code: def _train_implicit_cg(
      mat, this: np.ndarray, other: np.ndarray, reg: float
  ):
      "One half of an implicit ALS training round with conjugate gradient."
      nr = mat.nrows
      nc = other.shape[0]
  
      assert mat.ncols == nc
  
      OtOr = _implicit_otor(other, reg)
  
      frob = 0.0
  
      for i in prange(nr):
          cols = mat.row_cs(i)
          if len(cols) == 0:
              continue
  
          rates = mat.row_vs(i)
  
          # we can optimize by only considering the nonzero entries of Cu-I
          # this means we only need the corresponding matrix columns
          M = other[cols, :]
          # and solve
          w = this[i, :].copy()
          _cg_solve(OtOr, M, rates, w, 3)
  
          # update stats
          delta = this[i, :] - w
          frob += np.dot(delta, delta)
  
          # put back the result
          this[i, :] = w
  
      return np.sqrt(frob)
  
imports: import logging
  from collections import namedtuple
  import numpy
  from numba import njit, prange
  from csr import CSR
  from seedbank import numpy_rng
  from .bias import Bias
  from .mf_common import MFPredictor
  from ..data import sparse_ratings
  from ..None import util
  from ..math.solve import _dposv
label_desc: Use image node_id 9 for calling a global function with example usage: _train_implicit_cg(mat, this, other, reg) and returns: np