
import numpy as np







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase = 1E-1_2     ,    lowerCamelCase = 1_00     ,    )  ->	tuple[float, np.ndarray]:
				assert np.shape(lowerCamelCase						)[0] == np.shape(lowerCamelCase						)[1]
				# Ensure proper dimensionality.
				assert np.shape(lowerCamelCase						)[0] == np.shape(lowerCamelCase						)[0]
				# Ensure inputs are either both complex or both real
				assert np.iscomplexobj(lowerCamelCase						) == np.iscomplexobj(lowerCamelCase						)
				UpperCamelCase_:							Dict	=						np.iscomplexobj(lowerCamelCase						)
				if is_complex:
								# Ensure complex input_matrix is Hermitian
								assert np.array_equal(lowerCamelCase     ,    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.

				UpperCamelCase_:							Dict	=						False
				UpperCamelCase_:							Tuple	=						0
				UpperCamelCase_:							str	=						0
				UpperCamelCase_:							List[str]	=						1E1_2

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

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

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

								UpperCamelCase_:							str	=						lambda_

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

				return lambda_, vector







def        A__					(						)  ->	None:
				UpperCamelCase_:							List[str]	=						np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]						)
				UpperCamelCase_:							Tuple	=						np.array([41, 4, 20]						)
				UpperCamelCase_:							Tuple	=						real_input_matrix.astype(np.complexaaa						)
				UpperCamelCase_:							List[Any]	=						np.triu(1j * complex_input_matrix     ,    1						)
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				UpperCamelCase_:							List[Any]	=						np.array([41, 4, 20]						).astype(np.complexaaa						)

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

								# Our implementation.
								UpperCamelCase_,   UpperCamelCase_:							int	=						power_iteration(lowerCamelCase     ,    lowerCamelCase						)

								# Numpy implementation.

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


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							test_power_iteration()








lowerCamelCase_						: Optional[Any]         =  """
# Transformers installation
! pip install transformers datasets
# To install from source instead of the last release, comment the command above and uncomment the following one.
# ! pip install git+https://github.com/huggingface/transformers.git
"""

lowerCamelCase_						: Union[str, Any]         =  [{"""type""": """code""", """content""": INSTALL_CONTENT}]
lowerCamelCase_						: Optional[Any]         =  {
    """{processor_class}""": """FakeProcessorClass""",
    """{model_class}""": """FakeModelClass""",
    """{object_class}""": """FakeObjectClass""",
}






