class_docstr: Broyden-Fletcher-Goldfarb-Shanno (BFGS) Hessian update strategy.
  
  Parameters
  ----------
  exception_strategy : {'skip_update', 'damp_update'}, optional
      Define how to proceed when the curvature condition is violated.
      Set it to 'skip_update' to just skip the update. Or, alternatively,
      set it to 'damp_update' to interpolate between the actual BFGS
      result and the unmodified matrix. Both exceptions strategies
      are explained  in [1]_, p.536-537.
  min_curvature : float
      This number, scaled by a normalization factor, defines the
      minimum curvature ``dot(delta_grad, delta_x)`` allowed to go
      unaffected by the exception strategy. By default is equal to
      1e-8 when ``exception_strategy = 'skip_update'`` and equal
      to 0.2 when ``exception_strategy = 'damp_update'``.
  init_scale : {float, 'auto'}
      Matrix scale at first iteration. At the first
      iteration the Hessian matrix or its inverse will be initialized
      with ``init_scale*np.eye(n)``, where ``n`` is the problem dimension.
      Set it to 'auto' in order to use an automatic heuristic for choosing
      the initial scale. The heuristic is described in [1]_, p.143.
      By default uses 'auto'.
  
  Notes
  -----
  The update is based on the description in [1]_, p.140.
  
  References
  ----------
  .. [1] Nocedal, Jorge, and Stephen J. Wright. "Numerical optimization"
         Second Edition (2006).
class_docstr_tok: ["Broyden-Fletcher-Goldfarb-Shanno","(","BFGS",")","Hessian","update","strategy",".","Parameters","--","--","--","--","--","exception_strategy",":","{","'skip_update","'",",","'damp_update","'","}",",","optional","Define","how","to","proceed","when","the","curvature","condition","is","violated",".","Set","it","to","'skip_update","'","to","just","skip","the","update",".","Or",",","alternatively",",","set","it","to","'damp_update","'","to","interpolate","between","the","actual","BFGS","result","and","the","unmodified","matrix",".","Both","exceptions","strategies","are","explained","in","[","1","]","_",",","p.536-537",".","min_curvature",":","float","This","number",",","scaled","by","a","normalization","factor",",","defines","the","minimum","curvature","``","dot","(","delta_grad",",","delta_x",")","``","allowed","to","go","unaffected","by","the","exception","strategy",".","By","default","is","equal","to","1e-8","when","``","exception_strategy","=","'skip_update","'","``","and","equal","to","0.2","when","``","exception_strategy","=","'damp_update","'","``",".","init_scale",":","{","float",",","'auto","'","}","Matrix","scale","at","first","iteration",".","At","the","first","iteration","the","Hessian","matrix","or","its","inverse","will","be","initialized","with","``","init_scale","*","np.eye","(","n",")","``",",","where","``","n","``","is","the","problem","dimension",".","Set","it","to","'auto","'","in","order","to","use","an","automatic","heuristic","for","choosing","the","initial","scale",".","The","heuristic","is","described","in","[","1","]","_",",","p.143",".","By","default","uses","'auto","'",".","Notes","--","--","-","The","update","is","based","on","the","description","in","[","1","]","_",",","p.140",".","References","--","--","--","--","--","..","[","1","]","Nocedal",",","Jorge",",","and","Stephen","J.","Wright",".","``","Numerical","optimization","''","Second","Edition","(","2006",")","."]
code: def _update_implementation(self, delta_x, delta_grad):
      # Auxiliary variables w and z
      if self.approx_type == "hess":
          w = delta_x
          z = delta_grad
      else:
          w = delta_grad
          z = delta_x
      # Do some common operations
      wz = np.dot(w, z)
      Mw = self.dot(w)
      wMw = Mw.dot(w)
      # Guarantee that wMw > 0 by reinitializing matrix.
      # While this is always true in exact arithmetics,
      # indefinite matrix may appear due to roundoff errors.
      if wMw <= 0.0:
          scale = self._auto_scale(delta_x, delta_grad)
          # Reinitialize matrix
          if self.approx_type == "hess":
              self.B = scale * np.eye(self.n, dtype=float)
          else:
              self.H = scale * np.eye(self.n, dtype=float)
          # Do common operations for new matrix
          Mw = self.dot(w)
          wMw = Mw.dot(w)
      # Check if curvature condition is violated
      if wz <= self.min_curvature * wMw:
          # If the option 'skip_update' is set
          # we just skip the update when the condion
          # is violated.
          if self.exception_strategy == "skip_update":
              return
          # If the option 'damp_update' is set we
          # interpolate between the actual BFGS
          # result and the unmodified matrix.
          elif self.exception_strategy == "damp_update":
              update_factor = (1 - self.min_curvature) / (1 - wz / wMw)
              z = update_factor * z + (1 - update_factor) * Mw
              wz = np.dot(w, z)
      # Update matrix
      if self.approx_type == "hess":
          self._update_hessian(wz, Mw, wMw, z)
      else:
          self._update_inverse_hessian(wz, Mw, wMw, z)
  
imports: import numpy
  from numpy.linalg import norm
  from scipy.linalg import get_blas_funcs
  from warnings import warn
label_desc: Use image node_id 4 for calling the BFGS obj's underlying member method code with example usage: obj._update_implementation(delta_x, delta_grad) without return types