from typing import Any

import numpy as np

def        SCREAMING_SNAKE_CASE_					(						snake_case__							)					->     bool:
							assert np.shape(snake_case__							)[0] == np.shape(snake_case__							)[1]

							# If matrix not symmetric, exit right away.
							if np.allclose(snake_case__       ,  matrix.T							) is False:
														return False

							# Get eigenvalues and eignevectors for a symmetric matrix.
							lowerCAmelCase		,				lowerCAmelCase     		=		np.linalg.eigh(snake_case__							)

							# Check sign of all eigenvalues.
							# np.all returns a value of type np.bool_
							return bool(np.all(eigen_values > 0							)							)

def        SCREAMING_SNAKE_CASE_					(						snake_case__							)					->     Any:
							lowerCAmelCase     		=		np.random.randn(snake_case__       ,  snake_case__							)
							lowerCAmelCase     		=		np.dot(snake_case__       ,  random_matrix.T							)
							assert _is_matrix_spd(snake_case__							)
							return spd_matrix

def        SCREAMING_SNAKE_CASE_					(						snake_case__       ,  snake_case__       ,  snake_case__ = 1_0_0_0       ,  snake_case__ = 1E-8       ,  )					->     Any:
							assert np.shape(snake_case__							)[0] == np.shape(snake_case__							)[1]
							assert np.shape(snake_case__							)[0] == np.shape(snake_case__							)[0]
							assert _is_matrix_spd(snake_case__							)

							# Initialize solution guess, residual, search direction.
							lowerCAmelCase     		=		np.zeros((np.shape(snake_case__							)[0], 1)							)
							lowerCAmelCase     		=		np.copy(snake_case__							)
							lowerCAmelCase     		=		np.copy(snake_case__							)

							# Set initial errors in solution guess and residual.
							lowerCAmelCase     		=		1E9
							lowerCAmelCase     		=		1E9
							lowerCAmelCase     		=		1E9

							# Set iteration counter to threshold number of iterations.
							lowerCAmelCase     		=		0

							while error > tol:
														# Save this value so we only calculate the matrix-vector product once.
														lowerCAmelCase     		=		np.dot(snake_case__       ,  snake_case__							)

														# The main algorithm.

														# Update search direction magnitude.
														lowerCAmelCase     		=		np.dot(ra.T       ,  snake_case__							) / np.dot(pa.T       ,  snake_case__							)
														# Update solution guess.
														lowerCAmelCase     		=		xa + alpha * pa
														# Calculate new residual.
														lowerCAmelCase     		=		ra - alpha * w
														# Calculate new Krylov subspace scale.
														lowerCAmelCase     		=		np.dot(r.T       ,  snake_case__							) / np.dot(ra.T       ,  snake_case__							)
														# Calculate new A conjuage search direction.
														lowerCAmelCase     		=		r + beta * pa

														# Calculate errors.
														lowerCAmelCase     		=		np.linalg.norm(r - ra							)
														lowerCAmelCase     		=		np.linalg.norm(x - xa							)
														lowerCAmelCase     		=		np.maximum(snake_case__       ,  snake_case__							)

														# Update variables.
														lowerCAmelCase     		=		np.copy(snake_case__							)
														lowerCAmelCase     		=		np.copy(snake_case__							)
														lowerCAmelCase     		=		np.copy(snake_case__							)

														# Update number of iterations.
														iterations += 1
														if iterations > max_iterations:
																					break

							return x

def        SCREAMING_SNAKE_CASE_					(						)					->     None:
							lowerCAmelCase     		=		3
							lowerCAmelCase     		=		_create_spd_matrix(snake_case__							)
							lowerCAmelCase     		=		np.random.randn(snake_case__       ,  1							)
							lowerCAmelCase     		=		np.dot(snake_case__       ,  snake_case__							)

							# Numpy solution.
							lowerCAmelCase     		=		np.linalg.solve(snake_case__       ,  snake_case__							)

							# Our implementation.
							lowerCAmelCase     		=		conjugate_gradient(snake_case__       ,  snake_case__							)

							# Ensure both solutions are close to x_true (and therefore one another).
							assert np.linalg.norm(x_numpy - x_true							) <= 1E-6
							assert np.linalg.norm(x_conjugate_gradient - x_true							) <= 1E-6


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	test_conjugate_gradient()




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)}')



