code: def update_tr_radius(
      Delta, actual_reduction, predicted_reduction, step_norm, bound_hit
  ):
      """Update the radius of a trust region based on the cost reduction.
  
      Returns
      -------
      Delta : float
          New radius.
      ratio : float
          Ratio between actual and predicted reductions. Zero if predicted
          reduction is zero.
      """
      if predicted_reduction > 0:
          ratio = actual_reduction / predicted_reduction
      else:
          ratio = 0
  
      if ratio < 0.25:
          Delta = 0.25 * step_norm
      elif ratio > 0.75 and bound_hit:
          Delta *= 2.0
  
      return Delta, ratio
  
imports: from __future__ import division, print_function, absolute_import
  from math import copysign
  import numpy
  from numpy.linalg import norm
  from scipy.linalg import cho_factor, cho_solve, LinAlgError
  from scipy.sparse import issparse
  from scipy.sparse.linalg import LinearOperator, aslinearoperator
label_desc: Use image node_id 4 for calling a global function with example usage: update_tr_radius(Delta, actual_reduction, predicted_reduction, step_norm, bound_hit) and returns: Delta,  ratio