






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()










def 		_a					(   a				:int				)							->    int:
 if not isinstance(a			,							a				):
  raise TypeError('''recursive_lucas_number accepts only integer arguments.'''				)
 if n_th_number == 0:
  return 2
 if n_th_number == 1:
  return 1

 return recursive_lucas_number(n_th_number - 1				) + recursive_lucas_number(
     n_th_number - 2				)
def 		_a					(   a				:int				)							->    int:
 if not isinstance(a			,							a				):
  raise TypeError('''dynamic_lucas_number accepts only integer arguments.'''				)
 a						,  a							=   2, 1
 for _ in range(a				):
  a						,  a							=   b, a + b
 return a


if __name__ == "__main__":
     from doctest import testmod

     testmod()
     UpperCAmelCase__            =	int(input("Enter the number of terms in lucas series:\n").strip())
     print("Using recursive function to calculate lucas series:")
     print(" ".join(str(recursive_lucas_number(i)) for i in range(n)))
     print("\nUsing dynamic function to calculate lucas series:")
     print(" ".join(str(dynamic_lucas_number(i)) for i in range(n)))


