import numpy as np

def    SCREAMING_SNAKE_CASE_     (						snake_case__   ,  snake_case__   ,  snake_case__ = 1E-12   ,  snake_case__ = 1_0_0   ,  )     ->	Dict:
							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__							)
							lowerCAmelCase			=		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.

							lowerCAmelCase			=		False
							lowerCAmelCase			=		0
							lowerCAmelCase			=		0
							lowerCAmelCase			=		1E12

							while not convergence:
														# Multiple matrix by the vector.
														lowerCAmelCase			=		np.dot(A__   ,  A__							)
														# Normalize the resulting output vector.
														lowerCAmelCase			=		w / np.linalg.norm(A__							)
														# Find rayleigh quotient
														# (faster than usual b/c we know vector is normalized already)
														lowerCAmelCase			=		vector.conj().T if is_complex else vector.T
														lowerCAmelCase			=		np.dot(A__   ,  np.dot(A__   ,  A__							)							)

														# Check convergence.
														lowerCAmelCase			=		np.abs(lambda_ - lambda_previous							) / lambda_
														iterations += 1

														if error <= error_tol or iterations >= max_iterations:
																					lowerCAmelCase			=		True

														lowerCAmelCase			=		lambda_

							if is_complex:
														lowerCAmelCase			=		np.real(lambda_							)

							return lambda_, vector

def    SCREAMING_SNAKE_CASE_     (						)     ->	Optional[Any]:
							lowerCAmelCase			=		np.array([[4_1, 4, 2_0], [4, 2_6, 3_0], [2_0, 3_0, 5_0]]							)
							lowerCAmelCase			=		np.array([4_1, 4, 2_0]							)
							lowerCAmelCase			=		real_input_matrix.astype(np.complexaaa							)
							lowerCAmelCase			=		np.triu(1J * complex_input_matrix   ,  1							)
							complex_input_matrix += imag_matrix
							complex_input_matrix += -1 * imag_matrix.T
							lowerCAmelCase			=		np.array([4_1, 4, 2_0]							).astype(np.complexaaa							)

							for problem_type in ["real", "complex"]:
														if problem_type == "real":
																					lowerCAmelCase			=		real_input_matrix
																					lowerCAmelCase			=		real_vector
														elif problem_type == "complex":
																					lowerCAmelCase			=		complex_input_matrix
																					lowerCAmelCase			=		complex_vector

														# Our implementation.
														lowerCAmelCase			=		power_iteration(A__   ,  A__							)

														# Numpy implementation.

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











"""simple docstring"""
def 				snake_case      (   A__	,A__	,A__	,A__	,A__						):
			if index == number_of_items:
						return 0
			UpperCAmelCase_     : str			     =					0
			UpperCAmelCase_     : Dict			     =					0
			UpperCAmelCase_     : Tuple			     =					knapsack(A__	,A__	,A__	,A__	,index + 1						)
			if weights[index] <= max_weight:
						UpperCAmelCase_     : Union[str, Any]			     =					values[index] + knapsack(
						    A__	,A__	,A__	,max_weight - weights[index]	,index + 1						)
			return max(A__	,A__						)


if __name__ == "__main__":
			import doctest

			doctest.testmod()



