






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 typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_flax_available,
    is_tf_available,
    is_tokenizers_available,
    is_torch_available,
)


_UpperCAmelCase			:		str   				=    {
    "configuration_blenderbot_small": [
        "BLENDERBOT_SMALL_PRETRAINED_CONFIG_ARCHIVE_MAP",
        "BlenderbotSmallConfig",
        "BlenderbotSmallOnnxConfig",
    ],
    "tokenization_blenderbot_small": ["BlenderbotSmallTokenizer"],
}

try:
			if not is_tokenizers_available():
						raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
			pass
else:
			_UpperCAmelCase			:		List[str]   				=    ["BlenderbotSmallTokenizerFast"]

try:
			if not is_torch_available():
						raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
			pass
else:
			_UpperCAmelCase			:		List[str]   				=    [
			    "BLENDERBOT_SMALL_PRETRAINED_MODEL_ARCHIVE_LIST",
			    "BlenderbotSmallForCausalLM",
			    "BlenderbotSmallForConditionalGeneration",
			    "BlenderbotSmallModel",
			    "BlenderbotSmallPreTrainedModel",
			]

try:
			if not is_tf_available():
						raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
			pass
else:
			_UpperCAmelCase			:		Dict   				=    [
			    "TFBlenderbotSmallForConditionalGeneration",
			    "TFBlenderbotSmallModel",
			    "TFBlenderbotSmallPreTrainedModel",
			]

try:
			if not is_flax_available():
						raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
			pass
else:
			_UpperCAmelCase			:		List[str]   				=    [
			    "FlaxBlenderbotSmallForConditionalGeneration",
			    "FlaxBlenderbotSmallModel",
			    "FlaxBlenderbotSmallPreTrainedModel",
			]

if TYPE_CHECKING:
			from .configuration_blenderbot_small import (
			    BLENDERBOT_SMALL_PRETRAINED_CONFIG_ARCHIVE_MAP,
			    BlenderbotSmallConfig,
			    BlenderbotSmallOnnxConfig,
			)
			from .tokenization_blenderbot_small import BlenderbotSmallTokenizer

			try:
						if not is_tokenizers_available():
									raise OptionalDependencyNotAvailable()
			except OptionalDependencyNotAvailable:
						pass
			else:
						from .tokenization_blenderbot_small_fast import BlenderbotSmallTokenizerFast

			try:
						if not is_torch_available():
									raise OptionalDependencyNotAvailable()
			except OptionalDependencyNotAvailable:
						pass
			else:
						from .modeling_blenderbot_small import (
						    BLENDERBOT_SMALL_PRETRAINED_MODEL_ARCHIVE_LIST,
						    BlenderbotSmallForCausalLM,
						    BlenderbotSmallForConditionalGeneration,
						    BlenderbotSmallModel,
						    BlenderbotSmallPreTrainedModel,
						)

			try:
						if not is_tf_available():
									raise OptionalDependencyNotAvailable()
			except OptionalDependencyNotAvailable:
						pass
			else:
						from .modeling_tf_blenderbot_small import (
						    TFBlenderbotSmallForConditionalGeneration,
						    TFBlenderbotSmallModel,
						    TFBlenderbotSmallPreTrainedModel,
						)

			try:
						if not is_flax_available():
									raise OptionalDependencyNotAvailable()
			except OptionalDependencyNotAvailable:
						pass
			else:
						from .modeling_flax_blenderbot_small import (
						    FlaxBlenderbotSmallForConditionalGeneration,
						    FlaxBlenderbotSmallModel,
						    FlaxBlenderbotSmallPreTrainedModel,
						)

else:
			import sys

			_UpperCAmelCase			:		int   				=    _LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)





