
'''simple docstring'''
def 							A_  (    __SCREAMING_SNAKE_CASE    :	Dict			)					->							list[int]:





  """simple docstring"""

  if len(__lowerCamelCase			) > 1:
    __A		:  int       				=		len(__lowerCamelCase			) // 2  # Finds the middle of the array
    __A		:  Any       				=		arr[
        :middle_length
    ]  # Creates an array of the elements in the first half.
    __A		:  List[Any]       				=		arr[
        middle_length:
    ]  # Creates an array of the elements in the second half.
    __A		:  Any       				=		len(__lowerCamelCase			)
    __A		:  Union[str, Any]       				=		len(__lowerCamelCase			)
    merge(__lowerCamelCase			)  # Starts sorting the left.
    merge(__lowerCamelCase			)  # Starts sorting the right
    __A		:  Any       				=		0  # Left Counter
    __A		:  Dict       				=		0  # Right Counter
    __A		:  Any       				=		0  # Position Counter
    while (
        left_index < left_size and right_index < right_size
    ):  # Runs until the lowers size of the left and right are sorted.
      if left_array[left_index] < right_array[right_index]:
        __A		:  int       				=		left_array[left_index]
        left_index += 1
      else:
        __A		:  List[Any]       				=		right_array[right_index]
        right_index += 1
      index += 1
    while (
        left_index < left_size
    ):  # Adds the left over elements in the left half of the array
      __A		:  List[Any]       				=		left_array[left_index]
      left_index += 1
      index += 1
    while (
        right_index < right_size
    ):  # Adds the left over elements in the right half of the array
      __A		:  Optional[Any]       				=		right_array[right_index]
      right_index += 1
      index += 1
  return arr


if __name__ == "__main__":
     import doctest

     doctest.testmod()







from typing import Any

import numpy as np






def  _UpperCAmelCase						( __lowerCamelCase							)		->bool:


 '''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.
 snake_case	,       snake_case :			int							=   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  _UpperCAmelCase						( __lowerCamelCase							)		->Any:


 '''simple docstring'''
 snake_case :			Dict							=   np.random.randn(__lowerCamelCase			,  __lowerCamelCase							)
 snake_case :			List[Any]							=   np.dot(__lowerCamelCase			,  random_matrix.T							)
 assert _is_matrix_spd(__lowerCamelCase							)
 return spd_matrix






def  _UpperCAmelCase						( __lowerCamelCase			,  __lowerCamelCase			,  __lowerCamelCase = 1000			,  __lowerCamelCase = 1E-8			,  )		->Any:


 '''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.
 snake_case :			Any							=   np.zeros((np.shape(__lowerCamelCase							)[0], 1)							)
 snake_case :			Optional[int]							=   np.copy(__lowerCamelCase							)
 snake_case :			Optional[Any]							=   np.copy(__lowerCamelCase							)

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

 # Set iteration counter to threshold number of iterations.
 snake_case :			List[str]							=   0

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

  # The main algorithm.

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

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

  # Update variables.
  snake_case :			Tuple							=   np.copy(__lowerCamelCase							)
  snake_case :			Tuple							=   np.copy(__lowerCamelCase							)
  snake_case :			List[str]							=   np.copy(__lowerCamelCase							)

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

 return x






def  _UpperCAmelCase						( )		->None:


 '''simple docstring'''
 snake_case :			str							=   3
 snake_case :			Dict							=   _create_spd_matrix(__lowerCamelCase							)
 snake_case :			str							=   np.random.randn(__lowerCamelCase			,  1							)
 snake_case :			str							=   np.dot(__lowerCamelCase			,  __lowerCamelCase							)

 # Numpy solution.
 snake_case :			Any							=   np.linalg.solve(__lowerCamelCase			,  __lowerCamelCase							)

 # Our implementation.
 snake_case :			List[str]							=   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()






