

import numpy as np

def  lowerCamelCase__ (					_a							,			_a							,			_a = 1E-12							,			_a = 100							,			):
	assert np.shape(_a)[0] == np.shape(_a)[1]
	# Ensure proper dimensionality.
	assert np.shape(_a)[0] == np.shape(_a)[0]
	# Ensure inputs are either both complex or both real
	assert np.iscomplexobj(_a) == np.iscomplexobj(_a)
	SCREAMING_SNAKE_CASE   :      List[Any]				     =       np.iscomplexobj(_a)
	if is_complex:
		# Ensure complex input_matrix is Hermitian
		assert np.array_equal(_a							,			input_matrix.conj().T)

	# Set convergence to False. Will define convergence when we exceed max_iterations
	# or when we have small changes from one iteration to next.

	SCREAMING_SNAKE_CASE   :      Any				     =       False
	SCREAMING_SNAKE_CASE   :      List[Any]				     =       0
	SCREAMING_SNAKE_CASE   :      Optional[Any]				     =       0
	SCREAMING_SNAKE_CASE   :      Optional[Any]				     =       1E12

	while not convergence:
		# Multiple matrix by the vector.
		SCREAMING_SNAKE_CASE   :      List[str]				     =       np.dot(_a							,			_a)
		# Normalize the resulting output vector.
		SCREAMING_SNAKE_CASE   :      Any				     =       w / np.linalg.norm(_a)
		# Find rayleigh quotient
		# (faster than usual b/c we know vector is normalized already)
		SCREAMING_SNAKE_CASE   :      Tuple				     =       vector.conj().T if is_complex else vector.T
		SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       np.dot(_a							,			np.dot(_a							,			_a))

		# Check convergence.
		SCREAMING_SNAKE_CASE   :      Optional[Any]				     =       np.abs(lambda_ - lambda_previous) / lambda_
		iterations += 1

		if error <= error_tol or iterations >= max_iterations:
			SCREAMING_SNAKE_CASE   :      Dict				     =       True

		SCREAMING_SNAKE_CASE   :      List[Any]				     =       lambda_

	if is_complex:
		SCREAMING_SNAKE_CASE   :      List[Any]				     =       np.real(lambda_)

	return lambda_, vector

def  lowerCamelCase__ (					):
	SCREAMING_SNAKE_CASE   :      Any				     =       np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]])
	SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       np.array([41, 4, 20])
	SCREAMING_SNAKE_CASE   :      Optional[int]				     =       real_input_matrix.astype(np.complexaaa)
	SCREAMING_SNAKE_CASE   :      List[Any]				     =       np.triu(1j * complex_input_matrix							,			1)
	complex_input_matrix += imag_matrix
	complex_input_matrix += -1 * imag_matrix.T
	SCREAMING_SNAKE_CASE   :      Any				     =       np.array([41, 4, 20]).astype(np.complexaaa)

	for problem_type in ["real", "complex"]:
		if problem_type == "real":
			SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       real_input_matrix
			SCREAMING_SNAKE_CASE   :      int				     =       real_vector
		elif problem_type == "complex":
			SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       complex_input_matrix
			SCREAMING_SNAKE_CASE   :      List[Any]				     =       complex_vector

		# Our implementation.
		SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE   :      int				     =       power_iteration(_a							,			_a)

		# Numpy implementation.

		# Get eigenvalues and eigenvectors using built-in numpy
		# eigh (eigh used for symmetric or hermetian matrices).
		SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE   :      str				     =       np.linalg.eigh(_a)
		# Last eigenvalue is the maximum one.
		SCREAMING_SNAKE_CASE   :      str				     =       eigen_values[-1]
		# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
		SCREAMING_SNAKE_CASE   :      Optional[int]				     =       eigen_vectors[:, -1]

		# Check our implementation and numpy gives close answers.
		assert np.abs(eigen_value - eigen_value_max) <= 1E-6
		# Take absolute values element wise of each eigenvector.
		# as they are only unique to a minus sign.
		assert np.linalg.norm(np.abs(_a) - np.abs(_a)) <= 1E-6


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			test_power_iteration()


import numpy as np






class _UpperCamelCase							:
			'''simple docstring'''







			def __init__(			self							: Dict	)							->       str:

