






"""simple docstring"""



import math


def 	SCREAMING_SNAKE_CASE__       (  snake_case       : int				)->		bool:






     '''simple docstring'''



     if 1 < number < 4:
          # 2 and 3 are primes
          return True
     elif number < 2 or number % 2 == 0 or number % 3 == 0:
          # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
          return False

     # All primes number are in format of 6k +/- 1
     for i in range(5			,					int(math.sqrt(snake_case				) + 1				)			,					6				):
          if number % i == 0 or number % (i + 2) == 0:
               return False
     return True


def 	SCREAMING_SNAKE_CASE__       (  snake_case       : int = 1_0001				)->		int:






     '''simple docstring'''



     try:
          UpperCAmelCase__						:    Tuple							=							int(snake_case				)
     except (TypeError, ValueError):
          raise TypeError("Parameter nth must be int or castable to int."				) from None
     if nth <= 0:
          raise ValueError("Parameter nth must be greater than or equal to one."				)
     UpperCAmelCase__						:    list[int]							=							[]
     UpperCAmelCase__						:    Dict							=							2
     while len(snake_case				) < nth:
          if is_prime(snake_case				):
               primes.append(snake_case				)
               num += 1
          else:
               num += 1
     return primes[len(snake_case				) - 1]


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










"""simple docstring"""



import numpy as np


def 	SCREAMING_SNAKE_CASE__       (  snake_case       : np.ndarray			,					snake_case       : np.ndarray			,					snake_case       : float = 1E-1_2			,					snake_case       : int = 100			,					)->		tuple[float, np.ndarray]:






     '''simple docstring'''



     assert np.shape(snake_case				)[0] == np.shape(snake_case				)[1]
     # Ensure proper dimensionality.
     assert np.shape(snake_case				)[0] == np.shape(snake_case				)[0]
     # Ensure inputs are either both complex or both real
     assert np.iscomplexobj(snake_case				) == np.iscomplexobj(snake_case				)
     UpperCAmelCase__						:    str							=							np.iscomplexobj(snake_case				)
     if is_complex:
          # Ensure complex input_matrix is Hermitian
          assert np.array_equal(snake_case			,					input_matrix.conj().T				)

     # Set convergence to False. Will define convergence when we exceed max_iterations
     # or when we have small changes from one iteration to next.

     UpperCAmelCase__						:    Union[str, Any]							=							False
     UpperCAmelCase__						:    List[Any]							=							0
     UpperCAmelCase__						:    Any							=							0
     UpperCAmelCase__						:    List[str]							=							1E1_2

     while not convergence:
          # Multiple matrix by the vector.
          UpperCAmelCase__						:    Optional[Any]							=							np.dot(snake_case			,					snake_case				)
          # Normalize the resulting output vector.
          UpperCAmelCase__						:    Tuple							=							w / np.linalg.norm(snake_case				)
          # Find rayleigh quotient
          # (faster than usual b/c we know vector is normalized already)
          UpperCAmelCase__						:    Any							=							vector.conj().T if is_complex else vector.T
          UpperCAmelCase__						:    List[str]							=							np.dot(snake_case			,					np.dot(snake_case			,					snake_case				)				)

          # Check convergence.
          UpperCAmelCase__						:    Optional[Any]							=							np.abs(lambda_ - lambda_previous				) / lambda_
          iterations += 1

          if error <= error_tol or iterations >= max_iterations:
               UpperCAmelCase__						:    Tuple							=							True

          UpperCAmelCase__						:    Tuple							=							lambda_

     if is_complex:
          UpperCAmelCase__						:    str							=							np.real(lambda_				)

     return lambda_, vector


def 	SCREAMING_SNAKE_CASE__       (  )->		None:






     '''simple docstring'''



     UpperCAmelCase__						:    List[Any]							=							np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]				)
     UpperCAmelCase__						:    Dict							=							np.array([41, 4, 20]				)
     UpperCAmelCase__						:    Union[str, Any]							=							real_input_matrix.astype(np.complexaaa				)
     UpperCAmelCase__						:    Optional[int]							=							np.triu(1J * complex_input_matrix			,					1				)
     complex_input_matrix += imag_matrix
     complex_input_matrix += -1 * imag_matrix.T
     UpperCAmelCase__						:    int							=							np.array([41, 4, 20]				).astype(np.complexaaa				)

     for problem_type in ["real", "complex"]:
          if problem_type == "real":
               UpperCAmelCase__						:    List[str]							=							real_input_matrix
               UpperCAmelCase__						:    Any							=							real_vector
          elif problem_type == "complex":
               UpperCAmelCase__						:    str							=							complex_input_matrix
               UpperCAmelCase__						:    List[Any]							=							complex_vector

          # Our implementation.
          UpperCAmelCase__							,				UpperCAmelCase__						:    Optional[int]							=							power_iteration(snake_case			,					snake_case				)

          # Numpy implementation.

          # Get eigenvalues and eigenvectors using built-in numpy
          # eigh (eigh used for symmetric or hermetian matrices).
          UpperCAmelCase__							,				UpperCAmelCase__						:    List[Any]							=							np.linalg.eigh(snake_case				)
          # Last eigenvalue is the maximum one.
          UpperCAmelCase__						:    str							=							eigen_values[-1]
          # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
          UpperCAmelCase__						:    Optional[int]							=							eigen_vectors[:, -1]

          # Check our implementation and numpy gives close answers.
          assert np.abs(eigen_value - eigen_value_max				) <= 1E-6
          # Take absolute values element wise of each eigenvector.
          # as they are only unique to a minus sign.
          assert np.linalg.norm(np.abs(snake_case				) - np.abs(snake_case				)				) <= 1E-6


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 test_power_iteration()


