code: 
  '''simple docstring'''
  from typing import Any
  
  import numpy as np
  
  
  
  
  
  
  class   snake_case__					:
  
  
  
  
  
  
  
  		"""simple docstring"""
  
  		def __init__(			self							: List[str]     ,  UpperCamelCase__							: np.ndarray     ,  UpperCamelCase__							: int			)	-> None:
  
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  
  
  				if np.any(tableau[:, -1]     ,  where=tableau[:, -1] < 0			):
  						raise ValueError('''RHS must be > 0'''			)
  
  				snake_case     :					List[str]     =	tableau
  				snake_case ,snake_case     :					Any     =	tableau.shape
  
  				# Number of decision variables x1, x2, x3...
  				snake_case     :					List[str]     =	n_vars
  
  				# Number of artificial variables to be minimised
  				snake_case     :					Optional[Any]     =	len(np.where(tableau[self.n_vars : -1] == -1			)[0]			)
  
  				# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
  				snake_case     :					str     =	(self.n_art_vars > 0) + 1
  
  				# Number of slack variables added to make inequalities into equalities
  				snake_case     :					Union[str, Any]     =	self.n_rows - self.n_stages
  
  				# Objectives for each stage
  				snake_case     :					int     =	['''max''']
  
  				# In two stage simplex, first minimise then maximise
  				if self.n_art_vars:
  						self.objectives.append('''min'''			)
  
  				snake_case     :					str     =	['''''']
  
  				# Index of current pivot row and column
  				snake_case     :					Dict     =	None
  				snake_case     :					int     =	None
  
  				# Does objective row only contain (non)-negative values?
  				snake_case     :					Optional[int]     =	False
  
  
  
  		@staticmethod
  		def 			lowerCAmelCase							(			*UpperCamelCase__							: int			)	-> list[str]:
  
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  
  
  				if len(UpperCamelCase__			) != 3:
  						raise ValueError('''Must provide n_vars, n_slack, and n_art_vars'''			)
  
  				if not all(x >= 0 and isinstance(UpperCamelCase__     ,  UpperCamelCase__			) for x in args			):
  						raise ValueError('''All arguments must be non-negative integers'''			)
  
  				# decision | slack | artificial
  				snake_case     :					Tuple     =	['''x''', '''s''', '''a''']
  				snake_case     :					Optional[Any]     =	[]
  				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							: int     ,  UpperCamelCase__							: np.ndarray			)	-> tuple[Any, Any]:
  
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  
  
  				snake_case     :					Union[str, Any]     =	self.objectives[-1]
  
  				# Find entries of highest magnitude in objective rows
  				snake_case     :					Tuple     =	(objective == '''min''') - (objective == '''max''')
  				snake_case     :					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:
  						snake_case     :					Tuple     =	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
  				snake_case     :					int     =	slice(self.n_stages     ,  self.n_rows			)
  
  				# RHS
  				snake_case     :					Optional[int]     =	tableau[s, -1]
  
  				# Elements of pivot column within slice
  				snake_case     :					Dict     =	tableau[s, col_idx]
  
  				# Array filled with nans
  				snake_case     :					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
  				snake_case     :					Optional[int]     =	np.divide(UpperCamelCase__     ,  UpperCamelCase__     ,  out=UpperCamelCase__     ,  where=divisor > 0			)
  
  				# Arg of minimum quotient excluding the nan values. n_stages is added
  				# to compensate for earlier exclusion of objective columns
  				snake_case     :					List[str]     =	np.nanargmin(UpperCamelCase__			) + self.n_stages
  				return row_idx, col_idx
  
  
  
  		def 			lowerCAmelCase							(			self							: Optional[int]     ,  UpperCamelCase__							: np.ndarray     ,  UpperCamelCase__							: int     ,  UpperCamelCase__							: int			)	-> np.ndarray:
  
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  
  
  				snake_case     :					List[str]     =	tableau[row_idx].copy()
  
  				snake_case     :					Optional[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
  				snake_case     :					Tuple     =	piv_row
  				return tableau
  
  
  
  		def 			lowerCAmelCase							(			self							: List[str]     ,  UpperCamelCase__							: np.ndarray			)	-> np.ndarray:
  
