




import numpy as np







def      lowerCamelCase_						(			UpperCamelCase__							: np.ndarray					,	UpperCamelCase__							: np.ndarray					,	UpperCamelCase__							: float = 1E-12					,	UpperCamelCase__							: int = 100					,	)	->			tuple[float, np.ndarray]:





				"""simple docstring"""


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

				__lowerCamelCase									=       False
				__lowerCamelCase									=       0
				__lowerCamelCase									=       0
				__lowerCamelCase									=       1E12

				while not convergence:
								# Multiple matrix by the vector.
								__lowerCamelCase									=       np.dot(UpperCamelCase__					,	UpperCamelCase__ )
								# Normalize the resulting output vector.
								__lowerCamelCase									=       w / np.linalg.norm(UpperCamelCase__ )
								# Find rayleigh quotient
								# (faster than usual b/c we know vector is normalized already)
								__lowerCamelCase									=       vector.conj().T if is_complex else vector.T
								__lowerCamelCase									=       np.dot(UpperCamelCase__					,	np.dot(UpperCamelCase__					,	UpperCamelCase__ ) )

								# Check convergence.
								__lowerCamelCase									=       np.abs(lambda_ - lambda_previous ) / lambda_
								iterations += 1

								if error <= error_tol or iterations >= max_iterations:
												__lowerCamelCase									=       True

								__lowerCamelCase									=       lambda_

				if is_complex:
								__lowerCamelCase									=       np.real(lambda_ )

				return lambda_, vector







def      lowerCamelCase_						(			)	->			None:





				"""simple docstring"""


				__lowerCamelCase									=       np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]] )
				__lowerCamelCase									=       np.array([41, 4, 20] )
				__lowerCamelCase									=       real_input_matrix.astype(np.complexaaa )
				__lowerCamelCase									=       np.triu(1j * complex_input_matrix					,	1 )
				complex_input_matrix += imag_matrix
				complex_input_matrix += -1 * imag_matrix.T
				__lowerCamelCase									=       np.array([41, 4, 20] ).astype(np.complexaaa )

				for problem_type in ["real", "complex"]:
								if problem_type == "real":
												__lowerCamelCase									=       real_input_matrix
												__lowerCamelCase									=       real_vector
								elif problem_type == "complex":
												__lowerCamelCase									=       complex_input_matrix
												__lowerCamelCase									=       complex_vector

								# Our implementation.
								__lowerCamelCase			,		__lowerCamelCase									=       power_iteration(UpperCamelCase__					,	UpperCamelCase__ )

								# Numpy implementation.

								# Get eigenvalues and eigenvectors using built-in numpy
								# eigh (eigh used for symmetric or hermetian matrices).
								__lowerCamelCase			,		__lowerCamelCase									=       np.linalg.eigh(UpperCamelCase__ )
								# Last eigenvalue is the maximum one.
								__lowerCamelCase									=       eigen_values[-1]
								# Last column in this matrix is eigenvector corresponding to largest eigenvalue.
								__lowerCamelCase									=       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(UpperCamelCase__ ) - np.abs(UpperCamelCase__ ) ) <= 1E-6


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					test_power_iteration()








def      lowerCamelCase_						(			UpperCamelCase__							: str					,	UpperCamelCase__							: int )	->			list:





				"""simple docstring"""


				__lowerCamelCase									=       word.split()

				def justify(UpperCamelCase__							: list					,	UpperCamelCase__							: int					,	UpperCamelCase__							: int ) -> str:
								__lowerCamelCase									=       max_width - width
								__lowerCamelCase									=       len(UpperCamelCase__ )
								if len(UpperCamelCase__ ) == 1:
												# if there is only word in line
												# just insert overall_spaces_count for the remainder of line
												return line[0] + " " * overall_spaces_count
								else:
												__lowerCamelCase									=       words_count - 1
												# num_spaces_between_words_list[i] : tells you to insert
												# num_spaces_between_words_list[i] spaces
												# after word on line[i]
												__lowerCamelCase									=       spaces_to_insert_between_words * [
												    overall_spaces_count // spaces_to_insert_between_words
												]
												__lowerCamelCase									=       (
												    overall_spaces_count % spaces_to_insert_between_words
												)
												# distribute spaces via round robin to the left words
												for i in range(UpperCamelCase__ ):
																num_spaces_between_words_list[i] += 1
												__lowerCamelCase									=       []
												for i in range(UpperCamelCase__ ):
																# add the word
																aligned_words_list.append(line[i] )
																# add the spaces to insert
																aligned_words_list.append(num_spaces_between_words_list[i] * ' ' )
												# just add the last word to the sentence
												aligned_words_list.append(line[-1] )
												# join the aligned words list to form a justified line
												return "".join(UpperCamelCase__ )

				__lowerCamelCase									=       []
				__lowerCamelCase									=       []
				__lowerCamelCase									=       0
				for word in words:
								if width + len(UpperCamelCase__ ) + len(UpperCamelCase__ ) <= max_width:
												# keep adding words until we can fill out max_width
												# width = sum of length of all words (without overall_spaces_count)
												# len(word) = length of current word
												# len(line) = number of overall_spaces_count to insert between words
												line.append(UpperCamelCase__ )
												width += len(UpperCamelCase__ )
								else:
												# justify the line and add it to result
												answer.append(justify(UpperCamelCase__					,	UpperCamelCase__					,	UpperCamelCase__ ) )
												# reset new line and new width
												__lowerCamelCase			,		__lowerCamelCase									=       [word], len(UpperCamelCase__ )
				__lowerCamelCase									=       max_width - width - len(UpperCamelCase__ )
				answer.append(' '.join(UpperCamelCase__ ) + (remaining_spaces + 1) * ' ' )
				return answer


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

					testmod()


