
'''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     :					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     :					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()




'''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()


