






from typing import Any

import numpy as np
def 		_a					(   a				:np.ndarray				)							->    bool:
 assert np.shape(a				)[0] == np.shape(a				)[1]

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

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

 # Check sign of all eigenvalues.
 # np.all returns a value of type np.bool_
 return bool(np.all(eigen_values > 0				)				)
def 		_a					(   a				:int				)							->    Any:
 a							=   np.random.randn(a			,							a				)
 a							=   np.dot(a			,							random_matrix.T				)
 assert _is_matrix_spd(a				)
 return spd_matrix
def 		_a					(   a				:np.ndarray			,							a				:np.ndarray			,							a				:int = 1_000			,							a				:float = 1e-8			,							)							->    Any:
 assert np.shape(a				)[0] == np.shape(a				)[1]
 assert np.shape(a				)[0] == np.shape(a				)[0]
 assert _is_matrix_spd(a				)

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

 # Set initial errors in solution guess and residual.
 a							=   1e9
 a							=   1e9
 a							=   1e9

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

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

  # The main algorithm.

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

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

  # Update variables.
  a							=   np.copy(a				)
  a							=   np.copy(a				)
  a							=   np.copy(a				)

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

 return x
def 		_a					(   )							->    None:
 a							=   3
 a							=   _create_spd_matrix(a				)
 a							=   np.random.randn(a			,							1				)
 a							=   np.dot(a			,							a				)

 # Numpy solution.
 a							=   np.linalg.solve(a			,							a				)

 # Our implementation.
 a							=   conjugate_gradient(a			,							a				)

 # 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 numpy as np

UpperCAmelCase__            =	[
    ["a", "b", "c", "d", "e"],
    ["f", "g", "h", "i", "k"],
    ["l", "m", "n", "o", "p"],
    ["q", "r", "s", "t", "u"],
    ["v", "w", "x", "y", "z"],
]






class 	lowercase_      :






      '''simple docstring'''




      def __init__(   self   :		Optional[Any]						)					->None:




       """simple docstring"""


       a							=   np.array(__UpperCAmelCase						)







      def    __lowerCAmelCase      (   self   :		List[str]				,      __UpperCAmelCase   :		str						)					->np.ndarray:




       """simple docstring"""


       a						,  a							=   np.where(letter == self.SQUARE						)
       a							=   np.concatenate([indexa + 1, indexa + 1]						)
       return indexes







      def    __lowerCAmelCase      (   self   :		int				,      __UpperCAmelCase   :		int				,      __UpperCAmelCase   :		int						)					->str:




       """simple docstring"""


       return self.SQUARE[indexa - 1, indexa - 1]







      def    __lowerCAmelCase      (   self   :		Tuple				,      __UpperCAmelCase   :		str						)					->str:




       """simple docstring"""


       a							=   message.lower()
       a							=   message.replace('''j'''				,      '''i'''						)

       a							=   ''''''
       for letter_index in range(len(__UpperCAmelCase						)						):
        if message[letter_index] != " ":
         a							=   self.letter_to_numbers(message[letter_index]						)
         a							=   encoded_message + str(numbers[0]						) + str(numbers[1]						)
        elif message[letter_index] == " ":
         a							=   encoded_message + ''' '''

       return encoded_message







      def    __lowerCAmelCase      (   self   :		Any				,      __UpperCAmelCase   :		str						)					->str:




       """simple docstring"""


       a							=   message.replace(''' '''				,      '''  '''						)
       a							=   ''''''
       for numbers_index in range(int(len(__UpperCAmelCase						) / 2						)						):
        if message[numbers_index * 2] != " ":
         a							=   message[numbers_index * 2]
         a							=   message[numbers_index * 2 + 1]

         a							=   self.numbers_to_letter(int(__UpperCAmelCase						)				,      int(__UpperCAmelCase						)						)
         a							=   decoded_message + letter
        elif message[numbers_index * 2] == " ":
         a							=   decoded_message + ''' '''

       return decoded_message


