
"""simple docstring"""




import numpy as np

def lowercase_						(     _snake_case     ,_snake_case     ,_snake_case = 1E-12     ,_snake_case = 100     ,):
							assert np.shape(_snake_case		)[0] == np.shape(_snake_case		)[1]
							# Ensure proper dimensionality.
							assert np.shape(_snake_case		)[0] == np.shape(_snake_case		)[0]
							# Ensure inputs are either both complex or both real
							assert np.iscomplexobj(_snake_case		) == np.iscomplexobj(_snake_case		)
							SCREAMING_SNAKE_CASE__		: List[str]     =     np.iscomplexobj(_snake_case		)
							if is_complex:
														# Ensure complex input_matrix is Hermitian
														assert np.array_equal(_snake_case     ,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__		: Tuple     =     False
							SCREAMING_SNAKE_CASE__		: Dict     =     0
							SCREAMING_SNAKE_CASE__		: Tuple     =     0
							SCREAMING_SNAKE_CASE__		: Optional[Any]     =     1E12

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

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

														if error <= error_tol or iterations >= max_iterations:
																					SCREAMING_SNAKE_CASE__		: int     =     True

														SCREAMING_SNAKE_CASE__		: Any     =     lambda_

							if is_complex:
														SCREAMING_SNAKE_CASE__		: Union[str, Any]     =     np.real(lambda_		)

							return lambda_, vector

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

							for problem_type in ["real", "complex"]:
														if problem_type == "real":
																					SCREAMING_SNAKE_CASE__		: List[str]     =     real_input_matrix
																					SCREAMING_SNAKE_CASE__		: Dict     =     real_vector
														elif problem_type == "complex":
																					SCREAMING_SNAKE_CASE__		: Tuple     =     complex_input_matrix
																					SCREAMING_SNAKE_CASE__		: int     =     complex_vector

														# Our implementation.
														SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__		: Union[str, Any]     =     power_iteration(_snake_case     ,_snake_case		)

														# Numpy implementation.

														# Get eigenvalues and eigenvectors using built-in numpy
														# eigh (eigh used for symmetric or hermetian matrices).
														SCREAMING_SNAKE_CASE__   ,   SCREAMING_SNAKE_CASE__		: Optional[int]     =     np.linalg.eigh(_snake_case		)
														# Last eigenvalue is the maximum one.
														SCREAMING_SNAKE_CASE__		: Tuple     =     eigen_values[-1]
														# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
														SCREAMING_SNAKE_CASE__		: 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(_snake_case		) - np.abs(_snake_case		)		) <= 1E-6


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_power_iteration()






"""simple docstring"""




import baseaa

def lowercase_						(     _snake_case		):
							return baseaa.baaencode(string.encode("""utf-8"""		)		)

def lowercase_						(     _snake_case		):
							return baseaa.baadecode(_snake_case		).decode("""utf-8"""		)


if __name__ == "__main__":
					UpperCAmelCase__   :					Dict												=				'Hello World!'
					UpperCAmelCase__   :					Tuple												=				baseaa_encode(test)
					print(encoded)

					UpperCAmelCase__   :					Tuple												=				baseaa_decode(encoded)
					print(decoded)