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  
  
  				self.objectives.pop()
  
  				if not self.objectives:
  						return tableau
  
  				# Slice containing ids for artificial columns
  				snake_case     :					Any     =	slice(-self.n_art_vars - 1     ,  -1			)
  
  				# Delete the artificial variable columns
  				snake_case     :					Union[str, Any]     =	np.delete(UpperCamelCase__     ,  UpperCamelCase__     ,  axis=1			)
  
  				# Delete the objective row of the first stage
  				snake_case     :					Union[str, Any]     =	np.delete(UpperCamelCase__     ,  0     ,  axis=0			)
  
  				snake_case     :					str     =	1
  				self.n_rows -= 1
  				snake_case     :					str     =	0
  				snake_case     :					int     =	False
  				return tableau
  
  
  
  		def 			lowerCAmelCase							(			self							: List[str]			)	-> 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:
  								snake_case     :					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			)
  
  						snake_case ,snake_case     :					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
  								snake_case     :					Optional[int]     =	self.change_stage(self.tableau			)
  						else:
  								snake_case     :					List[str]     =	self.pivot(self.tableau     ,  UpperCamelCase__     ,  UpperCamelCase__			)
  				return {}
  
  
  
  		def 			lowerCAmelCase							(			self							: List[str]     ,  UpperCamelCase__							: np.ndarray     ,  UpperCamelCase__							: list[str]			)	-> dict[str, float]:
  
  
  
  
  
  
  				"""simple docstring"""
  
  
  
  
  
  
  
  				snake_case     :					List[Any]     =	{'''P''': abs(tableau[0, -1]			)}
  
  				for i in range(self.n_vars			):
  						# Gives ids of nonzero entries in the ith column
  						snake_case     :					Dict     =	np.nonzero(tableau[:, i]			)
  						snake_case     :					int     =	len(nonzero[0]			)
  
  						# First entry in the nonzero ids
  						snake_case     :					Any     =	nonzero[0][0]
  						snake_case     :					Union[str, Any]     =	tableau[nonzero_rowidx, i]
  
  						# If there is only one nonzero value in column, which is one
  						if n_nonzero == nonzero_val == 1:
  								snake_case     :					Dict     =	tableau[nonzero_rowidx, -1]
  								snake_case     :					str     =	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(UpperCamelCase__     ,  0			)
  				return output_dict
  
  
  if __name__ == "__main__":
  						import doctest
  
  						doctest.testmod()
  
  
  
style_context: 
  '''simple docstring'''
  import numpy as np
  
  
  
  
  def       _UpperCamelCase	(					SCREAMING_SNAKE_CASE__ )     ->       Optional[Any]:
  
  
  		'''simple docstring'''
  
  
  		snake_case ,snake_case     :					str     =	a.shape
  		snake_case     :					Union[str, Any]     =	min(SCREAMING_SNAKE_CASE__   ,					SCREAMING_SNAKE_CASE__ )
  		snake_case     :					List[Any]     =	np.eye(SCREAMING_SNAKE_CASE__ )
  		snake_case     :					Union[str, Any]     =	a.copy()
  
  		for k in range(t - 1 ):
  				# select a column of modified matrix A':
  				snake_case     :					Union[str, Any]     =	r[k:, [k]]
  				# construct first basis vector
  				snake_case     :					str     =	np.zeros_like(SCREAMING_SNAKE_CASE__ )
  				snake_case     :					List[str]     =	1.0
  				# determine scaling factor
  				snake_case     :					int     =	np.linalg.norm(SCREAMING_SNAKE_CASE__ )
  				# construct vector v for Householder reflection
  				snake_case     :					Optional[int]     =	x + np.sign(x[0] ) * alpha * ea
  				v /= np.linalg.norm(SCREAMING_SNAKE_CASE__ )
  
  				# construct the Householder matrix
  				snake_case     :					Tuple     =	np.eye(m - k ) - 2.0 * v @ v.T
  				# pad with ones and zeros as necessary
  				snake_case     :					int     =	np.block([[np.eye(SCREAMING_SNAKE_CASE__ ), np.zeros((k, m - k) )], [np.zeros((m - k, k) ), q_k]] )
  
  				snake_case     :					int     =	q @ q_k.T
  				snake_case     :					Dict     =	q_k @ r
  
  		return q, r
  
  
  if __name__ == "__main__":
  						import doctest
  
  						doctest.testmod()
  
  
  