code: 
  '''simple docstring'''
  
  from typing import Any
  
  import numpy as np
  
  
  class   a_  :
  
  
  
  
         def __init__(     self   ,   snake_case_   ,   snake_case_  ):
           # Check if RHS is negative
           if np.any(tableau[:, -1]   ,   where=tableau[:, -1] < 0  ):
             raise ValueError("""RHS must be > 0"""  )
  
           _lowerCAmelCase       :  Optional[Any]   				=					tableau
           _lowerCAmelCase				,					_lowerCAmelCase       :  Union[str, Any]   				=					tableau.shape
  
           # Number of decision variables x1, x2, x3...
           _lowerCAmelCase       :  Tuple   				=					n_vars
  
           # Number of artificial variables to be minimised
           _lowerCAmelCase       :  Tuple   				=					len(np.where(tableau[self.n_vars : -1] == -1  )[0]  )
  
           # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
           _lowerCAmelCase       :  int   				=					(self.n_art_vars > 0) + 1
  
           # Number of slack variables added to make inequalities into equalities
           _lowerCAmelCase       :  Union[str, Any]   				=					self.n_rows - self.n_stages
  
           # Objectives for each stage
           _lowerCAmelCase       :  Union[str, Any]   				=					["""max"""]
  
           # In two stage simplex, first minimise then maximise
           if self.n_art_vars:
             self.objectives.append("""min"""  )
  
           _lowerCAmelCase       :  Optional[Any]   				=					[""""""]
  
           # Index of current pivot row and column
           _lowerCAmelCase       :  List[str]   				=					None
           _lowerCAmelCase       :  List[str]   				=					None
  
           # Does objective row only contain (non)-negative values?
           _lowerCAmelCase       :  Tuple   				=					False
  
  
  
  
         @staticmethod
         def 							__UpperCamelCase       (     *snake_case_  ):
           if len(snake_case_  ) != 3:
             raise ValueError("""Must provide n_vars, n_slack, and n_art_vars"""  )
  
           if not all(x >= 0 and isinstance(snake_case_   ,   snake_case_  ) for x in args  ):
             raise ValueError("""All arguments must be non-negative integers"""  )
  
           # decision | slack | artificial
           _lowerCAmelCase       :  Tuple   				=					["""x""", """s""", """a"""]
           _lowerCAmelCase       :  Dict   				=					[]
           for i in range(3  ):
             for j in range(args[i]  ):
               titles.append(string_starts[i] + str(j + 1  )  )
           titles.append("""RHS"""  )
           return titles
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           _lowerCAmelCase       :  Dict   				=					self.objectives[-1]
  
           # Find entries of highest magnitude in objective rows
           _lowerCAmelCase       :  Optional[int]   				=					(objective == """min""") - (objective == """max""")
           _lowerCAmelCase       :  List[str]   				=					np.argmax(sign * tableau[0, : self.n_vars]  )
  
           # Choice is only valid if below 0 for maximise, and above for minimise
           if sign * self.tableau[0, col_idx] <= 0:
             _lowerCAmelCase       :  int   				=					True
             return 0, 0
  
           # Pivot row is chosen as having the lowest quotient when elements of
           # the pivot column divide the right-hand side
  
           # Slice excluding the objective rows
           _lowerCAmelCase       :  Any   				=					slice(self.n_stages   ,   self.n_rows  )
  
           # RHS
           _lowerCAmelCase       :  Any   				=					tableau[s, -1]
  
           # Elements of pivot column within slice
           _lowerCAmelCase       :  Optional[int]   				=					tableau[s, col_idx]
  
           # Array filled with nans
           _lowerCAmelCase       :  Optional[int]   				=					np.full(self.n_rows - self.n_stages   ,   np.nan  )
  
           # If element in pivot column is greater than zeron_stages, return
           # quotient or nan otherwise
           _lowerCAmelCase       :  int   				=					np.divide(snake_case_   ,   snake_case_   ,   out=snake_case_   ,   where=divisor > 0  )
  
           # Arg of minimum quotient excluding the nan values. n_stages is added
           # to compensate for earlier exclusion of objective columns
           _lowerCAmelCase       :  List[str]   				=					np.nanargmin(snake_case_  ) + self.n_stages
           return row_idx, col_idx
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_   ,   snake_case_   ,   snake_case_  ):
           _lowerCAmelCase       :  Tuple   				=					tableau[row_idx].copy()
  
