






'''simple docstring'''





from typing import Any

import numpy as np
class 					lowercase							:


    '''simple docstring'''

    def __init__(       self  :  Optional[Any]					,  __lowerCamelCase  :  np.ndarray					,  __lowerCamelCase  :  int  )				->				None:

     '''simple docstring'''




     if np.any(tableau[:, -1]					,  where=tableau[:, -1] < 0  ):
      raise ValueError("RHS must be > 0"  )

     lowerCamelCase__	=  tableau
     lowerCamelCase__	,						lowerCamelCase__	=  tableau.shape

     # Number of decision variables x1, x2, x3...
     lowerCamelCase__	=  n_vars

     # Number of artificial variables to be minimised
     lowerCamelCase__	=  len(np.where(tableau[self.n_vars : -1] == -1  )[0]  )

     # 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
     lowerCamelCase__	=  (self.n_art_vars > 0) + 1

     # Number of slack variables added to make inequalities into equalities
     lowerCamelCase__	=  self.n_rows - self.n_stages

     # Objectives for each stage
     lowerCamelCase__	=  ["max"]

     # In two stage simplex, first minimise then maximise
     if self.n_art_vars:
      self.objectives.append("min"  )

     lowerCamelCase__	=  [""]

     # Index of current pivot row and column
     lowerCamelCase__	=  None
     lowerCamelCase__	=  None

     # Does objective row only contain (non)-negative values?
     lowerCamelCase__	=  False

    @staticmethod
    def  a__					(       *__lowerCamelCase  :  int  )				->				list[str]:

     '''simple docstring'''




     if len(__lowerCamelCase  ) != 3:
      raise ValueError("Must provide n_vars, n_slack, and n_art_vars"  )

     if not all(x >= 0 and isinstance(__lowerCamelCase					,  __lowerCamelCase  ) for x in args  ):
      raise ValueError("All arguments must be non-negative integers"  )

     # decision | slack | artificial
     lowerCamelCase__	=  ["x", "s", "a"]
     lowerCamelCase__	=  []
     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  a__					(       self  :  Dict					,  __lowerCamelCase  :  np.ndarray  )				->				tuple[Any, Any]:

     '''simple docstring'''




     lowerCamelCase__	=  self.objectives[-1]

     # Find entries of highest magnitude in objective rows
     lowerCamelCase__	=  (objective == "min") - (objective == "max")
     lowerCamelCase__	=  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__	=  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__	=  slice(self.n_stages					,  self.n_rows  )

     # RHS
     lowerCamelCase__	=  tableau[s, -1]

     # Elements of pivot column within slice
     lowerCamelCase__	=  tableau[s, col_idx]

     # Array filled with nans
     lowerCamelCase__	=  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__	=  np.divide(__lowerCamelCase					,  __lowerCamelCase					,  out=__lowerCamelCase					,  where=divisor > 0  )

     # Arg of minimum quotient excluding the nan values. n_stages is added
     # to compensate for earlier exclusion of objective columns
     lowerCamelCase__	=  np.nanargmin(__lowerCamelCase  ) + self.n_stages
     return row_idx, col_idx

    def  a__					(       self  :  Optional[int]					,  __lowerCamelCase  :  np.ndarray					,  __lowerCamelCase  :  int					,  __lowerCamelCase  :  int  )				->				np.ndarray:

     '''simple docstring'''




     lowerCamelCase__	=  tableau[row_idx].copy()

     lowerCamelCase__	=  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__	=  piv_row
     return tableau

    def  a__					(       self  :  Optional[Any]					,  __lowerCamelCase  :  np.ndarray  )				->				np.ndarray:

     '''simple docstring'''




     self.objectives.pop()

     if not self.objectives:
      return tableau

     # Slice containing ids for artificial columns
     lowerCamelCase__	=  slice(-self.n_art_vars - 1					,  -1  )

     # Delete the artificial variable columns
     lowerCamelCase__	=  np.delete(__lowerCamelCase					,  __lowerCamelCase					,  axis=1  )

     # Delete the objective row of the first stage
     lowerCamelCase__	=  np.delete(__lowerCamelCase					,  0					,  axis=0  )

     lowerCamelCase__	=  1
     self.n_rows -= 1
     lowerCamelCase__	=  0
     lowerCamelCase__	=  False
     return tableau

    def  a__					(       self  :  Optional[int]  )				->				dict[Any, Any]:

     '''simple docstring'''




     for _ in range(100  ):
      # Completion of each stage removes an objective. If both stages
      # are complete, then no objectives are left
      if not self.objectives:
       lowerCamelCase__	=  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__	=  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__	=  self.change_stage(self.tableau  )
      else:
       lowerCamelCase__	=  self.pivot(self.tableau					,  __lowerCamelCase					,  __lowerCamelCase  )
     return {}




    def  a__					(       self  :  Optional[int]					,  __lowerCamelCase  :  np.ndarray					,  __lowerCamelCase  :  list[str]  )				->				dict[str, float]:

     '''simple docstring'''




     lowerCamelCase__	=  {"P": abs(tableau[0, -1]  )}

     for i in range(self.n_vars  ):
      # Gives ids of nonzero entries in the ith column
      lowerCamelCase__	=  np.nonzero(tableau[:, i]  )
      lowerCamelCase__	=  len(nonzero[0]  )

      # First entry in the nonzero ids
      lowerCamelCase__	=  nonzero[0][0]
      lowerCamelCase__	=  tableau[nonzero_rowidx, i]

      # If there is only one nonzero value in column, which is one
      if n_nonzero == nonzero_val == 1:
       lowerCamelCase__	=  tableau[nonzero_rowidx, -1]
       lowerCamelCase__	=  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(__lowerCamelCase					,  0  )
     return output_dict


if __name__ == "__main__":
 import doctest

 doctest.testmod()









'''simple docstring'''





from math import cos, sin






def  lowerCamelCase_  (     lowercase__):
 lowerCamelCase__	=  float(lowercase__)
 return [[scaling_factor * int(x == y) for x in range(2)] for y in range(2)]






def  lowerCamelCase_  (     lowercase__):
 lowerCamelCase__	,						lowerCamelCase__	=  cos(lowercase__), sin(lowercase__)
 return [[c, -s], [s, c]]






def  lowerCamelCase_  (     lowercase__):
 lowerCamelCase__	,						lowerCamelCase__	=  cos(lowercase__), sin(lowercase__)
 lowerCamelCase__	=  c * s
 return [[c * c, cs], [cs, s * s]]






def  lowerCamelCase_  (     lowercase__):
 lowerCamelCase__	,						lowerCamelCase__	=  cos(lowercase__), sin(lowercase__)
 lowerCamelCase__	=  c * s
 return [[2 * c - 1, 2 * cs], [2 * cs, 2 * s - 1]]


print(F"""    {scaling(5) = }""")
print(F"""  {rotation(45) = }""")
print(F"""{projection(45) = }""")
print(F"""{reflection(45) = }""")

