





from typing import Any

import numpy as np

def     UpperCAmelCase_						(						_UpperCAmelCase       :np.ndarray							)  ->							bool:
     '''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_					      =					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       :int							)  ->							Any:
     '''simple docstring'''


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

def     UpperCAmelCase_						(						_UpperCAmelCase       :np.ndarray     ,	_UpperCAmelCase       :np.ndarray     ,	_UpperCAmelCase       :int = 1000     ,	_UpperCAmelCase       :float = 1e-8     ,	)  ->							Any:
     '''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_					      =					np.zeros((np.shape(_UpperCAmelCase							)[0], 1)							)
     A_					      =					np.copy(_UpperCAmelCase							)
     A_					      =					np.copy(_UpperCAmelCase							)

     # 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(_UpperCAmelCase     ,	_UpperCAmelCase							)

          # The main algorithm.

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

          # Update variables.
          A_					      =					np.copy(_UpperCAmelCase							)
          A_					      =					np.copy(_UpperCAmelCase							)
          A_					      =					np.copy(_UpperCAmelCase							)

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

     return x

def     UpperCAmelCase_						(						)  ->							None:
     '''simple docstring'''


     A_					      =					3
     A_					      =					_create_spd_matrix(_UpperCAmelCase							)
     A_					      =					np.random.randn(_UpperCAmelCase     ,	1							)
     A_					      =					np.dot(_UpperCAmelCase     ,	_UpperCAmelCase							)

     # Numpy solution.
     A_					      =					np.linalg.solve(_UpperCAmelCase     ,	_UpperCAmelCase							)

     # Our implementation.
     A_					      =					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
from pathlib import Path

def     UpperCAmelCase_						(						_UpperCAmelCase       :list[str]							)  ->							int:
     '''simple docstring'''


     A_					      =					[tuple(_UpperCAmelCase							) for login in logins]

     A_					      =					{char for login in split_logins for char in login}

     for permutation in itertools.permutations(_UpperCAmelCase							):
          A_					      =					True
          for login in logins:
               if not (
                   permutation.index(login[0]							)
                   < permutation.index(login[1]							)
                   < permutation.index(login[2]							)
               ):
                    A_					      =					False
                    break

          if satisfied:
               return int(''''''.join(_UpperCAmelCase							)							)

     raise Exception('''Unable to find the secret passcode'''							)

def     UpperCAmelCase_						(						_UpperCAmelCase       :str = "keylog.txt"							)  ->							int:
     '''simple docstring'''


     A_					      =					Path(_UpperCAmelCase							).parent.joinpath(_UpperCAmelCase							).read_text().splitlines()

     return find_secret_passcode(_UpperCAmelCase							)


if __name__ == "__main__":
      print(f'''{solution() = }''')



