norm of a square matrix or batches of square matrices,
            when porder is in (1, -1, inf, -inf)
        """
        reduce_all = True if axis is None or axis == [] else False
        axis = axis if axis != None and axis != [] else [0]
        keepdim = False

        if in_dygraph_mode():
            abs_out = _C_ops.abs(input)
            sum_out = _C_ops.reduce_sum(abs_out, 'dim', axis, 'keepdim',
                                        keepdim, 'reduce_all', reduce_all)
            if porder == 1 or porder == np.inf:
                return _C_ops.reduce_max(sum_out, 'dim', [-1], 'keepdim',
                                         keepdim, 'reduce_all', reduce_all)
            if porder == -1 or porder == -np.inf:
                return _C_ops.reduce_min(sum_out, 'dim', [-1], 'keepdim',
                                         keepdim, 'reduce_all', reduce_all)

        block = LayerHelper('norm', **locals())
        abs_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        sum_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        block.append_op(
            type='abs', inputs={'X': input}, outputs={'Out': abs_out})
        block.append_op(
            type='reduce_sum',
            inputs={'X': abs_out},
            outputs={'Out': sum_out},
            attrs={'dim': axis,
                   'keep_dim': keepdim,
                   'reduce_all': reduce_all})
        if porder == 1 or porder == np.inf:
            block.append_op(
                type='reduce_max',
                inputs={'X': sum_out},
                outputs={'Out': out},
                attrs={
                    'dim': [-1],
                    'keep_dim': keepdim,
                    'reduce_all': reduce_all
                })
        if porder == -1 or porder == -np.inf:
            block.append_op(
                type='reduce_min',
                inputs={'X': sum_out},
                outputs={'Out': out},
                attrs={
                    'dim': [-1],
                    'keep_dim': keepdim,
                    'reduce_all': reduce_all
                })
        return out

    def fro_norm(input, porder=2, axis=[-1]):
        """
        NOTE:
            Calculate the frobenius norm of a square matrix or batches of square matrices.
        """
        reduce_all = True if axis is None or axis == [] else False
        keepdim = False

        if in_dygraph_mode():
            pow_out = _C_ops.pow(input, 'factor', porder)
            sum_out_1 = _C_ops.reduce_sum(pow_out, 'dim', axis, 'keepdim',
                                          keepdim, 'reduce_all', reduce_all)
            sum_out_2 = _C_ops.reduce_sum(sum_out_1, 'dim', axis, 'keepdim',
                                          keepdim, 'reduce_all', reduce_all)
            return _C_ops.pow(sum_out_2, 'factor', float(1. / porder))

        block = LayerHelper('norm', **locals())
        pow_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        sum_out_1 = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        sum_out_2 = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        block.append_op(
            type='pow',
            inputs={'X': input},
            outputs={'Out': pow_out},
            attrs={'factor': porder})
        block.append_op(
            type='reduce_sum',
            inputs={'X': pow_out},
            outputs={'Out': sum_out_1},
            attrs={'dim': axis,
                   'keep_dim': keepdim,
                   'reduce_all': reduce_all})
        block.append_op(
            type='reduce_sum',
            inputs={'X': sum_out_1},
            outputs={'Out': sum_out_2},
            attrs={'dim': axis,
                   'keep_dim': keepdim,
                   'reduce_all': reduce_all})
        block.append_op(
            type='pow',
            inputs={'X': sum_out_2},
            outputs={'Out': out},
            attrs={'factor': float(1. / porder)})
        return out

    def svd_norm(input, porder, axis=[-1]):
        """
        NOTE:
            Calculate the matrix norm, which is related to singular values, of a matrix
            or batches of matrices, including nuclear norm, 2-norm and (-2)-norm.
        """
        reduce_all = True if axis is None or axis == [] else False
        keepdim = False

        u, s, vh = svd(input, full_matrices=False)

        if in_dygraph_mode():
            if porder == "nuc":
                return _C_ops.reduce_sum(s, 'dim', axis, 'keepdim', keepdim,
                                         'reduce_all', reduce_all)
            max_out = _C_ops.reduce_max(s, 'dim', axis, 'keepdim', keepdim,
                                        'reduce_all', reduce_all)
            min_out = _C_ops.reduce_min(s, 'dim', axis, 'keepdim', keepdim,
                                        'reduce_all', reduce_all)
            if porder == 2:
                return _C_ops.elementwise_div(max_out, min_out, 'aixs', axis,
                                              'use_mkldnn', False)
            if porder == -2:
                return _C_ops.elementwise_div(min_out, max_out, 'aixs', axis,
                                              'use_mkldnn', False)

