
from typing import Any

import numpy as np






def        lowerCamelCase__    (    __lowerCAmelCase  :  np.ndarray    ):




			"""simple docstring"""


			assert np.shape(__lowerCAmelCase    )[0] == np.shape(__lowerCAmelCase    )[1]

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

			# Get eigenvalues and eignevectors for a symmetric matrix.
			lowerCAmelCase_      =     np.linalg.eigh(__lowerCAmelCase    )

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






def        lowerCamelCase__    (    __lowerCAmelCase  :  int    ):




			"""simple docstring"""


			lowerCAmelCase_      =     np.random.randn(__lowerCAmelCase   ,							__lowerCAmelCase    )
			lowerCAmelCase_      =     np.dot(__lowerCAmelCase   ,							random_matrix.T    )
			assert _is_matrix_spd(__lowerCAmelCase    )
			return spd_matrix






def        lowerCamelCase__    (    __lowerCAmelCase  :  np.ndarray   ,							__lowerCAmelCase  :  np.ndarray   ,							__lowerCAmelCase  :  int = 1000   ,							__lowerCAmelCase  :  float = 1e-8   ,							):




			"""simple docstring"""


			assert np.shape(__lowerCAmelCase    )[0] == np.shape(__lowerCAmelCase    )[1]
			assert np.shape(__lowerCAmelCase    )[0] == np.shape(__lowerCAmelCase    )[0]
			assert _is_matrix_spd(__lowerCAmelCase    )

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

			# 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(__lowerCAmelCase   ,							__lowerCAmelCase    )

						# The main algorithm.

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

						# Update variables.
						lowerCAmelCase_      =     np.copy(__lowerCAmelCase    )
						lowerCAmelCase_      =     np.copy(__lowerCAmelCase    )
						lowerCAmelCase_      =     np.copy(__lowerCAmelCase    )

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

			return x






def        lowerCamelCase__    (    ):




			"""simple docstring"""


			lowerCAmelCase_      =     3
			lowerCAmelCase_      =     _create_spd_matrix(__lowerCAmelCase    )
			lowerCAmelCase_      =     np.random.randn(__lowerCAmelCase   ,							1    )
			lowerCAmelCase_      =     np.dot(__lowerCAmelCase   ,							__lowerCAmelCase    )

			# Numpy solution.
			lowerCAmelCase_      =     np.linalg.solve(__lowerCAmelCase   ,							__lowerCAmelCase    )

			# Our implementation.
			lowerCAmelCase_      =     conjugate_gradient(__lowerCAmelCase   ,							__lowerCAmelCase    )

			# 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 math






def        lowerCamelCase__    (    __lowerCAmelCase  :  int    ):




			"""simple docstring"""


			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or number % 2 == 0 or number % 3 == 0:
						# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
						return False

			# All primes number are in format of 6k +/- 1
			for i in range(5   ,							int(math.sqrt(__lowerCAmelCase    ) + 1    )   ,							6    ):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True






def        lowerCamelCase__    (    __lowerCAmelCase  :  int = 600851475143    ):




			"""simple docstring"""


			try:
						lowerCAmelCase_      =     int(__lowerCAmelCase    )
			except (TypeError, ValueError):
						raise TypeError("Parameter n must be int or castable to int."    )
			if n <= 0:
						raise ValueError("Parameter n must be greater than or equal to one."    )
			lowerCAmelCase_      =     0
			if is_prime(__lowerCAmelCase    ):
						return n
			while n % 2 == 0:
						n //= 2
			if is_prime(__lowerCAmelCase    ):
						return n
			for i in range(3   ,							int(math.sqrt(__lowerCAmelCase    )    ) + 1   ,							2    ):
						if n % i == 0:
									if is_prime(n // i    ):
												lowerCAmelCase_      =     n // i
												break
									elif is_prime(__lowerCAmelCase    ):
												lowerCAmelCase_      =     i
			return max_number


if __name__ == "__main__":
			print(f"""{solution() = }""")



