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  """Namespace for ops used in imperative programming."""
  
  from __future__ import absolute_import
  from ..ndarray import numpy as _mx_nd_np
  
  __all__ = ['norm', 'svd', 'cholesky']
  
  
  def norm(x, ord=None, axis=None, keepdims=False):
      r"""Matrix or vector norm.
  
      This function can only support Frobenius norm for now.
      The Frobenius norm is given by [1]_:
  
          :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
  
      Parameters
      ----------
      x : ndarray
          Input array.
      ord : {'fro'}, optional
          Order of the norm.
      axis : {int, 2-tuple of ints, None}, optional
          If `axis` is an integer, it specifies the axis of `x` along which to
          compute the vector norms.  If `axis` is a 2-tuple, it specifies the
          axes that hold 2-D matrices, and the matrix norms of these matrices
          are computed.  If `axis` is None, the norm of the whole ndarray is
          returned.
  
      keepdims : bool, optional
          If this is set to True, the axes which are normed over are left in the
          result as dimensions with size one.  With this option the result will
          broadcast correctly against the original `x`.
  
      Returns
      -------
      n : float or ndarray
          Norm of the matrix or vector(s).
  
      Notes
      -----
      This operator differs from NumPy in the aspect that it always returns a
      zero-dim tensor for the cases where Python float values are expected
      in NumPy.
  
      References
      ----------
      .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
             Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
  
      Examples
      --------
      >>> from numpy import linalg as LA
      >>> a = np.arange(9) - 4
      >>> a
      array([-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.])
      >>> b = a.reshape((3, 3))
      >>> b
      array([[-4., -3., -2.],
             [-1.,  0.,  1.],
             [ 2.,  3.,  4.]])
      >>> LA.norm(a)
      array(7.745967)
      >>>
      >>> LA.norm(b)
      array(7.745967)
      >>> LA.norm(b, 'fro')
      array(7.745967)
      """
      return _mx_nd_np.linalg.norm(x, ord, axis, keepdims)
  
  
  def svd(a):
      r"""
      Singular Value Decomposition.
  
      When `a` is a 2D array, it is factorized as ``ut @ np.diag(s) @ v``,
      where `ut` and `v` are 2D orthonormal arrays and `s` is a 1D
      array of `a`'s singular values. When `a` is higher-dimensional, SVD is
      applied in stacked mode as explained below.
  
      Parameters
      ----------
      a : (..., M, N) ndarray
          A real array with ``a.ndim >= 2`` and ``M <= N``.
  
      Returns
      -------
      ut: (..., M, M) ndarray
          Orthonormal array(s). The first ``a.ndim - 2`` dimensions have the same
          size as those of the input `a`.
      s : (..., M) ndarray
          Vector(s) with the singular values, within each vector sorted in
          descending order. The first ``a.ndim - 2`` dimensions have the same
          size as those of the input `a`.
      v : (..., M, N) ndarray
          Orthonormal array(s). The first ``a.ndim - 2`` dimensions have the same
          size as those of the input `a`.
  
      Notes
      -----
  
      The decomposition is performed using LAPACK routine ``_gesvd``.
  
      SVD is usually described for the factorization of a 2D matrix :math:`A`.
      The higher-dimensional case will be discussed below. In the 2D case, SVD is
      written as :math:`A = U^T S V`, where :math:`A = a`, :math:`U^T = ut`,
      :math:`S= \mathtt{np.diag}(s)` and :math:`V = v`. The 1D array `s`
      contains the singular values of `a` and `ut` and `v` are orthonormal. The rows
      of `v` are the eigenvectors of :math:`A^T A` and the columns of `ut` are
      the eigenvectors of :math:`A A^T`. In both cases the corresponding
      (possibly non-zero) eigenvalues are given by ``s**2``.
  
      The sign of rows of `u` and `v` are determined as described in
      `Auto-Differentiating Linear Algebra <https://arxiv.org/pdf/1710.08717.pdf>`_.
  
      If `a` has more than two dimensions, then broadcasting rules apply.
      This means that SVD is working in "stacked" mode: it iterates over
      all indices of the first ``a.ndim - 2`` dimensions and for each
      combination SVD is applied to the last two indices. The matrix `a`
      can be reconstructed from the decomposition with either
      ``(ut * s[..., None, :]) @ v`` or
      ``ut @ (s[..., None] * v)``. (The ``@`` operator denotes batch matrix multiplication)
  
      This function differs from the original `numpy.linalg.svd
      <https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.svd.html>`_ in
      the following way(s):
       - The sign of rows of `u` and `v` may differ.
       - Does not support complex input.
  
      Examples
      --------
      >>> a = np.arange(54).reshape(6, 9)
      >>> ut, s, v = np.linalg.svd(a)
      >>> ut.shape, s.shape, v.shape
      ((6, 6), (6,), (6, 9))
      >>> s = s.reshape(6, 1)
      >>> ret = np.dot(ut, s * v)
      >>> (ret - a > 1e-3).sum()
      array(0.)
      >>> (ret - a < -1e-3).sum()
      array(0.)
      """
      return _mx_nd_np.linalg.svd(a)
  
  
  def cholesky(a):
      r"""
      Cholesky decomposition.
  
      Return the Cholesky decomposition, `L * L.T`, of the square matrix `a`,
      where `L` is lower-triangular and .T is the transpose operator. `a` must be
      symmetric and positive-definite. Only `L` is actually returned. Complex-valued
      input is currently not supported.
  
      Parameters
      ----------
      a : (..., M, M) ndarray
          Symmetric, positive-definite input matrix.
  
      Returns
      -------
      L : (..., M, M) ndarray
          Lower-triangular Cholesky factor of `a`.
  
      Raises
      ------
      MXNetError
          If the decomposition fails, for example, if `a` is not positive-definite.
  
      Notes
      -----
      Broadcasting rules apply.
  
      The Cholesky decomposition is often used as a fast way of solving
  
      .. math:: A \mathbf{x} = \mathbf{b}
  
      (when `A` is both symmetric and positive-definite).
  
      First, we solve for :math:`\mathbf{y}` in
  
      .. math:: L \mathbf{y} = \mathbf{b},
  
      and then for :math:`\mathbf{x}` in
  
      .. math:: L.T \mathbf{x} = \mathbf{y}.
  
      Examples
      --------
      >>> A = np.array([[16, 4], [4, 10]])
      >>> A
      array([[16.,  4.],
             [ 4., 10.]])
      >>> L = np.linalg.cholesky(A)
      >>> L
      array([[4., 0.],
             [1., 3.]])
      >>> np.dot(L, L.T)
      array([[16.,  4.],
             [ 4., 10.]])
      """
      return _mx_nd_np.linalg.cholesky(a)
  