        block = LayerHelper('norm', **locals())
        out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        if porder == "nuc":
            block.append_op(
                type='reduce_sum',
                inputs={'X': s},
                outputs={'Out': out},
                attrs={
                    'dim': axis,
                    'keep_dim': keepdim,
                    'reduce_all': reduce_all
                })
            return out
        max_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        min_out = block.create_variable_for_type_inference(
            dtype=block.input_dtype())
        block.append_op(
            type='reduce_max',
            inputs={'X': s},
            outputs={'Out': max_out},
            attrs={'dim': axis,
                   'keep_dim': keepdim,
                   'reduce_all': reduce_all})
        block.append_op(
            type='reduce_min',
            inputs={'X': s},
            outputs={'Out': min_out},
            attrs={'dim': axis,
                   'keep_dim': keepdim,
                   'reduce_all': reduce_all})
        if porder == 2:
            block.append_op(
                type='elementwise_div',
                inputs={'X': max_out,
                        'Y': min_out},
                outputs={'Out': out},
                attrs={'aixs': axis,
                       'use_mkldnn': False})
            return out
        if porder == -2:
            block.append_op(
                type='elementwise_div',
                inputs={'X': min_out,
                        'Y': max_out},
                outputs={'Out': out},
                attrs={'aixs': axis,
                       'use_mkldnn': False})
            return out

    def empty_tensor(input, shape):
        if in_dygraph_mode():
            return input.reshape(shape)
        raise ValueError("only support x is nonempty tensor in static mode")

    x_shape = list(x.shape)
    if not len(x_shape) >= 2:
        raise ValueError("input should be a matrix or batches of matrices, " +
                         "but the dimention of received input is {}".format(
                             len(x_shape)))
    if p == None:
        p = 2
    x_size = 0 if (0 in x_shape) else 1
    if p in ("fro", "nuc", 1, -1, np.inf, -np.inf):
        if x_shape[len(x_shape) - 1] == x_shape[len(x_shape) - 2]:
            if x_size == 0:
                return empty_tensor(x, x_shape[:-2])
            x_inv = x.inverse()
            if p == "fro":
                return fro_norm(x) * fro_norm(x_inv)
            if p == "nuc":
                return svd_norm(x, p) * svd_norm(x_inv, p)
            if p in (1, -1):
                return mat_norm(
                    x, porder=p, axis=[-2]) * mat_norm(
                        x_inv, porder=p, axis=[-2])
            if p in (np.inf, -np.inf):
                return mat_norm(
                    x, porder=p, axis=[-1]) * mat_norm(
                        x_inv, porder=p, axis=[-1])
        else:
            raise ValueError("only support p is {} when input is a ".format(p) +
                             "square matrix or batches of square matrices")
    elif p in (2, -2):
        if x_size == 0:
            return empty_tensor(x, x_shape[:-2])
        return svd_norm(x, porder=p)
    else:
        raise ValueError(
            "unsupported {} for p, only supporting ('fro', 'nuc', ".format(
                p) + "1, -1, 2, -2, inf, -inf) or none")


def dot(x, y, name=None):
    """
    This operator calculates inner product for vectors.

    .. note::
       Support 1-d and 2-d Tensor. When it is 2d, the first dimension of this matrix
       is the batch dimension, which means that the vectors of multiple batches are dotted.

    Parameters:
        x(Tensor): 1-D or 2-D ``Tensor``. Its dtype should be ``float32``, ``float64``, ``int32``, ``int64``
        y(Tensor): 1-D or 2-D ``Tensor``. Its dtype soulde be ``float32``, ``float64``, ``int32``, ``int64``
        name(str, optional): Name of the output. Default is None. It's used to print debug info for developers. Details: :ref:`api_guide_Name`

    Returns:
        Tensor: the calculated result Tensor.

