
from typing import Any

import numpy as np


def 		_lowerCamelCase       (   _a  ):
  """simple docstring"""




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

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

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

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


def 		_lowerCamelCase       (   _a  ):
  """simple docstring"""




  _lowerCamelCase      =						np.random.randn(lowercase__			,   lowercase__  )
  _lowerCamelCase      =						np.dot(lowercase__			,   random_matrix.T  )
  assert _is_matrix_spd(lowercase__  )
  return spd_matrix


def 		_lowerCamelCase       (   _a			,   _a			,   _a = 1_0_0_0			,   _a = 1E-8			,   ):
  """simple docstring"""




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

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

  # Set initial errors in solution guess and residual.
  _lowerCamelCase      =						1E9
  _lowerCamelCase      =						1E9
  _lowerCamelCase      =						1E9

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

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

    # The main algorithm.

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

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

    # Update variables.
    _lowerCamelCase      =						np.copy(lowercase__  )
    _lowerCamelCase      =						np.copy(lowercase__  )
    _lowerCamelCase      =						np.copy(lowercase__  )

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

  return x


def 		_lowerCamelCase       (   ):
  """simple docstring"""




  _lowerCamelCase      =						3
  _lowerCamelCase      =						_create_spd_matrix(lowercase__  )
  _lowerCamelCase      =						np.random.randn(lowercase__			,   1  )
  _lowerCamelCase      =						np.dot(lowercase__			,   lowercase__  )

  # Numpy solution.
  _lowerCamelCase      =						np.linalg.solve(lowercase__			,   lowercase__  )

  # Our implementation.
  _lowerCamelCase      =						conjugate_gradient(lowercase__			,   lowercase__  )

  # 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 _LazyModule


_lowercase				:					List[Any]     					={"""tokenization_bert_japanese""": ["""BertJapaneseTokenizer""", """CharacterTokenizer""", """MecabTokenizer"""]}


if TYPE_CHECKING:
	from .tokenization_bert_japanese import BertJapaneseTokenizer, CharacterTokenizer, MecabTokenizer

else:
	import sys

	_lowercase				:					Union[str, Any]     					=_LazyModule(__name__, globals()["""__file__"""], _import_structure, module_spec=__spec__)

