




"""simple docstring"""





def        _UpperCamelCase     (      UpperCamelCase					,							UpperCamelCase				)      ->      str:


			"""simple docstring"""
			__UpperCAmelCase						:			list[list[str]]												=       [[] for _ in range(UpperCamelCase				)]
			__UpperCAmelCase						:			Union[str, Any]												=       key - 1

			if key <= 0:
						raise ValueError("Height of grid can't be 0 or negative"				)
			if key == 1 or len(UpperCamelCase				) <= key:
						return input_string

			for position, character in enumerate(UpperCamelCase				):
						__UpperCAmelCase						:			Dict												=       position % (lowest * 2)  # puts it in bounds
						__UpperCAmelCase						:			List[str]												=       min(UpperCamelCase					,							lowest * 2 - num				)  # creates zigzag pattern
						temp_grid[num].append(UpperCamelCase				)
			__UpperCAmelCase						:			Union[str, Any]												=       ["".join(UpperCamelCase				) for row in temp_grid]
			__UpperCAmelCase						:			Any												=       "".join(UpperCamelCase				)

			return output_string

def        _UpperCamelCase     (      UpperCamelCase					,							UpperCamelCase				)      ->      str:


			"""simple docstring"""
			__UpperCAmelCase						:			Tuple												=       []
			__UpperCAmelCase						:			Union[str, Any]												=       key - 1

			if key <= 0:
						raise ValueError("Height of grid can't be 0 or negative"				)
			if key == 1:
						return input_string

			__UpperCAmelCase						:			list[list[str]]												=       [[] for _ in range(UpperCamelCase				)]  # generates template
			for position in range(len(UpperCamelCase				)				):
						__UpperCAmelCase						:			Optional[int]												=       position % (lowest * 2)  # puts it in bounds
						__UpperCAmelCase						:			str												=       min(UpperCamelCase					,							lowest * 2 - num				)  # creates zigzag pattern
						temp_grid[num].append("*"				)

			__UpperCAmelCase						:			Union[str, Any]												=       0
			for row in temp_grid:  # fills in the characters
						__UpperCAmelCase						:			Tuple												=       input_string[counter : counter + len(UpperCamelCase				)]
						grid.append(list(UpperCamelCase				)				)
						counter += len(UpperCamelCase				)

			__UpperCAmelCase						:			List[str]												=       ""  # reads as zigzag
			for position in range(len(UpperCamelCase				)				):
						__UpperCAmelCase						:			Dict												=       position % (lowest * 2)  # puts it in bounds
						__UpperCAmelCase						:			Union[str, Any]												=       min(UpperCamelCase					,							lowest * 2 - num				)  # creates zigzag pattern
						output_string += grid[num][0]
						grid[num].pop(0				)
			return output_string

def        _UpperCamelCase     (      UpperCamelCase				)      ->      dict[int, str]:


			"""simple docstring"""
			__UpperCAmelCase						:			Tuple												=       {}
			for key_guess in range(1					,							len(UpperCamelCase				)				):  # tries every key
						__UpperCAmelCase						:			str												=       decrypt(UpperCamelCase					,							UpperCamelCase				)
			return results


if __name__ == "__main__":
					import doctest

					doctest.testmod()










"""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()