    Examples:

    .. code-block:: python

        import paddle
        import numpy as np

        x_data = np.random.uniform(0.1, 1, [10]).astype(np.float32)
        y_data = np.random.uniform(1, 3, [10]).astype(np.float32)
        x = paddle.to_tensor(x_data)
        y = paddle.to_tensor(y_data)
        z = paddle.dot(x, y)
        print(z)

    """
    op_type = 'dot'
    # skip var type check in dygraph mode to improve efficiency
    if in_dygraph_mode():
        op = getattr(_C_ops, op_type)
        return op(x, y)

    assert x is not None, 'x cannot be None in {}'.format(op_type)
    assert y is not None, 'y cannot be None in {}'.format(op_type)

    check_variable_and_dtype(x, 'x', ['float32', 'float64', 'int32', 'int64'],
                             op_type)
    check_variable_and_dtype(y, 'y', ['float32', 'float64', 'int32', 'int64'],
                             op_type)

    helper = LayerHelper(op_type, **locals())
    if name is None:
        out = helper.create_variable_for_type_inference(dtype=x.dtype)
    else:
        out = helper.create_variable(
            name=name, dtype=x.dtype, persistable=False)
    helper.append_op(
        type="dot", inputs={'X': x,
                            'Y': y}, attrs={}, outputs={"Out": out})
    return out


def t(input, name=None):
    """
    Transpose <=2-D tensor.
    0-D and 1-D tensors are returned as it is and 2-D tensor is equal to
    the paddle.transpose function which perm dimensions set 0 and 1.

    Args:
        input (Tensor): The input Tensor. It is a N-D (N<=2) Tensor of data types float16, float32, float64, int32.
        name(str, optional): The default value is None.  Normally there is no need for
            user to set this property.  For more information, please refer to :ref:`api_guide_Name`
    Returns:
        Tensor: A transposed n-D Tensor, with data type being float16, float32, float64, int32, int64.

    For Example:

        .. code-block:: text

             # Example 1 (0-D tensor)
             x = tensor([0.79])
             paddle.t(x) = tensor([0.79])

             # Example 2 (1-D tensor)
             x = tensor([0.79, 0.84, 0.32])
             paddle.t(x) = tensor([0.79, 0.84, 0.32])

             # Example 3 (2-D tensor)
             x = tensor([0.79, 0.84, 0.32],
                        [0.64, 0.14, 0.57])
             paddle.t(x) = tensor([0.79, 0.64],
                                  [0.84, 0.14],
                                  [0.32, 0.57])

     Examples:

        .. code-block:: python

            import paddle
            x = paddle.ones(shape=[2, 3], dtype='int32')
            x_transposed = paddle.t(x)
            print(x_transposed.shape)
            # [3, 2]
    """
    if len(input.shape) > 2:
        raise ValueError(
            "Input(input) only support N-D (N<=2) tensor, but received "
            "length of Input(input) is %s. Perhaps you can use paddle."
            "tensor.transpose() instead." % len(input.shape))
    if in_dygraph_mode():
        if len(input.shape) == 1:
            return input
        # 2-D tensor
        perm = [1, 0]
        out, _ = _C_ops.transpose2(input, 'axis', perm)
        return out

    check_variable_and_dtype(
        input, 'input', ['float16', 'float32', 'float64', 'int32', 'int64'],
        'transpose')

    helper = LayerHelper('t', **locals())
    out = helper.create_variable_for_type_inference(input.dtype)
    input_shape = helper.create_variable_for_type_inference(input.dtype)
    if len(input.shape) == 1:
        out = input
    else:
        helper.append_op(
            type='transpose2',
            inputs={'X': [input]},
            outputs={'Out': [out],
                     'XShape': [input_shape]},
            attrs={'axis': [1, 0]})
    return out


def cross(x, y, axis=None, name=None):
    """
    Computes the cross product between two tensors along an axis.

    Inputs must have the same shape, and the length of their axes should be equal to 3.
    If `axis` is not given, it defaults to the first axis found with the length 3.

    Args:
        x (Tensor): The first input tensor.
        y (Tensor): The second input tensor.
        axis (int, optional): The axis along which to compute the cross product. It defaults to the first axis found with the length 3.
        name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tensor. A Tensor with same data type as `x`.

    Examples:
        .. code-block:: python

            import paddle

            x = paddle.to_tensor([[1.0, 1.0, 1.0],
                                  [2.0, 2.0, 2.0],
                                  [3.0, 3.0, 3.0]])
            y = paddle.to_tensor([[1.0, 1.0, 1.0],
                                  [1.0, 1.0, 1.0],
                                  [1.0, 1.0, 1.0]])

            z1 = paddle.cross(x, y)
            # [[-1. -1. -1.]
            #  [ 2.  2.  2.]
            #  [-1. -1. -1.]]

