
from typing import Any

import numpy as np




def 						A_			(				snake_case				: np.ndarray			)   ->     bool:



						'''simple docstring'''

						assert np.shape(snake_case			)[0] == np.shape(snake_case			)[1]

						# If matrix not symmetric, exit right away.
						if np.allclose(snake_case			,   matrix.T			) is False:
												return False

						# Get eigenvalues and eignevectors for a symmetric matrix.
						__UpperCamelCase	,				__UpperCamelCase  	=      np.linalg.eigh(snake_case			)

						# Check sign of all eigenvalues.
						# np.all returns a value of type np.bool_
						return bool(np.all(eigen_values > 0			)			)




def 						A_			(				snake_case				: int			)   ->     Any:



						'''simple docstring'''

						__UpperCamelCase  	=      np.random.randn(snake_case			,   snake_case			)
						__UpperCamelCase  	=      np.dot(snake_case			,   random_matrix.T			)
						assert _is_matrix_spd(snake_case			)
						return spd_matrix




def 						A_			(				snake_case				: np.ndarray			,   snake_case				: np.ndarray			,   snake_case				: int = 1000			,   snake_case				: float = 1e-8			,   )   ->     Any:



						'''simple docstring'''

						assert np.shape(snake_case			)[0] == np.shape(snake_case			)[1]
						assert np.shape(snake_case			)[0] == np.shape(snake_case			)[0]
						assert _is_matrix_spd(snake_case			)

						# Initialize solution guess, residual, search direction.
						__UpperCamelCase  	=      np.zeros((np.shape(snake_case			)[0], 1)			)
						__UpperCamelCase  	=      np.copy(snake_case			)
						__UpperCamelCase  	=      np.copy(snake_case			)

						# Set initial errors in solution guess and residual.
						__UpperCamelCase  	=      1e9
						__UpperCamelCase  	=      1e9
						__UpperCamelCase  	=      1e9

						# Set iteration counter to threshold number of iterations.
						__UpperCamelCase  	=      0

						while error > tol:
												# Save this value so we only calculate the matrix-vector product once.
												__UpperCamelCase  	=      np.dot(snake_case			,   snake_case			)

												# The main algorithm.

												# Update search direction magnitude.
												__UpperCamelCase  	=      np.dot(ra.T			,   snake_case			) / np.dot(pa.T			,   snake_case			)
												# Update solution guess.
												__UpperCamelCase  	=      xa + alpha * pa
												# Calculate new residual.
												__UpperCamelCase  	=      ra - alpha * w
												# Calculate new Krylov subspace scale.
												__UpperCamelCase  	=      np.dot(r.T			,   snake_case			) / np.dot(ra.T			,   snake_case			)
												# Calculate new A conjuage search direction.
												__UpperCamelCase  	=      r + beta * pa

												# Calculate errors.
												__UpperCamelCase  	=      np.linalg.norm(r - ra			)
												__UpperCamelCase  	=      np.linalg.norm(x - xa			)
												__UpperCamelCase  	=      np.maximum(snake_case			,   snake_case			)

												# Update variables.
												__UpperCamelCase  	=      np.copy(snake_case			)
												__UpperCamelCase  	=      np.copy(snake_case			)
												__UpperCamelCase  	=      np.copy(snake_case			)

												# Update number of iterations.
												iterations += 1
												if iterations > max_iterations:
																		break

						return x




def 						A_			(				)   ->     None:



						'''simple docstring'''

						__UpperCamelCase  	=      3
						__UpperCamelCase  	=      _create_spd_matrix(snake_case			)
						__UpperCamelCase  	=      np.random.randn(snake_case			,   1			)
						__UpperCamelCase  	=      np.dot(snake_case			,   snake_case			)

						# Numpy solution.
						__UpperCamelCase  	=      np.linalg.solve(snake_case			,   snake_case			)

						# Our implementation.
						__UpperCamelCase  	=      conjugate_gradient(snake_case			,   snake_case			)

						# Ensure both solutions are close to x_true (and therefore one another).
						assert np.linalg.norm(x_numpy - x_true			) <= 1e-6
						assert np.linalg.norm(x_conjugate_gradient - x_true			) <= 1e-6


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				test_conjugate_gradient()







from typing import TYPE_CHECKING

from ...utils import OptionalDependencyNotAvailable, _LazyModule, is_sentencepiece_available, is_torch_available


lowerCAmelCase__							=			{
    "configuration_xlm_prophetnet": ["XLM_PROPHETNET_PRETRAINED_CONFIG_ARCHIVE_MAP", "XLMProphetNetConfig"],
}

try:
      if not is_sentencepiece_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      lowerCAmelCase__							=			["XLMProphetNetTokenizer"]

try:
      if not is_torch_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      lowerCAmelCase__							=			[
          "XLM_PROPHETNET_PRETRAINED_MODEL_ARCHIVE_LIST",
          "XLMProphetNetDecoder",
          "XLMProphetNetEncoder",
          "XLMProphetNetForCausalLM",
          "XLMProphetNetForConditionalGeneration",
          "XLMProphetNetModel",
          "XLMProphetNetPreTrainedModel",
      ]


if TYPE_CHECKING:
      from .configuration_xlm_prophetnet import XLM_PROPHETNET_PRETRAINED_CONFIG_ARCHIVE_MAP, XLMProphetNetConfig

      try:
            if not is_sentencepiece_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .tokenization_xlm_prophetnet import XLMProphetNetTokenizer

      try:
            if not is_torch_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .modeling_xlm_prophetnet import (
                XLM_PROPHETNET_PRETRAINED_MODEL_ARCHIVE_LIST,
                XLMProphetNetDecoder,
                XLMProphetNetEncoder,
                XLMProphetNetForCausalLM,
                XLMProphetNetForConditionalGeneration,
                XLMProphetNetModel,
                XLMProphetNetPreTrainedModel,
            )

else:
      import sys

      lowerCAmelCase__							=			_LazyModule(__name__, globals()["__file__"], _import_structure, module_spec=__spec__)