				"""simple docstring"""

				SCREAMING_SNAKE_CASE   :      List[str]				     =       (0, 0)
				SCREAMING_SNAKE_CASE   :      Any				     =       None
				SCREAMING_SNAKE_CASE   :      Any				     =       0
				SCREAMING_SNAKE_CASE   :      List[str]				     =       0
				SCREAMING_SNAKE_CASE   :      Any				     =       0
			def __eq__(			self							: List[str]      ,      a							: Any	)							->       Tuple:

				"""simple docstring"""

				return self.position == cell.position






			def 							__UpperCamelCase (			self							: Optional[Any]	)							->       str:

				"""simple docstring"""

				print(self.position	)






class _UpperCamelCase							:
			'''simple docstring'''







			def __init__(			self							: List[Any]      ,      a							: List[Any]=(5, 5)	)							->       Optional[int]:

				"""simple docstring"""

				SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       np.zeros(a	)
				SCREAMING_SNAKE_CASE   :      str				     =       world_size[0]
				SCREAMING_SNAKE_CASE   :      int				     =       world_size[1]
			def 							__UpperCamelCase (			self							: Union[str, Any]	)							->       Optional[int]:

				"""simple docstring"""

				print(self.w	)






			def 							__UpperCamelCase (			self							: Dict      ,      a							: Dict	)							->       Tuple:

				"""simple docstring"""

				SCREAMING_SNAKE_CASE   :      Optional[Any]				     =       [
				    (-1, -1),
				    (-1, 0),
				    (-1, 1),
				    (0, -1),
				    (0, 1),
				    (1, -1),
				    (1, 0),
				    (1, 1),
				]
				SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       cell.position[0]
				SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       cell.position[1]
				SCREAMING_SNAKE_CASE   :      Optional[int]				     =       []
				for n in neughbour_cord:
					SCREAMING_SNAKE_CASE   :      Optional[Any]				     =       current_x + n[0]
					SCREAMING_SNAKE_CASE   :      int				     =       current_y + n[1]
					if 0 <= x < self.world_x_limit and 0 <= y < self.world_y_limit:
						SCREAMING_SNAKE_CASE   :      int				     =       Cell()
						SCREAMING_SNAKE_CASE   :      str				     =       (x, y)
						SCREAMING_SNAKE_CASE   :      Any				     =       cell
						neighbours.append(a	)
				return neighbours

def  lowerCamelCase__ (					_a							,			_a							,			_a):
	SCREAMING_SNAKE_CASE   :      Tuple				     =       []
	SCREAMING_SNAKE_CASE   :      Tuple				     =       []
	_open.append(_a)

	while _open:
		SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       np.argmin([n.f for n in _open])
		SCREAMING_SNAKE_CASE   :      List[str]				     =       _open[min_f]
		_closed.append(_open.pop(_a))
		if current == goal:
			break
		for n in world.get_neigbours(_a):
			for c in _closed:
				if c == n:
					continue
			SCREAMING_SNAKE_CASE   :      Tuple				     =       current.g + 1
			SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE   :      List[str]				     =       n.position
			SCREAMING_SNAKE_CASE     ,SCREAMING_SNAKE_CASE   :      str				     =       goal.position
			SCREAMING_SNAKE_CASE   :      Any				     =       (ya - ya) ** 2 + (xa - xa) ** 2
			SCREAMING_SNAKE_CASE   :      List[str]				     =       n.h + n.g

			for c in _open:
				if c == n and c.f < n.f:
					continue
			_open.append(_a)
	SCREAMING_SNAKE_CASE   :      Tuple				     =       []
	while current.parent is not None:
		path.append(current.position)
		SCREAMING_SNAKE_CASE   :      Union[str, Any]				     =       current.parent
	path.append(current.position)
	return path[::-1]


if __name__ == "__main__":
			a_						     =    Gridworld()
			# Start position and goal
			a_						     =    Cell()
			a_						     =    (0, 0)
			a_						     =    Cell()
			a_						     =    (4, 4)
			print(F'''path from {start.position} to {goal.position}''')
			a_						     =    astar(world, start, goal)
			#   Just for visual reasons.
			for i in s:
						a_						     =    1
			print(world.w)