            z2 = paddle.cross(x, y, axis=1)
            # [[0. 0. 0.]
            #  [0. 0. 0.]
            #  [0. 0. 0.]]
    """
    if in_dygraph_mode():
        if axis is not None:
            return _C_ops.cross(x, y, 'dim', axis)
        else:
            return _C_ops.cross(x, y)

    helper = LayerHelper("cross", **locals())
    out = helper.create_variable_for_type_inference(x.dtype)
    attrs = dict()
    attrs['dim'] = axis

    helper.append_op(
        type='cross',
        inputs={'X': x,
                'Y': y},
        outputs={'Out': out},
        attrs=attrs)
    return out


def cholesky(x, upper=False, name=None):
    r"""
    Computes the Cholesky decomposition of one symmetric positive-definite
    matrix or batches of symmetric positive-definite matrice.

    If `upper` is `True`, the decomposition has the form :math:`A = U^{T}U` ,
    and the returned matrix :math:`U` is upper-triangular. Otherwise, the
    decomposition has the form  :math:`A = LL^{T}` , and the returned matrix
    :math:`L` is lower-triangular.

    Args:
        x (Tensor): The input tensor. Its shape should be `[*, M, M]`,
            where * is zero or more batch dimensions, and matrices on the
            inner-most 2 dimensions all should be symmetric positive-definite.
            Its data type should be float32 or float64.
        upper (bool): The flag indicating whether to return upper or lower
            triangular matrices. Default: False.

    Returns:
        Tensor: A Tensor with same shape and data type as `x`. It represents \
            triangular matrices generated by Cholesky decomposition.

    Examples:
        .. code-block:: python

            import paddle
            import numpy as np

            a = np.random.rand(3, 3)
            a_t = np.transpose(a, [1, 0])
            x_data = np.matmul(a, a_t) + 1e-03
            x = paddle.to_tensor(x_data)
            out = paddle.cholesky(x, upper=False)
            print(out)
            # [[1.190523   0.         0.        ]
            #  [0.9906703  0.27676893 0.        ]
            #  [1.25450498 0.05600871 0.06400121]]

    """
    if in_dygraph_mode():
        return _C_ops.cholesky(x, "upper", upper)
    check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'cholesky')
    check_type(upper, 'upper', bool, 'cholesky')
    helper = LayerHelper('cholesky', **locals())
    out = helper.create_variable_for_type_inference(dtype=x.dtype)
    helper.append_op(
        type='cholesky',
        inputs={'X': [x]},
        outputs={'Out': out},
        attrs={'upper': upper})
    return out


def matrix_rank(x, tol=None, hermitian=False, name=None):
    r"""
    Computes the rank of a matrix.

    The rank of a matrix is the number of singular values that are greater than the specified tol threshold when hermitian=False,
    or the number of eigenvalues in absolute value that are greater than the specified tol threshold when hermitian=True.

    Args:
        x (Tensor): The input tensor.
            Its shape should be [..., m, n], where ... is zero or more batch dimensions. If x is a batch of matrices then the output
            has the same batch dimensions. The data type of x should be float32 or float64.
        tol (float,Tensor,optional): the tolerance value. Default: None.
            If tol is not specified, and sigma is the largest singular value (or eigenvalue in absolute value), and eps is the
            epsilon value for the dtype of x, then tol is computed with formula tol=sigma * max(m,n) * eps. Note that if x is
            a batch of matrices, tol is computed this way for every batch.
        hermitian (bool,optional): indicates whether x is Hermitian. Default: False.
            When hermitian=True, x is assumed to be Hermitian, but x is not checked inside the function. Instead, We just use the
            lower triangular of the matrix to compute.
        name (str, optional): Name for the operation (optional, default is None). For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tensor: Rank of tensor x.

    Examples:
        .. code-block:: python

            import paddle

            a = paddle.eye(10)
            b = paddle.linalg.matrix_rank(a)
            print(b)
            # b = [10]

            c = paddle.ones(shape=[3, 4, 5, 5])
            d = paddle.linalg.matrix_rank(c, tol=0.01, hermitian=True)
            print(d)
            # d = [[1, 1, 1, 1],
            #      [1, 1, 1, 1],
            #      [1, 1, 1, 1]]

