import numpy as np






def        __UpperCAmelCase    (       UpperCAmelCase,		UpperCAmelCase,		UpperCAmelCase = 1e-12,		UpperCAmelCase = 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       )
			lowercase							=		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.

			lowercase							=		False
			lowercase							=		0
			lowercase							=		0
			lowercase							=		1e12

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

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

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

						lowercase							=		lambda_

			if is_complex:
						lowercase							=		np.real(lambda_       )

			return lambda_, vector






def        __UpperCAmelCase    (       )->       None:



			"""simple docstring"""







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

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

						# Our implementation.
						lowercase		,lowercase							=		power_iteration(UpperCAmelCase,		UpperCAmelCase       )

						# Numpy implementation.

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






from __future__ import annotations

import math






def        __UpperCAmelCase    (       UpperCAmelCase       )->       list[int]:



			"""simple docstring"""







			if num <= 0:
						lowercase							=		f'{num}: Invalid input, please enter a positive integer.'
						raise ValueError(UpperCAmelCase       )

			lowercase							=		[True] * (num + 1)
			lowercase							=		[]
			lowercase							=		2
			lowercase							=		int(math.sqrt(UpperCAmelCase       )       )

			while start <= end:
						# If start is a prime
						if sieve[start] is True:
									prime.append(UpperCAmelCase       )

									# Set multiples of start be False
									for i in range(start * start,		num + 1,		UpperCAmelCase       ):
												if sieve[i] is True:
															lowercase							=		False

						start += 1

			for j in range(end + 1,		num + 1       ):
						if sieve[j] is True:
									prime.append(UpperCAmelCase       )

			return prime


if __name__ == "__main__":
						print(prime_sieve(int(input("Enter a positive integer: ").strip())))





