'''simple docstring'''


from typing import Any

import numpy as np


def 					_lowerCamelCase    (      lowerCamelCase_					:							np.ndarray   ):



     """simple docstring"""





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

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

     # Get eigenvalues and eignevectors for a symmetric matrix.
     UpperCAmelCase_   ,			UpperCAmelCase_							:   List[str]   =      np.linalg.eigh(lowerCamelCase_   )

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


def 					_lowerCamelCase    (      lowerCamelCase_					:							int   ):



     """simple docstring"""





     UpperCAmelCase_							:   Tuple   =      np.random.randn(lowerCamelCase_	,     lowerCamelCase_   )
     UpperCAmelCase_							:   Optional[Any]   =      np.dot(lowerCamelCase_	,     random_matrix.T   )
     assert _is_matrix_spd(lowerCamelCase_   )
     return spd_matrix


def 					_lowerCamelCase    (      lowerCamelCase_					:							np.ndarray	,     lowerCamelCase_					:							np.ndarray	,     lowerCamelCase_					:							int = 1000	,     lowerCamelCase_					:							float = 1e-8	,     ):



     """simple docstring"""





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

     # Initialize solution guess, residual, search direction.
     UpperCAmelCase_							:   List[Any]   =      np.zeros((np.shape(lowerCamelCase_   )[0], 1)   )
     UpperCAmelCase_							:   Any   =      np.copy(lowerCamelCase_   )
     UpperCAmelCase_							:   Tuple   =      np.copy(lowerCamelCase_   )

     # Set initial errors in solution guess and residual.
     UpperCAmelCase_							:   int   =      1e9
     UpperCAmelCase_							:   List[str]   =      1e9
     UpperCAmelCase_							:   List[str]   =      1e9

     # Set iteration counter to threshold number of iterations.
     UpperCAmelCase_							:   str   =      0

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

          # The main algorithm.

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

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

          # Update variables.
          UpperCAmelCase_							:   Tuple   =      np.copy(lowerCamelCase_   )
          UpperCAmelCase_							:   Tuple   =      np.copy(lowerCamelCase_   )
          UpperCAmelCase_							:   Tuple   =      np.copy(lowerCamelCase_   )

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

     return x


def 					_lowerCamelCase    (      ):



     """simple docstring"""





     UpperCAmelCase_							:   List[Any]   =      3
     UpperCAmelCase_							:   Dict   =      _create_spd_matrix(lowerCamelCase_   )
     UpperCAmelCase_							:   Optional[Any]   =      np.random.randn(lowerCamelCase_	,     1   )
     UpperCAmelCase_							:   Dict   =      np.dot(lowerCamelCase_	,     lowerCamelCase_   )

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

     # Our implementation.
     UpperCAmelCase_							:   int   =      conjugate_gradient(lowerCamelCase_	,     lowerCamelCase_   )

     # 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_tokenizers_available, is_torch_available


snake_case__     :			int      			=  {
    '''configuration_markuplm''': ['''MARKUPLM_PRETRAINED_CONFIG_ARCHIVE_MAP''', '''MarkupLMConfig'''],
    '''feature_extraction_markuplm''': ['''MarkupLMFeatureExtractor'''],
    '''processing_markuplm''': ['''MarkupLMProcessor'''],
    '''tokenization_markuplm''': ['''MarkupLMTokenizer'''],
}

try:
 if not is_tokenizers_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 snake_case__     :			List[str]      			=  ['''MarkupLMTokenizerFast''']

try:
 if not is_torch_available():
  raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
 pass
else:
 snake_case__     :			Dict      			=  [
     '''MARKUPLM_PRETRAINED_MODEL_ARCHIVE_LIST''',
     '''MarkupLMForQuestionAnswering''',
     '''MarkupLMForSequenceClassification''',
     '''MarkupLMForTokenClassification''',
     '''MarkupLMModel''',
     '''MarkupLMPreTrainedModel''',
 ]


if TYPE_CHECKING:
 from .configuration_markuplm import MARKUPLM_PRETRAINED_CONFIG_ARCHIVE_MAP, MarkupLMConfig
 from .feature_extraction_markuplm import MarkupLMFeatureExtractor
 from .processing_markuplm import MarkupLMProcessor
 from .tokenization_markuplm import MarkupLMTokenizer

 try:
  if not is_tokenizers_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .tokenization_markuplm_fast import MarkupLMTokenizerFast

 try:
  if not is_torch_available():
   raise OptionalDependencyNotAvailable()
 except OptionalDependencyNotAvailable:
  pass
 else:
  from .modeling_markuplm import (
      MARKUPLM_PRETRAINED_MODEL_ARCHIVE_LIST,
      MarkupLMForQuestionAnswering,
      MarkupLMForSequenceClassification,
      MarkupLMForTokenClassification,
      MarkupLMModel,
      MarkupLMPreTrainedModel,
  )


else:
 import sys

 snake_case__     :			Optional[Any]      			=  _LazyModule(__name__, globals()['''__file__'''], _import_structure)






