
'''simple docstring'''







from __future__ import annotations

import math







def        SCREAMING_SNAKE_CASE__			(     __A			) ->			list[int]:

	if num <= 0:
		_snake_case     		=	F'{num}: Invalid input, please enter a positive integer.'
		raise ValueError(__A			)

	_snake_case     		=	[True] * (num + 1)
	_snake_case     		=	[]
	_snake_case     		=	2
	_snake_case     		=	int(math.sqrt(__A			)			)

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

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

		start += 1

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

	return prime


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



'''simple docstring'''







from typing import Any

import numpy as np







def        SCREAMING_SNAKE_CASE__			(     __A			) ->			bool:
	return np.array_equal(__A    ,       matrix.conjugate().T			)







def        SCREAMING_SNAKE_CASE__			(     __A    ,       __A			) ->			Any:
	_snake_case     		=	v.conjugate().T
	_snake_case     		=	v_star.dot(__A			)
	assert isinstance(__A    ,       np.ndarray			)
	return (v_star_dot.dot(__A			)) / (v_star.dot(__A			))







def        SCREAMING_SNAKE_CASE__			(     ) ->			None:
	_snake_case     		=	np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]			)
	_snake_case     		=	np.array([[1], [2], [3]]			)
	assert is_hermitian(__A			), F'{a} is not hermitian.'
	print(rayleigh_quotient(__A    ,       __A			)			)

	_snake_case     		=	np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]			)
	assert is_hermitian(__A			), F'{a} is not hermitian.'
	assert rayleigh_quotient(__A    ,       __A			) == float(3			)


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					tests()