    """

    if in_dygraph_mode():
        if tol is None:
            tol_tensor = None
            tol_attr = 0.0
            use_default_tol = True
        elif isinstance(tol, Variable):
            if tol.dtype != x.dtype:
                tol_tensor = cast(tol, x.dtype)
            else:
                tol_tensor = tol
            tol_attr = 0.0
            use_default_tol = False
        else:
            tol_tensor = None
            tol_attr = float(tol)
            use_default_tol = False
        return _C_ops.matrix_rank(x, tol_tensor, "tol", tol_attr, 'hermitian',
                                  hermitian, 'use_default_tol', use_default_tol)

    inputs = {}
    attrs = {}
    check_variable_and_dtype(x, 'x', ['float32', 'float64'], 'matrix_rank')
    inputs['X'] = x
    if tol is None:
        attrs['use_default_tol'] = True
    elif isinstance(tol, Variable):
        check_variable_and_dtype(tol, 'tol', ['float32'], 'matrix_rank')
        attrs['use_default_tol'] = False
        if tol.dtype != x.dtype:
            inputs['TolTensor'] = cast(tol, x.dtype)
        else:
            inputs['TolTensor'] = tol
    else:
        check_type(tol, 'tol', float, 'matrix_rank')
        attrs['use_default_tol'] = False
        attrs['tol'] = tol
    check_type(hermitian, 'hermitian', bool, 'matrix_rank')
    attrs['hermitian'] = hermitian

    helper = LayerHelper('matrix_rank', **locals())
    out = helper.create_variable_for_type_inference(dtype='int32')
    helper.append_op(
        type='matrix_rank', inputs=inputs, outputs={'Out': out}, attrs=attrs)
    return out


def bmm(x, y, name=None):
    """
    Applies batched matrix multiplication to two tensors.

    Both of the two input tensors must be three-dementional and share the same batch size.

    if x is a (b, m, k) tensor, y is a (b, k, n) tensor, the output will be a (b, m, n) tensor.

    Args:
        x (Tensor): The input Tensor.
        y (Tensor): The input Tensor.
        name(str|None): A name for this layer(optional). If set None, the layer
            will be named automatically.

    Returns:
        Tensor: The product Tensor.

    Examples:
        .. code-block:: python

            import paddle

            # In imperative mode:
            # size x: (2, 2, 3) and y: (2, 3, 2)
            x = paddle.to_tensor([[[1.0, 1.0, 1.0],
                                [2.0, 2.0, 2.0]],
                                [[3.0, 3.0, 3.0],
                                [4.0, 4.0, 4.0]]])
            y = paddle.to_tensor([[[1.0, 1.0],[2.0, 2.0],[3.0, 3.0]],
                                [[4.0, 4.0],[5.0, 5.0],[6.0, 6.0]]])
            out = paddle.bmm(x, y)
            #output size: (2, 2, 2)
            #output value:
            #[[[6.0, 6.0],[12.0, 12.0]],[[45.0, 45.0],[60.0, 60.0]]]
            out_np = out.numpy()
            
    """
    x_shape = x.shape
    y_shape = y.shape
    if not len(x_shape) == len(y_shape) == 3:
        raise ValueError(
            "x and y should be 3-dimensional. But received x's dimention: {}, y's dimention: {}".
            format(x_shape, y_shape))
    if x_shape[2] != y_shape[1]:
        raise ValueError(
            "x's width must be equal with y's height. But received x's shape: {}, y's shape: {}".
            format(x_shape, y_shape))
    if x_shape[0] != y_shape[0]:
        raise ValueError(
            "x's batch (shape[0]) must be equal with y's batch (shape[0]). But received x's shape: {}, y's shape: {}".
            format(x_shape, y_shape))

    if in_dygraph_mode():
        return _C_ops.bmm(x, y)

    helper = LayerHelper('bmm', **locals())
    out = helper.create_variable_for_type_inference(dtype=x.dtype)
    helper.append_op(type='bmm', inputs={'X': x, 'Y': y}, outputs={'Out': out})
    return out


def histogram(input, bins=100, min=0, max=0):
    """
    Computes the histogram of a tensor. The elements are sorted into equal width bins between min and max.
    If min and max are both zero, the minimum and maximum values of the data are used.

    Args:
        input (Tensor): A Tensor(or LoDTensor) with shape :math:`[N_1, N_2,..., N_k]` . The data type of the input Tensor
            should be float32, float64, int32, int64.
        bins (int): number of histogram bins
        min (int): lower end of the range (inclusive)
        max (int): upper end of the range (inclusive)

    Returns:
        Tensor: data type is int64, shape is (nbins,).

