





from typing import Any

import numpy as np


def        A__						(							SCREAMING_SNAKE_CASE_		) ->  Dict:
		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 :			List[Any]    		=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        A__						(							SCREAMING_SNAKE_CASE_		) ->  int:
		lowerCamelCase :			int    		=np.random.randn(lowerCamelCase_				,							lowerCamelCase_		)
		lowerCamelCase :			Union[str, Any]    		=np.dot(lowerCamelCase_				,							random_matrix.T		)
		assert _is_matrix_spd(lowerCamelCase_		)
		return spd_matrix


def        A__						(							SCREAMING_SNAKE_CASE_				,							SCREAMING_SNAKE_CASE_				,							SCREAMING_SNAKE_CASE_ = 1_0_0_0				,							SCREAMING_SNAKE_CASE_ = 1E-8				,							) ->  int:
		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 :			Union[str, Any]    		=np.zeros((np.shape(lowerCamelCase_		)[0], 1)		)
		lowerCamelCase :			Union[str, Any]    		=np.copy(lowerCamelCase_		)
		lowerCamelCase :			str    		=np.copy(lowerCamelCase_		)

		# Set initial errors in solution guess and residual.
		lowerCamelCase :			Tuple    		=1E9
		lowerCamelCase :			int    		=1E9
		lowerCamelCase :			Union[str, Any]    		=1E9

		# Set iteration counter to threshold number of iterations.
		lowerCamelCase :			Union[str, Any]    		=0

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

				# The main algorithm.

				# Update search direction magnitude.
				lowerCamelCase :			Any    		=np.dot(ra.T				,							lowerCamelCase_		) / np.dot(pa.T				,							lowerCamelCase_		)
				# Update solution guess.
				lowerCamelCase :			Dict    		=xa + alpha * pa
				# Calculate new residual.
				lowerCamelCase :			Dict    		=ra - alpha * w
				# Calculate new Krylov subspace scale.
				lowerCamelCase :			str    		=np.dot(r.T				,							lowerCamelCase_		) / np.dot(ra.T				,							lowerCamelCase_		)
				# Calculate new A conjuage search direction.
				lowerCamelCase :			Tuple    		=r + beta * pa

				# Calculate errors.
				lowerCamelCase :			Dict    		=np.linalg.norm(r - ra		)
				lowerCamelCase :			str    		=np.linalg.norm(x - xa		)
				lowerCamelCase :			Union[str, Any]    		=np.maximum(lowerCamelCase_				,							lowerCamelCase_		)

				# Update variables.
				lowerCamelCase :			Optional[int]    		=np.copy(lowerCamelCase_		)
				lowerCamelCase :			List[str]    		=np.copy(lowerCamelCase_		)
				lowerCamelCase :			Any    		=np.copy(lowerCamelCase_		)

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

		return x


def        A__						(							) ->  Any:
		lowerCamelCase :			Union[str, Any]    		=3
		lowerCamelCase :			List[str]    		=_create_spd_matrix(lowerCamelCase_		)
		lowerCamelCase :			List[Any]    		=np.random.randn(lowerCamelCase_				,							1		)
		lowerCamelCase :			int    		=np.dot(lowerCamelCase_				,							lowerCamelCase_		)

		# Numpy solution.
		lowerCamelCase :			Dict    		=np.linalg.solve(lowerCamelCase_				,							lowerCamelCase_		)

		# Our implementation.
		lowerCamelCase :			Union[str, Any]    		=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()


'''simple docstring'''


from itertools import combinations, product


def    _lowerCamelCase       ( lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int      ,	lowerCamelCase_   :				int  ):



     """simple docstring"""





     if xa == ya == 0 or xa == ya == 0:
          return False
     UpperCAmelCase_				:  Union[str, Any]  = xa * xa + ya * ya
     UpperCAmelCase_				:  str  = xa * xa + ya * ya
     UpperCAmelCase_				:  str  = (xa - xa) * (xa - xa) + (ya - ya) * (ya - ya)
     return (
         a_square + b_square == c_square
         or a_square + c_square == b_square
         or b_square + c_square == a_square
     )


def    _lowerCamelCase       ( lowerCamelCase_   :				int = 50  ):



     """simple docstring"""





     return sum(
         1
         for pta, pta in combinations(product(range(limit + 1  )      ,	repeat=2  )      ,	2  )
         if is_right(*lowerCamelCase_      ,	*lowerCamelCase_  )  )


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






