


'''simple docstring'''



from typing import Any

import numpy as np



def       UpperCAmelCase__ (     UpperCAmelCase_   :    np.ndarray							)					->		bool:
					assert np.shape(UpperCAmelCase_							)[0] == np.shape(UpperCAmelCase_							)[1]

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

					# Get eigenvalues and eignevectors for a symmetric matrix.
					__lowerCamelCase       :     Dict        =		np.linalg.eigh(UpperCAmelCase_							)

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



def       UpperCAmelCase__ (     UpperCAmelCase_   :    int							)					->		Any:
					__lowerCamelCase       :     Optional[int]        =		np.random.randn(UpperCAmelCase_				,   UpperCAmelCase_							)
					__lowerCamelCase       :     Tuple        =		np.dot(UpperCAmelCase_				,   random_matrix.T							)
					assert _is_matrix_spd(UpperCAmelCase_							)
					return spd_matrix



def       UpperCAmelCase__ (     UpperCAmelCase_   :    np.ndarray				,   UpperCAmelCase_   :    np.ndarray				,   UpperCAmelCase_   :    int = 10_00				,   UpperCAmelCase_   :    float = 1e-8				,   )					->		Any:
					assert np.shape(UpperCAmelCase_							)[0] == np.shape(UpperCAmelCase_							)[1]
					assert np.shape(UpperCAmelCase_							)[0] == np.shape(UpperCAmelCase_							)[0]
					assert _is_matrix_spd(UpperCAmelCase_							)

					# Initialize solution guess, residual, search direction.
					__lowerCamelCase       :     str        =		np.zeros((np.shape(UpperCAmelCase_							)[0], 1)							)
					__lowerCamelCase       :     str        =		np.copy(UpperCAmelCase_							)
					__lowerCamelCase       :     Union[str, Any]        =		np.copy(UpperCAmelCase_							)

					# Set initial errors in solution guess and residual.
					__lowerCamelCase       :     Optional[Any]        =		1e9
					__lowerCamelCase       :     Optional[int]        =		1e9
					__lowerCamelCase       :     int        =		1e9

					# Set iteration counter to threshold number of iterations.
					__lowerCamelCase       :     Any        =		0

					while error > tol:
										# Save this value so we only calculate the matrix-vector product once.
										__lowerCamelCase       :     Optional[Any]        =		np.dot(UpperCAmelCase_				,   UpperCAmelCase_							)

										# The main algorithm.

										# Update search direction magnitude.
										__lowerCamelCase       :     int        =		np.dot(ra.T				,   UpperCAmelCase_							) / np.dot(pa.T				,   UpperCAmelCase_							)
										# Update solution guess.
										__lowerCamelCase       :     Optional[Any]        =		xa + alpha * pa
										# Calculate new residual.
										__lowerCamelCase       :     Tuple        =		ra - alpha * w
										# Calculate new Krylov subspace scale.
										__lowerCamelCase       :     Union[str, Any]        =		np.dot(r.T				,   UpperCAmelCase_							) / np.dot(ra.T				,   UpperCAmelCase_							)
										# Calculate new A conjuage search direction.
										__lowerCamelCase       :     List[Any]        =		r + beta * pa

										# Calculate errors.
										__lowerCamelCase       :     Union[str, Any]        =		np.linalg.norm(r - ra							)
										__lowerCamelCase       :     List[Any]        =		np.linalg.norm(x - xa							)
										__lowerCamelCase       :     List[Any]        =		np.maximum(UpperCAmelCase_				,   UpperCAmelCase_							)

										# Update variables.
										__lowerCamelCase       :     List[str]        =		np.copy(UpperCAmelCase_							)
										__lowerCamelCase       :     str        =		np.copy(UpperCAmelCase_							)
										__lowerCamelCase       :     Optional[int]        =		np.copy(UpperCAmelCase_							)

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

					return x



def       UpperCAmelCase__ (     )					->		None:
					__lowerCamelCase       :     Dict        =		3
					__lowerCamelCase       :     int        =		_create_spd_matrix(UpperCAmelCase_							)
					__lowerCamelCase       :     Any        =		np.random.randn(UpperCAmelCase_				,   1							)
					__lowerCamelCase       :     Tuple        =		np.dot(UpperCAmelCase_				,   UpperCAmelCase_							)

					# Numpy solution.
					__lowerCamelCase       :     int        =		np.linalg.solve(UpperCAmelCase_				,   UpperCAmelCase_							)

					# Our implementation.
					__lowerCamelCase       :     Tuple        =		conjugate_gradient(UpperCAmelCase_				,   UpperCAmelCase_							)

					# 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()










'''simple docstring'''



from typing import TYPE_CHECKING

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


A__		: str												=  {"""configuration_hubert""": ["""HUBERT_PRETRAINED_CONFIG_ARCHIVE_MAP""", """HubertConfig"""]}

try:
      if not is_torch_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      A__		: List[str]												=  [
          """HUBERT_PRETRAINED_MODEL_ARCHIVE_LIST""",
          """HubertForCTC""",
          """HubertForSequenceClassification""",
          """HubertModel""",
          """HubertPreTrainedModel""",
      ]


try:
      if not is_tf_available():
            raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
      pass
else:
      A__		: Optional[int]												=  [
          """TF_HUBERT_PRETRAINED_MODEL_ARCHIVE_LIST""",
          """TFHubertForCTC""",
          """TFHubertModel""",
          """TFHubertPreTrainedModel""",
      ]

if TYPE_CHECKING:
      from .configuration_hubert import HUBERT_PRETRAINED_CONFIG_ARCHIVE_MAP, HubertConfig

      try:
            if not is_torch_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .modeling_hubert import (
                HUBERT_PRETRAINED_MODEL_ARCHIVE_LIST,
                HubertForCTC,
                HubertForSequenceClassification,
                HubertModel,
                HubertPreTrainedModel,
            )

      try:
            if not is_tf_available():
                  raise OptionalDependencyNotAvailable()
      except OptionalDependencyNotAvailable:
            pass
      else:
            from .modeling_tf_hubert import (
                TF_HUBERT_PRETRAINED_MODEL_ARCHIVE_LIST,
                TFHubertForCTC,
                TFHubertModel,
                TFHubertPreTrainedModel,
            )


else:
      import sys

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






