code: def cov(
      x,
      y,
      mean_x=None,
      mean_y=None,
      return_gram=False,
      return_mean=False,
  ):
      """
      Computes a covariance between two matrices using
      the form Cov(X, Y) = E(XY) - E(X)E(Y)
  
      This function prevents the need to explicitly
      compute the outer product E(X)E(Y) by taking
      advantage of the symmetry of that matrix
      and computes per element in a kernel.
  
      When E(XY) is approximately equal to E(X)E(Y),
      this method is prone to catastrophic cancellation.
      In such cases, a rectangular solver should be
      preferred.
  
      Parameters
      ----------
  
      x : device-array or cupyx.scipy.sparse of size (m, n)
      y : device-array or cupyx.scipy.sparse of size (m, n)
      mean_x : float (default = None)
          device-array of size (n, ) which is the mean
          of x across rows
      mean_x : float (default = None)
          device-array of size (n, ) which is the mean
          of x across rows
      return_gram : boolean (default = False)
          If True, gram matrix of the form (1 / n) * X.T.dot(Y)
          will be returned.
          When True, a copy will be created
          to store the results of the covariance.
          When False, the local gram matrix result
          will be overwritten
      return_mean: boolean (default = False)
          If True, the Maximum Likelihood Estimate used to
          calculate the mean of X and Y will be returned,
          of the form (1 / n) * mean(X) and (1 / n) * mean(Y)
  
      Returns
      -------
  
      result : cov(X, Y) when return_gram and return_mean are False
               cov(X, Y), gram(X, Y) when return_gram is True,
                  return_mean is False
               cov(X, Y), mean(X), mean(Y) when return_gram is False,
                  return_mean is True
               cov(X, Y), gram(X, Y), mean(X), mean(Y)
                  when return_gram is True and return_mean is True
      """
  
      if x.dtype != y.dtype:
          raise ValueError(
              "X and Y must have same dtype (%s != %s)"
              % (x.dtype, y.dtype)
          )
  
      if x.shape != y.shape:
          raise ValueError(
              "X and Y must have same shape %s != %s"
              % (x.shape, y.shape)
          )
  
      # Fix for cuml issue #5475 & cupy issue #7699
      # addressing problems with sparse matrix multiplication (spGEMM)
      if (
          x is y
          and cupyx.scipy.sparse.issparse(x)
          and mean_x is None
          and mean_y is None
      ):
          return _cov_sparse(
              x, return_gram=return_gram, return_mean=return_mean
          )
  
      if mean_x is not None and mean_y is not None:
          if mean_x.dtype != mean_y.dtype:
              raise ValueError(
                  "Mean of X and Mean of Y must have same dtype"
                  "(%s != %s)" % (mean_x.dtype, mean_y.dtype)
              )
  
          if mean_x.shape != mean_y.shape:
              raise ValueError(
                  "Mean of X and Mean of Y must have same shape"
                  "%s != %s" % (mean_x.shape, mean_y.shape)
              )
  
      gram_matrix = x.T.dot(y) * (1 / x.shape[0])
  
      if cupyx.scipy.sparse.issparse(gram_matrix):
          gram_matrix = gram_matrix.todense()
  
      if mean_x is None:
          mean_x = x.sum(axis=0) * (1 / x.shape[0])
  
      if mean_y is None:
          mean_y = y.sum(axis=0) * (1 / y.shape[0])
  
      if return_gram:
          cov_result = cp.zeros(
              (gram_matrix.shape[0], gram_matrix.shape[1]),
              dtype=gram_matrix.dtype,
          )
      else:
          cov_result = gram_matrix
  
      compute_cov = _cov_kernel(x.dtype)
  
      block_size = (32, 32)
      grid_size = (
          math.ceil(gram_matrix.shape[0] / 32),
          math.ceil(gram_matrix.shape[1] / 32),
      )
  
      compute_cov(
          grid_size,
          block_size,
          (
              cov_result,
              gram_matrix,
              mean_x,
              mean_y,
              gram_matrix.shape[0],
          ),
      )
  
      if not return_gram and not return_mean:
          return cov_result
      elif return_gram and not return_mean:
          return cov_result, gram_matrix
      elif not return_gram and return_mean:
          return cov_result, mean_x, mean_y
      elif return_gram and return_mean:
          return cov_result, gram_matrix, mean_x, mean_y
  
file_path: cuml/python/cuml/prims/stats/covariance.py
imports: from cuml.common.kernel_utils import cuda_kernel_factory
  import cuml.internals
  import math
  from cuml.internals.safe_imports import gpu_only_import
label_desc: Use image node_id 5 for calling a global function with example usage: cov(x, y, mean_x, mean_y, return_gram, return_mean) and returns: _cov_sparse, cov_result, cov_result,  gram_matrix, cov_result,  mean_x,  mean_y, cov_result,  gram_matrix,  mean_x,  mean_y