


'''simple docstring'''






from typing import Any

import numpy as np


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

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

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

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


def 					_snake_case   (     A       )						->    Any:
							lowerCAmelCase__          =	np.random.randn(A ,    A       )
							lowerCAmelCase__          =	np.dot(A ,    random_matrix.T       )
							assert _is_matrix_spd(A       )
							return spd_matrix


def 					_snake_case   (     A ,    A ,    A = 1000 ,    A = 1E-8 ,    )						->    Any:
							assert np.shape(A       )[0] == np.shape(A       )[1]
							assert np.shape(A       )[0] == np.shape(A       )[0]
							assert _is_matrix_spd(A       )

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

							# 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(A ,    A       )

														# The main algorithm.

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

														# Update variables.
														lowerCAmelCase__          =	np.copy(A       )
														lowerCAmelCase__          =	np.copy(A       )
														lowerCAmelCase__          =	np.copy(A       )

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

							return x


def 					_snake_case   (     )						->    None:
							lowerCAmelCase__          =	3
							lowerCAmelCase__          =	_create_spd_matrix(A       )
							lowerCAmelCase__          =	np.random.randn(A ,    1       )
							lowerCAmelCase__          =	np.dot(A ,    A       )

							# Numpy solution.
							lowerCAmelCase__          =	np.linalg.solve(A ,    A       )

							# Our implementation.
							lowerCAmelCase__          =	conjugate_gradient(A ,    A       )

							# 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()



'''simple docstring'''






from .hash_table import HashTable
from .number_theory.prime_numbers import is_prime, next_prime




class 		a__				(							a__  ):





							'''simple docstring'''






							def __init__(		self					,    *lowerCamelCase_					,    **lowerCamelCase_   )       ->							Optional[int]:
														super().__init__(*lowerCamelCase_					,    **lowerCamelCase_   )







							def   __SCREAMING_SNAKE_CASE							(		self					,    lowerCamelCase_					,    lowerCamelCase_   )       ->							Optional[int]:
														lowerCAmelCase__          =	(
														    next_prime(value % self.size_table   )
														    if not is_prime(value % self.size_table   )
														    else value % self.size_table
														)  # gt = bigger than
														return next_prime_gt - (data % next_prime_gt)







							def   __SCREAMING_SNAKE_CASE							(		self					,    lowerCamelCase_					,    lowerCamelCase_					,    lowerCamelCase_   )       ->							Optional[int]:
														return (increment * self.__hash_function_a(lowerCamelCase_					,    lowerCamelCase_   )) % self.size_table







							def   __SCREAMING_SNAKE_CASE							(		self					,    lowerCamelCase_					,    lowerCamelCase_=None   )       ->							Any:
														lowerCAmelCase__          =	1
														lowerCAmelCase__          =	self.hash_function(lowerCamelCase_   )

														while self.values[new_key] is not None and self.values[new_key] != key:
																					lowerCAmelCase__          =	(
																					    self.__hash_double_function(lowerCamelCase_					,    lowerCamelCase_					,    lowerCamelCase_   )
																					    if self.balanced_factor() >= self.lim_charge
																					    else None
																					)
																					if new_key is None:
																												break
																					else:
																												i += 1

														return new_key