import numpy as np

def        SCREAMING_SNAKE_CASE_					(						snake_case__							)					->     str:
							lowerCAmelCase		,				lowerCAmelCase     		=		a.shape
							lowerCAmelCase     		=		min(snake_case__       ,  snake_case__							)
							lowerCAmelCase     		=		np.eye(snake_case__							)
							lowerCAmelCase     		=		a.copy()

							for k in range(t - 1							):
														# select a column of modified matrix A':
														lowerCAmelCase     		=		r[k:, [k]]
														# construct first basis vector
														lowerCAmelCase     		=		np.zeros_like(snake_case__							)
														lowerCAmelCase     		=		1.0
														# determine scaling factor
														lowerCAmelCase     		=		np.linalg.norm(snake_case__							)
														# construct vector v for Householder reflection
														lowerCAmelCase     		=		x + np.sign(x[0]							) * alpha * ea
														v /= np.linalg.norm(snake_case__							)

														# construct the Householder matrix
														lowerCAmelCase     		=		np.eye(m - k							) - 2.0 * v @ v.T
														# pad with ones and zeros as necessary
														lowerCAmelCase     		=		np.block([[np.eye(snake_case__							), np.zeros((k, m - k)							)], [np.zeros((m - k, k)							), q_k]]							)

														lowerCAmelCase     		=		q @ q_k.T
														lowerCAmelCase     		=		q_k @ r

							return q, r


if __name__ == "__main__":
	import doctest

	doctest.testmod()




import random




class        lowercase_  :



	"""simple docstring"""





	def __init__(							self		,      __SCREAMING_SNAKE_CASE		,      __SCREAMING_SNAKE_CASE    )						->None:
								lowerCAmelCase     		=		x
								lowerCAmelCase     		=		y


	def       SCREAMING_SNAKE_CASE_    (							self    )						->bool:
								return (self.x**2 + self.y**2) <= 1




	@classmethod
	def       SCREAMING_SNAKE_CASE_    (							cls    )						->Tuple:
								return cls(x=random.random()		,      y=random.random()    )

def        SCREAMING_SNAKE_CASE_					(						snake_case__							)					->     float:
							if number_of_simulations < 1:
														raise ValueError('''At least one simulation is necessary to estimate PI.'''							)

							lowerCAmelCase     		=		0
							for _ in range(snake_case__							):
														lowerCAmelCase     		=		Point.random_unit_square()

														if random_point.is_in_unit_circle():
																					number_in_unit_circle += 1

							return 4 * number_in_unit_circle / number_of_simulations


if __name__ == "__main__":
	# import doctest

	# doctest.testmod()
	from math import pi

	lowercase__      :    int          =     '''Please enter the desired number of Monte Carlo simulations: '''
	lowercase__      :    Dict          =     estimate_pi(int(input(prompt).strip()))
	print(f'An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}')



