code: 
  
  
  '''simple docstring'''
  
  
  
  
  
  from typing import Any
  
  import numpy as np
  
  
  
  
  
  
  def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   np.ndarray   ):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			assert np.shape(_SCREAMING_SNAKE_CASE   )[0] == np.shape(_SCREAMING_SNAKE_CASE   )[1]
  
  			# If matrix not symmetric, exit right away.
  			if np.allclose(_SCREAMING_SNAKE_CASE,matrix.T   ) is False:
  						return False
  
  			# Get eigenvalues and eignevectors for a symmetric matrix.
  			__A,  __A=  np.linalg.eigh(_SCREAMING_SNAKE_CASE   )
  
  			# Check sign of all eigenvalues.
  			# np.all returns a value of type np.bool_
  			return bool(np.all(eigen_values > 0   )   )
  
  
  
  
  
  
  def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   int   ):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			__A=  np.random.randn(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  			__A=  np.dot(_SCREAMING_SNAKE_CASE,random_matrix.T   )
  			assert _is_matrix_spd(_SCREAMING_SNAKE_CASE   )
  			return spd_matrix
  
  
  
  
  
  
  def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   int = 1000,_SCREAMING_SNAKE_CASE				:   float = 1e-8,):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			assert np.shape(_SCREAMING_SNAKE_CASE   )[0] == np.shape(_SCREAMING_SNAKE_CASE   )[1]
  			assert np.shape(_SCREAMING_SNAKE_CASE   )[0] == np.shape(_SCREAMING_SNAKE_CASE   )[0]
  			assert _is_matrix_spd(_SCREAMING_SNAKE_CASE   )
  
  			# Initialize solution guess, residual, search direction.
  			__A=  np.zeros((np.shape(_SCREAMING_SNAKE_CASE   )[0], 1)   )
  			__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  			__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  
  			# Set initial errors in solution guess and residual.
  			__A=  1e9
  			__A=  1e9
  			__A=  1e9
  
  			# Set iteration counter to threshold number of iterations.
  			__A=  0
  
  			while error > tol:
  						# Save this value so we only calculate the matrix-vector product once.
  						__A=  np.dot(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  						# The main algorithm.
  
  						# Update search direction magnitude.
  						__A=  np.dot(ra.T,_SCREAMING_SNAKE_CASE   ) / np.dot(pa.T,_SCREAMING_SNAKE_CASE   )
  						# Update solution guess.
  						__A=  xa + alpha * pa
  						# Calculate new residual.
  						__A=  ra - alpha * w
  						# Calculate new Krylov subspace scale.
  						__A=  np.dot(r.T,_SCREAMING_SNAKE_CASE   ) / np.dot(ra.T,_SCREAMING_SNAKE_CASE   )
  						# Calculate new A conjuage search direction.
  						__A=  r + beta * pa
  
  						# Calculate errors.
  						__A=  np.linalg.norm(r - ra   )
  						__A=  np.linalg.norm(x - xa   )
  						__A=  np.maximum(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  						# Update variables.
  						__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  						__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  						__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  
  						# Update number of iterations.
  						iterations += 1
  						if iterations > max_iterations:
  									break
  
  			return x
  
  
  
  
  
  
  def 	UpperCAmelCase__(							):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			__A=  3
  			__A=  _create_spd_matrix(_SCREAMING_SNAKE_CASE   )
  			__A=  np.random.randn(_SCREAMING_SNAKE_CASE,1   )
  			__A=  np.dot(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  			# Numpy solution.
  			__A=  np.linalg.solve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  			# Our implementation.
  			__A=  conjugate_gradient(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  			# Ensure both solutions are close to x_true (and therefore one another).
  			assert np.linalg.norm(x_numpy - x_true   ) <= 1e-6
  			assert np.linalg.norm(x_conjugate_gradient - x_true   ) <= 1e-6
  
  
  if __name__ == "__main__":
  					import doctest
  
  					doctest.testmod()
  					test_conjugate_gradient()
  
  
  
  
style_context: 
  
  
  '''simple docstring'''
  
  
  
  
  
  UpperCAmelCase__      =							[[0, 1, 1], [0, 0, 1], [1, 0, 0]]
  class 	a__				:
  
  
  
  
  
  			'''simple docstring'''
  
  			def __init__(     self       : Any ,						lowerCAmelCase_       : Optional[int]  )			->		Tuple:
  						__A=  name
  						__A=  []
  						__A=  []
  			def 				lowerCAmelCase    (     self       : Any ,						lowerCAmelCase_       : Dict  )			->		Any:
  						self.inbound.append(lowerCAmelCase_  )
  			def 				lowerCAmelCase    (     self       : str ,						lowerCAmelCase_       : Any  )			->		List[Any]:
  						self.outbound.append(lowerCAmelCase_  )
  
  			def __repr__(     self       : int  )			->		Union[str, Any]:
  						return F"""<node={self.name} inbound={self.inbound} outbound={self.outbound}>"""
  
  
  
  
  
  
  def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   Any,_SCREAMING_SNAKE_CASE				:   Optional[int]=3,_SCREAMING_SNAKE_CASE				:   Dict=0.85   ):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			__A=  {}
  			for node in nodes:
  						__A=  1
  
  			__A=  {}
  			for node in nodes:
  						__A=  len(node.outbound   )
  
  			for i in range(_SCREAMING_SNAKE_CASE   ):
  						print(f"""======= Iteration {i + 1} ======="""   )
  						for _, node in enumerate(_SCREAMING_SNAKE_CASE   ):
  									__A=  (1 - d) + d * sum(
  									    ranks[ib] / outbounds[ib] for ib in node.inbound   )
  						print(_SCREAMING_SNAKE_CASE   )
  
  
  
  
  
  
  def 	UpperCAmelCase__(							):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			__A=  list(input('Enter Names of the Nodes: '   ).split()   )
  
  			__A=  [Node(_SCREAMING_SNAKE_CASE   ) for name in names]
  
  			for ri, row in enumerate(_SCREAMING_SNAKE_CASE   ):
  						for ci, col in enumerate(_SCREAMING_SNAKE_CASE   ):
  									if col == 1:
  												nodes[ci].add_inbound(names[ri]   )
  												nodes[ri].add_outbound(names[ci]   )
  
  			print('======= Nodes ======='   )
  			for node in nodes:
  						print(_SCREAMING_SNAKE_CASE   )
  
  			page_rank(_SCREAMING_SNAKE_CASE   )
  
  
  if __name__ == "__main__":
  					main()
  
  
  
  