    Examples:
        .. code-block:: python

            import paddle

            inputs = paddle.to_tensor([1, 2, 1])
            result = paddle.histogram(inputs, bins=4, min=0, max=3)
            print(result) # [0, 2, 1, 0]
    """
    if in_dygraph_mode():
        return _C_ops.histogram(input, "bins", bins, "min", min, "max", max)

    helper = LayerHelper('histogram', **locals())
    check_variable_and_dtype(
        input, 'X', ['int32', 'int64', 'float32', 'float64'], 'histogram')
    out = helper.create_variable_for_type_inference(VarDesc.VarType.INT64)
    helper.append_op(
        type='histogram',
        inputs={'X': input},
        outputs={'Out': out},
        attrs={'bins': bins,
               'min': min,
               'max': max})
    return out


def mv(x, vec, name=None):
    """
    Performs a matrix-vector product of the matrix x and the vector vec.

    Args:
        x (Tensor): A tensor with shape :math:`[M, N]` , The data type of the input Tensor x
            should be one of float32, float64.
        vec (Tensor): A tensor with shape :math:`[N]` , The data type of the input Tensor x
            should be one of float32, float64.
        name(str, optional): The default value is None.  Normally there is no need for user to set this
            property.  For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tensor: The tensor which is producted by x and vec.

    Examples:
        .. code-block:: python

            # x: [M, N], vec: [N]
            # paddle.mv(x, vec)  # out: [M]

            import numpy as np
            import paddle

            x_data = np.array([[2, 1, 3], [3, 0, 1]]).astype("float64")
            x = paddle.to_tensor(x_data)
            vec_data = np.array([3, 5, 1])
            vec = paddle.to_tensor(vec_data).astype("float64")
            out = paddle.mv(x, vec)
    """
    if in_dygraph_mode():
        out = _C_ops.mv(x, vec)
        return out

    def __check_input(x, vec):
        var_names = {'x': x, 'vec': vec}
        for name, val in var_names.items():
            check_variable_and_dtype(val, name, ['float32', 'float64'], 'mv')
        x_shape = list(x.shape)
        vec_shape = list(vec.shape)
        if len(x_shape) != 2:
            raise ValueError(
                "x should be 2-dimensional. But received x's dimention: {}".
                format(x_shape))
        if len(vec_shape) != 1:
            raise ValueError(
                "vec should be 1-dimensional. But received vec's dimention: {}".
                format(vec_shape))

    __check_input(x, vec)

    helper = LayerHelper('mv', **locals())
    out = helper.create_variable_for_type_inference(dtype=x.dtype)
    helper.append_op(
        type='mv', inputs={'X': x,
                           'Vec': vec}, outputs={'Out': out})
    return out


def svd(x, full_matrices=False, name=None):
    r"""
    Computes the singular value decomposition of one matrix or a batch of regular matrices.

    Let :math:`X` be the input matrix or a batch of input matrices, the output should satisfies:

    .. math::
        X = U * diag(S) * VT 
 
    Args:
        x (Tensor): The input tensor. Its shape should be `[..., N, M]`,
            where `...` is zero or more batch dimensions. N and M can be arbitraty
            positive number. Note that if x is sigular matrices, the grad is numerical 
            instable. The data type of x should be float32 or float64. 
        full_matrices (bool): A flag to control the behavor of svd. 
            If full_matrices = True, svd op will compute full U and V matrics, 
            which means shape of U is `[..., N, N]`, shape of V is `[..., M, M]`. K = min(M, N).
            If full_matrices = False, svd op will use a economic method to store U and V. 
            which means shape of U is `[..., N, K]`, shape of V is `[..., M, K]`. K = min(M, N).
        name (str, optional): Name for the operation (optional, default is None). 
            For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tuple of 3 tensors: (U, S, VH). VH is the conjugate transpose of V. S is the singlar value vectors of matrics with shape `[..., K]`

    Examples:
        .. code-block:: python

            import paddle

            x = paddle.to_tensor([[1.0, 2.0], [1.0, 3.0], [4.0, 6.0]]).astype('float64')
            x = x.reshape([3, 2])
            u, s, vh = paddle.linalg.svd(x)
            print (u)
            #U = [[ 0.27364809, -0.21695147  ],
            #      [ 0.37892198, -0.87112408 ],
            #      [ 0.8840446 ,  0.44053933 ]]

            print (s)
            #S = [8.14753743, 0.78589688]
            print (vh)
            #VT= [[ 0.51411221,  0.85772294],
            #     [ 0.85772294, -0.51411221]]
            
