



from typing import Any

import numpy as np

class      snake_case__       :







						"""simple docstring"""




						def __init__(     self      ,     __lowercase      ,     __lowercase						)      ->   None:
									"""simple docstring"""




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

									a__     :		Optional[int]		 =						tableau
									a__ ,		a__     :		int		 =						tableau.shape

									# Number of decision variables x1, x2, x3...
									a__     :		Optional[Any]		 =						n_vars

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

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

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

									# Objectives for each stage
									a__     :		Optional[int]		 =						["""max"""]

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

									a__     :		int		 =						[""""""]

									# Index of current pivot row and column
									a__     :		Tuple		 =						None
									a__     :		Dict		 =						None

									# Does objective row only contain (non)-negative values?
									a__     :		List[Any]		 =						False

						@staticmethod
						def 							SCREAMING_SNAKE_CASE__(     *__lowercase						)      ->   list[str]:
									"""simple docstring"""




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

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

									# decision | slack | artificial
									a__     :		Optional[Any]		 =						["""x""", """s""", """a"""]
									a__     :		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 							SCREAMING_SNAKE_CASE__(     self      ,     __lowercase						)      ->   tuple[Any, Any]:
									"""simple docstring"""




									a__     :		Optional[Any]		 =						self.objectives[-1]

									# Find entries of highest magnitude in objective rows
									a__     :		Optional[int]		 =						(objective == """min""") - (objective == """max""")
									a__     :		Union[str, Any]		 =						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:
												a__     :		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
									a__     :		Optional[Any]		 =						slice(self.n_stages      ,     self.n_rows						)

									# RHS
									a__     :		List[Any]		 =						tableau[s, -1]

									# Elements of pivot column within slice
									a__     :		List[Any]		 =						tableau[s, col_idx]

									# Array filled with nans
									a__     :		List[Any]		 =						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
									a__     :		Optional[Any]		 =						np.divide(__lowercase      ,     __lowercase      ,     out=__lowercase      ,     where=divisor > 0						)

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

						def 							SCREAMING_SNAKE_CASE__(     self      ,     __lowercase      ,     __lowercase      ,     __lowercase						)      ->   np.ndarray:
									"""simple docstring"""




									a__     :		Union[str, Any]		 =						tableau[row_idx].copy()

									a__     :		Union[str, Any]		 =						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
									a__     :		Tuple		 =						piv_row
									return tableau

						def 							SCREAMING_SNAKE_CASE__(     self      ,     __lowercase						)      ->   np.ndarray:
									"""simple docstring"""




									self.objectives.pop()

									if not self.objectives:
												return tableau

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

									# Delete the artificial variable columns
									a__     :		Dict		 =						np.delete(__lowercase      ,     __lowercase      ,     axis=1						)

									# Delete the objective row of the first stage
									a__     :		Optional[Any]		 =						np.delete(__lowercase      ,     0      ,     axis=0						)

									a__     :		Any		 =						1
									self.n_rows -= 1
									a__     :		Dict		 =						0
									a__     :		Union[str, Any]		 =						False
									return tableau

						def 							SCREAMING_SNAKE_CASE__(     self						)      ->   dict[Any, Any]:
									"""simple docstring"""




									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:
															a__     :		Union[str, 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						)

												a__ ,		a__     :		int		 =						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
															a__     :		List[Any]		 =						self.change_stage(self.tableau						)
												else:
															a__     :		List[Any]		 =						self.pivot(self.tableau      ,     __lowercase      ,     __lowercase						)
									return {}

						def 							SCREAMING_SNAKE_CASE__(     self      ,     __lowercase      ,     __lowercase						)      ->   dict[str, float]:
									"""simple docstring"""




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

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

												# First entry in the nonzero ids
												a__     :		List[Any]		 =						nonzero[0][0]
												a__     :		Any		 =						tableau[nonzero_rowidx, i]

												# If there is only one nonzero value in column, which is one
												if n_nonzero == nonzero_val == 1:
															a__     :		Optional[Any]		 =						tableau[nonzero_rowidx, -1]
															a__     :		Union[str, 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(__lowercase      ,     0						)
									return output_dict


if __name__ == "__main__":
		import doctest

		doctest.testmod()









import decimal

def 			lowerCAmelCase_  (						)							-> int:







			"""simple docstring"""
			a__     :		Optional[Any]		 =						0
			a__     :		Optional[int]		 =						decimal.Context(prec=105)
			for i in range(2		,     100):
						a__     :		int		 =						decimal.Decimal(_lowercase)
						a__     :		Optional[int]		 =						number.sqrt(_lowercase)
						if len(str(_lowercase)) > 1:
									answer += int(str(_lowercase)[0])
									a__     :		Tuple		 =						str(_lowercase)[2:101]
									answer += sum(int(_lowercase) for x in sqrt_number_str)
			return answer


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		print(f'{solution() = }')