           _lowerCAmelCase       :  List[str]   				=					piv_row[col_idx]
  
           # Entry becomes 1
           piv_row *= 1 / piv_val
  
           # Variable in pivot column becomes basic, ie the only non-zero entry
           for idx, coeff in enumerate(tableau[:, col_idx]  ):
             tableau[idx] += -coeff * piv_row
           _lowerCAmelCase       :  Optional[Any]   				=					piv_row
           return tableau
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_  ):
           self.objectives.pop()
  
           if not self.objectives:
             return tableau
  
           # Slice containing ids for artificial columns
           _lowerCAmelCase       :  List[str]   				=					slice(-self.n_art_vars - 1   ,   -1  )
  
           # Delete the artificial variable columns
           _lowerCAmelCase       :  List[Any]   				=					np.delete(snake_case_   ,   snake_case_   ,   axis=1  )
  
           # Delete the objective row of the first stage
           _lowerCAmelCase       :  str   				=					np.delete(snake_case_   ,   0   ,   axis=0  )
  
           _lowerCAmelCase       :  Union[str, Any]   				=					1
           self.n_rows -= 1
           _lowerCAmelCase       :  List[str]   				=					0
           _lowerCAmelCase       :  int   				=					False
           return tableau
  
  
  
  
         def 							__UpperCamelCase       (     self  ):
           # Stop simplex algorithm from cycling.
           for _ in range(1_0_0  ):
             # Completion of each stage removes an objective. If both stages
             # are complete, then no objectives are left
             if not self.objectives:
               _lowerCAmelCase       :  Optional[Any]   				=					self.generate_col_titles(
                   self.n_vars   ,   self.n_slack   ,   self.n_art_vars  )
  
               # Find the values of each variable at optimal solution
               return self.interpret_tableau(self.tableau   ,   self.col_titles  )
  
             _lowerCAmelCase				,					_lowerCAmelCase       :  List[Any]   				=					self.find_pivot(self.tableau  )
  
             # If there are no more negative values in objective row
             if self.stop_iter:
               # Delete artificial variable columns and rows. Update attributes
               _lowerCAmelCase       :  Any   				=					self.change_stage(self.tableau  )
             else:
               _lowerCAmelCase       :  Optional[int]   				=					self.pivot(self.tableau   ,   snake_case_   ,   snake_case_  )
           return {}
  
  
  
  
  
  
  
         def 							__UpperCamelCase       (     self   ,   snake_case_   ,   snake_case_  ):
           _lowerCAmelCase       :  Dict   				=					{"""P""": abs(tableau[0, -1]  )}
  
           for i in range(self.n_vars  ):
             # Gives ids of nonzero entries in the ith column
             _lowerCAmelCase       :  int   				=					np.nonzero(tableau[:, i]  )
             _lowerCAmelCase       :  Optional[int]   				=					len(nonzero[0]  )
  
             # First entry in the nonzero ids
             _lowerCAmelCase       :  Optional[Any]   				=					nonzero[0][0]
             _lowerCAmelCase       :  List[Any]   				=					tableau[nonzero_rowidx, i]
  
             # If there is only one nonzero value in column, which is one
             if n_nonzero == nonzero_val == 1:
               _lowerCAmelCase       :  Optional[int]   				=					tableau[nonzero_rowidx, -1]
               _lowerCAmelCase       :  List[Any]   				=					rhs_val
  
          # Check for basic variables
           for title in col_titles:
             # Don't add RHS or slack variables to output dict
             if title[0] not in "R-s-a":
               output_dict.setdefault(snake_case_   ,   0  )
           return output_dict
  
  
  if __name__ == "__main__":
       import doctest
  
       doctest.testmod()
  
  
  
style_context: 
  '''simple docstring'''
  
  def 	_UpperCAmelCase				(  _lowerCamelCase						:       int  ,      _lowerCamelCase						:       int  ,      _lowerCamelCase						:       int		)    ->						float:
    if common_ratio == 1:
      # Formula for sum if common ratio is 1
      return num_of_terms * first_term
  
    # Formula for finding sum of n terms of a GeometricProgression
    return (first_term / (1 - common_ratio)) * (1 - common_ratio**num_of_terms)
  
  
  