






import numpy as np






def 					lowerCamelCase__					(_UpperCAmelCase				,							_UpperCAmelCase				,							_UpperCAmelCase = 1e-12				,							_UpperCAmelCase = 100				,							):
			assert np.shape(_UpperCAmelCase)[0] == np.shape(_UpperCAmelCase)[1]
			# Ensure proper dimensionality.
			assert np.shape(_UpperCAmelCase)[0] == np.shape(_UpperCAmelCase)[0]
			# Ensure inputs are either both complex or both real
			assert np.iscomplexobj(_UpperCAmelCase) == np.iscomplexobj(_UpperCAmelCase)
			SCREAMING_SNAKE_CASE							=			np.iscomplexobj(_UpperCAmelCase)
			if is_complex:
						# Ensure complex input_matrix is Hermitian
						assert np.array_equal(_UpperCAmelCase				,							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							=			False
			SCREAMING_SNAKE_CASE							=			0
			SCREAMING_SNAKE_CASE							=			0
			SCREAMING_SNAKE_CASE							=			1e12

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

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

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

						SCREAMING_SNAKE_CASE							=			lambda_

			if is_complex:
						SCREAMING_SNAKE_CASE							=			np.real(lambda_)

			return lambda_, vector






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

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

						# Our implementation.
						SCREAMING_SNAKE_CASE					,    SCREAMING_SNAKE_CASE							=			power_iteration(_UpperCAmelCase				,							_UpperCAmelCase)

						# Numpy implementation.

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


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				test_power_iteration()






"""simple docstring"""


from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_torch_available


_lowercase				 =					{
    '''configuration_xlm_roberta_xl''': [
        '''XLM_ROBERTA_XL_PRETRAINED_CONFIG_ARCHIVE_MAP''',
        '''XLMRobertaXLConfig''',
        '''XLMRobertaXLOnnxConfig''',
    ],
}

try:
		if not is_torch_available():
				raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
		pass
else:
		_lowercase				 =					[
		    '''XLM_ROBERTA_XL_PRETRAINED_MODEL_ARCHIVE_LIST''',
		    '''XLMRobertaXLForCausalLM''',
		    '''XLMRobertaXLForMaskedLM''',
		    '''XLMRobertaXLForMultipleChoice''',
		    '''XLMRobertaXLForQuestionAnswering''',
		    '''XLMRobertaXLForSequenceClassification''',
		    '''XLMRobertaXLForTokenClassification''',
		    '''XLMRobertaXLModel''',
		    '''XLMRobertaXLPreTrainedModel''',
		]

if TYPE_CHECKING:
		from .configuration_xlm_roberta_xl import (
		    XLM_ROBERTA_XL_PRETRAINED_CONFIG_ARCHIVE_MAP,
		    XLMRobertaXLConfig,
		    XLMRobertaXLOnnxConfig,
		)

		try:
				if not is_torch_available():
						raise OptionalDependencyNotAvailable()
		except OptionalDependencyNotAvailable:
				pass
		else:
				from .modeling_xlm_roberta_xl import (
				    XLM_ROBERTA_XL_PRETRAINED_MODEL_ARCHIVE_LIST,
				    XLMRobertaXLForCausalLM,
				    XLMRobertaXLForMaskedLM,
				    XLMRobertaXLForMultipleChoice,
				    XLMRobertaXLForQuestionAnswering,
				    XLMRobertaXLForSequenceClassification,
				    XLMRobertaXLForTokenClassification,
				    XLMRobertaXLModel,
				    XLMRobertaXLPreTrainedModel,
				)

else:
		import sys

		_lowercase				 =					_LazyModule(__name__, globals()['''__file__'''], _import_structure)