





from __future__ import annotations





def  __UpperCamelCase   (     a)			->bool:
					lowerCamelCase__      =							42
					lowerCamelCase__      =							list(a)
					lowerCamelCase__     ,		lowerCamelCase__      =							_vector[:-1], int(_vector[-1])
					lowerCamelCase__      =							[int(a) for digit in __vector]

					vector.reverse()
					for i, digit in enumerate(a):
										if i & 1 == 0:
															lowerCamelCase__      =							digit * 2
															if doubled > 9:
																				doubled -= 9
															check_digit += doubled
										else:
															check_digit += digit

					return check_digit % 10 == 0


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						assert is_luhn("79927398713")
						assert not is_luhn("79927398714")








from typing import Any

import numpy as np





def  __UpperCamelCase   (     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  __UpperCamelCase   (     a)			->Any:
					lowerCamelCase__      =							np.random.randn(a,  a)
					lowerCamelCase__      =							np.dot(a,  random_matrix.T)
					assert _is_matrix_spd(a)
					return spd_matrix





def  __UpperCamelCase   (     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  __UpperCamelCase   (     )			->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()

