

"""simple docstring"""




from typing import Any

import numpy as np


class 		_A							:





      def __init__(	self     ,    SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__		) ->       None:
         # Check if RHS is negative
         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        _lowerCamelCase	(	*SCREAMING_SNAKE_CASE__		) ->       list[str]:
         if len(SCREAMING_SNAKE_CASE__		) != 3:
            raise ValueError("Must provide n_vars, n_slack, and n_art_vars"		)

         if not all(x >= 0 and isinstance(SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__		) 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        _lowerCamelCase	(	self     ,    SCREAMING_SNAKE_CASE__		) ->       tuple[Any, Any]:
         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(SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__     ,    out=SCREAMING_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__						       =      np.nanargmin(SCREAMING_SNAKE_CASE__		) + self.n_stages
         return row_idx, col_idx





      def        _lowerCamelCase	(	self     ,    SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__		) ->       np.ndarray:
         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        _lowerCamelCase	(	self     ,    SCREAMING_SNAKE_CASE__		) ->       np.ndarray:
         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(SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__     ,    axis=1		)

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

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





      def        _lowerCamelCase	(	self		) ->       dict[Any, Any]:
         # Stop simplex algorithm from cycling.
         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     ,    SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__		)
         return {}





      def        _lowerCamelCase	(	self     ,    SCREAMING_SNAKE_CASE__     ,    SCREAMING_SNAKE_CASE__		) ->       dict[str, float]:
         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(SCREAMING_SNAKE_CASE__     ,    0		)
         return output_dict


if __name__ == "__main__":
     import doctest

     doctest.testmod()







"""simple docstring"""




from __future__ import annotations



def 						UpperCAmelCase__ (					A__						,     A__						,     A__						,     A__ )   ->   None:




   """simple docstring"""
   lowerCamelCase__       ,							lowerCamelCase__						       =      m - 1, 0
   while i >= 0 and j < n:
      if key == mat[i][j]:
         print(f'Key {key} found at row- {i + 1} column- {j + 1}' )
         return
      if key < mat[i][j]:
         i -= 1
      else:
         j += 1
   print(f'Key {key} not found' )



def 						UpperCAmelCase__ (					)   ->   None:




   """simple docstring"""
   lowerCamelCase__						       =      [[2, 5, 7], [4, 8, 13], [9, 11, 15], [12, 17, 20]]
   lowerCamelCase__						       =      int(input("Enter the element to be searched:" ) )
   print(A__ )
   search_in_a_sorted_matrix(A__						,     len(A__ )						,     len(mat[0] )						,     A__ )


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     main()




