





"""simple docstring"""




from typing import Any

import numpy as np





def 		UpperCAmelCase__						(	_UpperCAmelCase   ):






  """simple docstring"""





  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.
  A_		,			A_    :			Any									=    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   ):






  """simple docstring"""





  A_    :			str									=    np.random.randn(_UpperCAmelCase					,   _UpperCAmelCase   )
  A_    :			str									=    np.dot(_UpperCAmelCase					,   random_matrix.T   )
  assert _is_matrix_spd(_UpperCAmelCase   )
  return spd_matrix





def 		UpperCAmelCase__						(	_UpperCAmelCase					,   _UpperCAmelCase					,   _UpperCAmelCase = 1000					,   _UpperCAmelCase = 1E-8					,   ):






  """simple docstring"""





  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.
  A_    :			Optional[int]									=    np.zeros((np.shape(_UpperCAmelCase   )[0], 1)   )
  A_    :			Dict									=    np.copy(_UpperCAmelCase   )
  A_    :			List[str]									=    np.copy(_UpperCAmelCase   )

  # Set initial errors in solution guess and residual.
  A_    :			Tuple									=    1E9
  A_    :			List[Any]									=    1E9
  A_    :			Optional[Any]									=    1E9

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

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

    # The main algorithm.

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

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

    # Update variables.
    A_    :			str									=    np.copy(_UpperCAmelCase   )
    A_    :			Union[str, Any]									=    np.copy(_UpperCAmelCase   )
    A_    :			Union[str, Any]									=    np.copy(_UpperCAmelCase   )

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

  return x





def 		UpperCAmelCase__						(	):






  """simple docstring"""





  A_    :			Dict									=    3
  A_    :			Optional[int]									=    _create_spd_matrix(_UpperCAmelCase   )
  A_    :			str									=    np.random.randn(_UpperCAmelCase					,   1   )
  A_    :			str									=    np.dot(_UpperCAmelCase					,   _UpperCAmelCase   )

  # Numpy solution.
  A_    :			Optional[Any]									=    np.linalg.solve(_UpperCAmelCase					,   _UpperCAmelCase   )

  # Our implementation.
  A_    :			str									=    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()


import itertools






def        __lowercase   (  _A = 1000000      )  ->    int:
      SCREAMING_SNAKE_CASE			:   Tuple	  = [1]

      for i in itertools.count(len(_A      )      ):
            SCREAMING_SNAKE_CASE			:   Optional[int]	  = 0
            for j in itertools.count(1      ):
                  SCREAMING_SNAKE_CASE			:   str	  = -1 if j % 2 == 0 else +1
                  SCREAMING_SNAKE_CASE			:   Any	  = (j * j * 3 - j) // 2
                  if index > i:
                        break
                  item += partitions[i - index] * sign
                  item %= number
                  index += j
                  if index > i:
                        break
                  item += partitions[i - index] * sign
                  item %= number

            if item == 0:
                  return i
            partitions.append(_A      )

      return 0


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  print(F"""{solution() = }""")




