






'''simple docstring'''




def       __lowercase		(    __lowercase     ,						__lowercase ) ->		int:

      '''simple docstring'''


      if len(str(__lowercase ) ) == 1 or len(str(__lowercase ) ) == 1:
            return a * b

      _A      =			max(len(str(__lowercase ) )     ,						len(str(__lowercase ) ) )
      _A      =			ma // 2

      _A  ,			_A      =			divmod(__lowercase     ,						10**ma )
      _A  ,			_A      =			divmod(__lowercase     ,						10**ma )

      _A      =			karatsuba(__lowercase     ,						__lowercase )
      _A      =			karatsuba((aa + aa)     ,						(ba + ba) )
      _A      =			karatsuba(__lowercase     ,						__lowercase )

      return (z * 10 ** (2 * ma)) + ((y - z - x) * 10 ** (ma)) + (x)

def       __lowercase		(    ) ->		Optional[int]:

      '''simple docstring'''


      print(karatsuba(1_5463     ,						2_3489 ) )


if __name__ == "__main__":
   main()













'''simple docstring'''




from typing import Any

import numpy as np

def       __lowercase		(    __lowercase ) ->		bool:

      '''simple docstring'''


      assert np.shape(__lowercase )[0] == np.shape(__lowercase )[1]

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

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

      # Check sign of all eigenvalues.
      # np.all returns a value of type np.bool_
      return bool(np.all(eigen_values > 0 ) )

def       __lowercase		(    __lowercase ) ->		Any:

      '''simple docstring'''


      _A      =			np.random.randn(__lowercase     ,						__lowercase )
      _A      =			np.dot(__lowercase     ,						random_matrix.T )
      assert _is_matrix_spd(__lowercase )
      return spd_matrix

def       __lowercase		(    __lowercase     ,						__lowercase     ,						__lowercase = 1000     ,						__lowercase = 1e-8     ,						) ->		Any:

      '''simple docstring'''


      assert np.shape(__lowercase )[0] == np.shape(__lowercase )[1]
      assert np.shape(__lowercase )[0] == np.shape(__lowercase )[0]
      assert _is_matrix_spd(__lowercase )

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

      # 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(__lowercase     ,						__lowercase )

            # The main algorithm.

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

            # Update variables.
            _A      =			np.copy(__lowercase )
            _A      =			np.copy(__lowercase )
            _A      =			np.copy(__lowercase )

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

      return x

def       __lowercase		(    ) ->		None:

      '''simple docstring'''


      _A      =			3
      _A      =			_create_spd_matrix(__lowercase )
      _A      =			np.random.randn(__lowercase     ,						1 )
      _A      =			np.dot(__lowercase     ,						__lowercase )

      # Numpy solution.
      _A      =			np.linalg.solve(__lowercase     ,						__lowercase )

      # Our implementation.
      _A      =			conjugate_gradient(__lowercase     ,						__lowercase )

      # 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()