            # one can verify : U * S * VT == X
            #                  U * UH == I 
            #                  V * VH == I
    """

    if in_dygraph_mode():
        return _C_ops.svd(x, 'full_matrices', full_matrices)
    check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'svd')
    check_type(full_matrices, 'full_matrices', bool, 'svd')
    helper = LayerHelper('svd', **locals())
    u = helper.create_variable_for_type_inference(dtype=x.dtype)
    vh = helper.create_variable_for_type_inference(dtype=x.dtype)
    s = helper.create_variable_for_type_inference(dtype=x.dtype)
    attrs = dict()
    attrs['full_matrices'] = full_matrices
    helper.append_op(
        type='svd',
        inputs={'X': [x]},
        outputs={'U': u,
                 'VH': vh,
                 'S': s},
        attr=attrs, )
    return u, s, vh


def matrix_power(x, n, name=None):
    r"""
    Computes the n-th power of a square matrix or a batch of square matrices.
    
    Let :math:`X` be a sqaure matrix or a batch of square matrices, :math:`n` be
    an exponent, the equation should be:

    .. math::
        Out = X ^ {n}

    Specifically,

    - If `n > 0`, it returns the matrix or a batch of matrices raised to the power
    of `n`.

    - If `n = 0`, it returns the identity matrix or a batch of identity matrices.

    - If `n < 0`, it returns the inverse of each matrix (if invertible) raised to
    the power of `abs(n)`.

    Args:
        x (Tensor): A square matrix or a batch of square matrices to be raised
            to power `n`. Its shape should be `[*, M, M]`, where `*` is zero or
            more batch dimensions. Its data type should be float32 or float64.
        n (int): The exponent. It can be any positive, negative integer or zero.
        name (str, optional): Name for the operation (optional, default is None).
            For more information, please refer to :ref:`api_guide_Name`.

    Returns:
        Tensor: The n-th power of the matrix (or the batch of matrices) `x`. Its
            data type should be the same as that of `x`.

    Examples:
        .. code-block:: python

            import paddle

            x = paddle.to_tensor([[1, 2, 3],
                                  [1, 4, 9],
                                  [1, 8, 27]], dtype='float64')
            print(paddle.matrix_power(x, 2))
            # [[6.  , 34. , 102.],
            #  [14. , 90. , 282.],
            #  [36. , 250., 804.]]

            print(paddle.matrix_power(x, 0))
            # [[1., 0., 0.],
            #  [0., 1., 0.],
            #  [0., 0., 1.]]

            print(paddle.matrix_power(x, -2))
            # [[ 12.91666667, -12.75000000,  2.83333333 ],
            #  [-7.66666667 ,  8.         , -1.83333333 ],
            #  [ 1.80555556 , -1.91666667 ,  0.44444444 ]]
    """
    if in_dygraph_mode():
        return core.ops.matrix_power(x, "n", n)

    check_variable_and_dtype(x, 'dtype', ['float32', 'float64'], 'matrix_power')
    check_type(n, 'n', int, 'matrix_power')
    helper = LayerHelper('matrix_power', **locals())
    out = helper.create_variable_for_type_inference(dtype=x.dtype)
    helper.append_op(
        type='matrix_power',
        inputs={'X': x},
        outputs={'Out': out},
        attrs={'n': n})
    return out


def multi_dot(x, name=None):
    """
    Multi_dot is an operator that calculates multiple matrix multiplications.

    Supports inputs of float, double and float16 dtypes. This function does not
    support batched inputs.

    The input tensor in [x] must be 2-D except for the first and last can be 1-D.
    If the first tensor is a 1-D vector of shape(n, ) it is treated as row vector
    of shape(1, n), similarly if the last tensor is a 1D vector of shape(n, ), it
    is treated as a column vector of shape(n, 1).

    If the first and last tensor are 2-D matrix, then the output is also 2-D matrix,
    otherwise the output is a 1-D vector.

    Multi_dot will select the lowest cost multiplication order for calculation. The
    cost of multiplying two matrices with shapes (a, b) and (b, c) is a * b * c.
    Given matrices A, B, C with shapes (20, 5), (5, 100), (100, 10) respectively,
    we can calculate the cost of different multiplication orders as follows:
    - Cost((AB)C) = 20x5x100 + 20x100x10 = 30000
    - Cost(A(BC)) = 5x100x10 + 20x5x10 = 6000

    In this case, multiplying B and C first, then multiply A, which is 5 times faster
    than sequential calculation.

