code: def _subgradient_update_barycenter(
      X, list_diag_v_k, list_w_k, weights_sum, barycenter, eta
  ):
      """Update barycenters using the formula from Algorithm 1 in [1]_.
  
      Parameters
      ----------
      X : numpy.array of shape (n, sz, d)
          Time-series to be averaged
  
      list_diag_v_k : list of numpy.array of shape (barycenter_size, )
          list of weighted :math:`V^{(k)}` diagonals (as vectors)
  
      list_w_k : list of numpy.array of shape (barycenter_size, sz_k)
          list of weighted :math:`W^{(k)}` matrices
  
      weights_sum : float
          sum of weights applied to matrices :math:`V^{(k)}` and :math:`W^{(k)}`
  
      barycenter : numpy.array of shape (barycenter_size, d)
          Barycenter as computed at the previous iteration of the algorithm
  
      eta : float
          Step-size for the subgradient descent algorithm
  
      Returns
      -------
      numpy.array of shape (barycenter_size, d)
          Updated barycenter
  
      References
      ----------
  
      .. [1] D. Schultz and B. Jain. Nonsmooth Analysis and Subgradient Methods
         for Averaging in Dynamic Time Warping Spaces.
         Pattern Recognition, 74, 340-358.
      """
      d = X.shape[2]
      barycenter_size = barycenter.shape[0]
      delta_bar = numpy.zeros((barycenter_size, d))
      for k, (v_k, w_k, x_k) in enumerate(
          zip(list_diag_v_k, list_w_k, X)
      ):
          delta_bar += v_k.reshape((-1, 1)) * barycenter
          delta_bar -= w_k.dot(x_k[: ts_size(x_k)])
      barycenter -= (2.0 * eta / weights_sum) * delta_bar
      return barycenter
  
imports: import numpy
  import warnings
  from scipy.interpolate import interp1d
  from sklearn.exceptions import ConvergenceWarning
  from sklearn.utils import check_random_state
  from ..metrics import dtw_path
  from ..utils import to_time_series_dataset, ts_size
  from .utils import _set_weights
label_desc: Use image node_id 10 for calling a global function with example usage: _subgradient_update_barycenter(X, list_diag_v_k, list_w_k, weights_sum, barycenter, eta) and returns: barycenter