code: def _graphical_lasso(
      emp_cov,
      alpha,
      *,
      cov_init=None,
      mode="cd",
      tol=1e-4,
      enet_tol=1e-4,
      max_iter=100,
      verbose=False,
      eps=np.finfo(np.float64).eps,
  ):
      _, n_features = emp_cov.shape
      if alpha == 0:
          # Early return without regularization
          precision_ = linalg.inv(emp_cov)
          cost = -2.0 * log_likelihood(emp_cov, precision_)
          cost += n_features * np.log(2 * np.pi)
          d_gap = np.sum(emp_cov * precision_) - n_features
          return emp_cov, precision_, (cost, d_gap), 0
  
      if cov_init is None:
          covariance_ = emp_cov.copy()
      else:
          covariance_ = cov_init.copy()
      # As a trivial regularization (Tikhonov like), we scale down the
      # off-diagonal coefficients of our starting point: This is needed, as
      # in the cross-validation the cov_init can easily be
      # ill-conditioned, and the CV loop blows. Beside, this takes
      # conservative stand-point on the initial conditions, and it tends to
      # make the convergence go faster.
      covariance_ *= 0.95
      diagonal = emp_cov.flat[:: n_features + 1]
      covariance_.flat[:: n_features + 1] = diagonal
      precision_ = linalg.pinvh(covariance_)
  
      indices = np.arange(n_features)
      i = 0  # initialize the counter to be robust to `max_iter=0`
      costs = list()
      # The different l1 regression solver have different numerical errors
      if mode == "cd":
          errors = dict(over="raise", invalid="ignore")
      else:
          errors = dict(invalid="raise")
      try:
          # be robust to the max_iter=0 edge case, see:
          # https://github.com/scikit-learn/scikit-learn/issues/4134
          d_gap = np.inf
          # set a sub_covariance buffer
          sub_covariance = np.copy(covariance_[1:, 1:], order="C")
          for i in range(max_iter):
              for idx in range(n_features):
                  # To keep the contiguous matrix `sub_covariance` equal to
                  # covariance_[indices != idx].T[indices != idx]
                  # we only need to update 1 column and 1 line when idx changes
                  if idx > 0:
                      di = idx - 1
                      sub_covariance[di] = covariance_[di][
                          indices != idx
                      ]
                      sub_covariance[:, di] = covariance_[:, di][
                          indices != idx
                      ]
                  else:
                      sub_covariance[:] = covariance_[1:, 1:]
                  row = emp_cov[idx, indices != idx]
                  with np.errstate(**errors):
                      if mode == "cd":
                          # Use coordinate descent
                          coefs = -(
                              precision_[indices != idx, idx]
                              / (precision_[idx, idx] + 1000 * eps)
                          )
                          (
                              coefs,
                              _,
                              _,
                              _,
                          ) = cd_fast.enet_coordinate_descent_gram(
                              coefs,
                              alpha,
                              0,
                              sub_covariance,
                              row,
                              row,
                              max_iter,
                              enet_tol,
                              check_random_state(None),
                              False,
                          )
                      else:  # mode == "lars"
                          _, _, coefs = lars_path_gram(
                              Xy=row,
                              Gram=sub_covariance,
                              n_samples=row.size,
                              alpha_min=alpha / (n_features - 1),
                              copy_Gram=True,
                              eps=eps,
                              method="lars",
                              return_path=False,
                          )
                  # Update the precision matrix
                  precision_[idx, idx] = 1.0 / (
                      covariance_[idx, idx]
                      - np.dot(covariance_[indices != idx, idx], coefs)
                  )
                  precision_[indices != idx, idx] = (
                      -precision_[idx, idx] * coefs
                  )
                  precision_[idx, indices != idx] = (
                      -precision_[idx, idx] * coefs
                  )
                  coefs = np.dot(sub_covariance, coefs)
                  covariance_[idx, indices != idx] = coefs
                  covariance_[indices != idx, idx] = coefs
              if not np.isfinite(precision_.sum()):
                  raise FloatingPointError(
                      "The system is too ill-conditioned for this solver"
                  )
              d_gap = _dual_gap(emp_cov, precision_, alpha)
              cost = _objective(emp_cov, precision_, alpha)
              if verbose:
                  print(
                      "[graphical_lasso] Iteration % 3i, cost % 3.2e, dual gap %.3e"
                      % (i, cost, d_gap)
                  )
              costs.append((cost, d_gap))
              if np.abs(d_gap) < tol:
                  break
              if not np.isfinite(cost) and i > 0:
                  raise FloatingPointError(
                      "Non SPD result: the system is too ill-conditioned for this solver"
                  )
          else:
              warnings.warn(
                  "graphical_lasso: did not converge after %i iteration: dual gap: %.3e"
                  % (max_iter, d_gap),
                  ConvergenceWarning,
              )
      except FloatingPointError as e:
          e.args = (
              e.args[0]
              + ". The system is too ill-conditioned for this solver",
          )
          raise e
  
      return covariance_, precision_, costs, i + 1
  
imports: import operator
  import sys
  import time
  import warnings
  from numbers import Integral, Real
  import numpy
  from scipy import linalg
  from ..base import _fit_context
  from ..exceptions import ConvergenceWarning
  from ..linear_model import _cd_fast
  from ..linear_model import lars_path_gram
  from ..model_selection import check_cv, cross_val_score
  from ..utils._param_validation import Interval, StrOptions, validate_params
  from ..utils.parallel import Parallel, delayed
  from ..utils.validation import _is_arraylike_not_scalar, check_random_state, check_scalar
  from .None import EmpiricalCovariance, empirical_covariance, log_likelihood
label_desc: Use image node_id 3 for calling a global function with example usage: _graphical_lasso(emp_cov, alpha) and returns: covariance_,  precision_,  costs,  unknown, emp_cov,  precision_,  cost,  d_gap,  int