






import numpy as np





def 	lowerCAmelCase_		(lowercase__						:       np.ndarray     ,					lowercase__						:       np.ndarray     ,					lowercase__						:       float = 1e-1_2     ,					lowercase__						:       int = 1_00     ,					)		->    tuple[float, np.ndarray]:


				'''simple docstring'''




				assert np.shape(lowercase__       )[0] == np.shape(lowercase__       )[1]
				# Ensure proper dimensionality.
				assert np.shape(lowercase__       )[0] == np.shape(lowercase__       )[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(lowercase__       ) == np.iscomplexobj(lowercase__       )
				lowerCAmelCase__      			=						np.iscomplexobj(lowercase__       )
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(lowercase__     ,					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__      			=						1e1_2

				while not convergence:
								# Multiple matrix by the vector.
								lowerCAmelCase__      			=						np.dot(lowercase__     ,					lowercase__       )
								# Normalize the resulting output vector.
								lowerCAmelCase__      			=						w / np.linalg.norm(lowercase__       )
								# 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(lowercase__     ,					np.dot(lowercase__     ,					lowercase__       )       )

								# 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 	lowerCAmelCase_		()		->    None:


				'''simple docstring'''




				lowerCAmelCase__      			=						np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]       )
				lowerCAmelCase__      			=						np.array([41, 4, 20]       )
				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([41, 4, 20]       ).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__			,		lowerCAmelCase__      			=						power_iteration(lowercase__     ,					lowercase__       )

								# Numpy implementation.

								# Get eigenvalues and eigenvectors using built-in numpy
								# eigh (eigh used for symmetric or hermetian matrices).
								lowerCAmelCase__			,		lowerCAmelCase__      			=						np.linalg.eigh(lowercase__       )
								# 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(lowercase__       ) - np.abs(lowercase__       )       ) <= 1e-6


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			test_power_iteration()













from __future__ import annotations





def 	lowerCAmelCase_		(lowercase__						:       list[int]       )		->    bool:


				'''simple docstring'''




				return len(set(lowercase__       )       ) == len(lowercase__       )


if __name__ == "__main__":
			import doctest

			doctest.testmod()





