code: def _train_implicit_lu(
      mat, this: np.ndarray, other: np.ndarray, reg: float
  ):
      "One half of an implicit ALS training round."
      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, :]
          # Compute M^T (C_u-I) M, restricted to these nonzero entries
          MMT = (M.T.copy() * rates) @ M
          # assert MMT.shape[0] == ctx.n_features
          # assert MMT.shape[1] == ctx.n_features
          # Build the matrix for solving
          A = OtOr + MMT
          # Compute RHS - only used columns (p_ui != 0) values needed
          # Cu is rates + 1 for the cols, so just trim Ot
          y = other.T[:, cols] @ (rates + 1.0)
          # and solve
          _dposv(A, y, True)
          # assert len(uv) == ctx.n_features
          delta = this[i, :] - y
          frob += np.dot(delta, delta)
          this[i, :] = y
  
      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 10 for calling a global function with example usage: _train_implicit_lu(mat, this, other, reg) and returns: np