    Args:
        x ([Tensor]): The input tensors which is a list Tensor.
        name(str|None): A name for this layer(optional). If set None, the layer
            will be named automatically.

    Returns:
        Tensor: The output Tensor.


    Examples:

    .. code-block:: python

        import paddle
        import numpy as np

        # A * B
        A_data = np.random.random([3, 4]).astype(np.float32)
        B_data = np.random.random([4, 5]).astype(np.float32)
        A = paddle.to_tensor(A_data)
        B = paddle.to_tensor(B_data)
        out = paddle.multi_dot([A, B])
        print(out.numpy().shape)
        # [3, 5]

        # A * B * C
        A_data = np.random.random([10, 5]).astype(np.float32)
        B_data = np.random.random([5, 8]).astype(np.float32)
        C_data = np.random.random([8, 7]).astype(np.float32)
        A = paddle.to_tensor(A_data)
        B = paddle.to_tensor(B_data)
        C = paddle.to_tensor(C_data)
        out = paddle.multi_dot([A, B, C])
        print(out.numpy().shape)
        # [10, 7]

    """
    if in_dygraph_mode():
        return _C_ops.multi_dot(x)

    check_type(x, 'x', (list, tuple), 'multi_dot')
    for id, item in enumerate(x):
        check_variable_and_dtype(item, 'x[' + str(id) + ']',
                                 ['float16', 'float32', 'float64'], 'multi_dot')
        if item.dtype != x[0].dtype:
            raise TypeError(
                "All the Tensors in the input must have the same data type.")

    helper = LayerHelper('multi_dot', **locals())
    dtype = helper.input_dtype(input_param_name='x')
    out = helper.create_variable_for_type_inference(dtype)
    helper.append_op(type='multi_dot', inputs={"X": x}, outputs={"Out": out})
    return out


def eigh(x, UPLO='L', name=None):
    """
    Compute the eigenvalues and eigenvectors of a 
    complex Hermitian (conjugate symmetric) or a real symmetric matrix.

    Args:
        x (Tensor): A tensor with shape :math:`[*, N, N]` , The data type of the input Tensor x
            should be one of float32, float64, complex64, complex128.
        UPLO(str, optional): (string, default 'L'), 'L' represents the lower triangular matrix,
                        "'U' represents the upper triangular matrix.".
        name(str, optional): The default value is None.  Normally there is no need for user to set this
            property.  For more information, please refer to :ref:`api_guide_Name`.

    Returns:

        out_value(Tensor):  A Tensor with shape [*, N] and data type of float32 and float64. The eigenvalues of eigh op.
        out_vector(Tensor): A Tensor with shape [*, N, N] and data type of float32,float64,complex64 and complex128. The eigenvectors of eigh op.

    Examples:
        .. code-block:: python

            import numpy as np
            import paddle

            x_data = np.array([[1, -2j], [2j, 5]])
            x = paddle.to_tensor(x_data)
            out_value, out_vector = paddle.eigh(x, UPLO='L')
            print(out_value)
            #[0.17157288, 5.82842712]
            print(out_vector)
            #[(-0.9238795325112867+0j), (-0.3826834323650898+0j)],
            #[ 0.3826834323650898j    , -0.9238795325112867j    ]]

    """
    if in_dygraph_mode():
        return _C_ops.eigh(x, 'UPLO', UPLO)

    def __check_input(x, UPLO):
        x_shape = list(x.shape)
        if len(x.shape) < 2:
            raise ValueError(
                "Input(input) only support >=2 tensor, but received "
                "length of Input(input) is %s." % len(x.shape))
        if x_shape[-1] != x_shape[-2]:
            raise ValueError(
                "The input matrix must be batches of square matrices. But received x's dimention: {}".
                format(x_shape))
        if UPLO is not 'L' and UPLO is not 'U':
            raise ValueError(
                "UPLO must be L or U. But received UPLO is: {}".format(UPLO))

    __check_input(x, UPLO)

    helper = LayerHelper('eigh', **locals())
    check_variable_and_dtype(
        x, 'dtype', ['float32', 'float64', 'complex64', 'complex128'], 'eigh')

    out_value = helper.create_variable_for_type_inference(dtype=x.dtype)
    out_vector = helper.create_variable_for_type_inference(dtype=x.dtype)

    helper.append_op(
        type='eigh',
        inputs={'X': x},
        outputs={'Eigenvalues': out_value,
                 'Eigenvectors': out_vector},
        attrs={'UPLO': UPLO})
    return out_value, out_vector