


from typing import Any

import numpy as np

def  lowerCAmelCase__				(    a__:	np.ndarray )						->	bool:


							'''simple docstring'''





							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.
							_UpperCAmelCase	, _UpperCAmelCase					=    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  lowerCAmelCase__				(    a__:	int )						->	Any:


							'''simple docstring'''





							_UpperCAmelCase					=    np.random.randn(a__    ,						a__ )
							_UpperCAmelCase					=    np.dot(a__    ,						random_matrix.T )
							assert _is_matrix_spd(a__ )
							return spd_matrix

def  lowerCAmelCase__				(    a__:	np.ndarray    ,						a__:	np.ndarray    ,						a__:	int = 1_0_0_0    ,						a__:	float = 1e-8    ,						)						->	Any:


							'''simple docstring'''





							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.
							_UpperCAmelCase					=    np.zeros((np.shape(a__ )[0], 1) )
							_UpperCAmelCase					=    np.copy(a__ )
							_UpperCAmelCase					=    np.copy(a__ )

							# Set initial errors in solution guess and residual.
							_UpperCAmelCase					=    1e9
							_UpperCAmelCase					=    1e9
							_UpperCAmelCase					=    1e9

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

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

														# The main algorithm.

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

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

														# Update variables.
														_UpperCAmelCase					=    np.copy(a__ )
														_UpperCAmelCase					=    np.copy(a__ )
														_UpperCAmelCase					=    np.copy(a__ )

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

							return x

def  lowerCAmelCase__				(    )						->	None:


							'''simple docstring'''





							_UpperCAmelCase					=    3
							_UpperCAmelCase					=    _create_spd_matrix(a__ )
							_UpperCAmelCase					=    np.random.randn(a__    ,						1 )
							_UpperCAmelCase					=    np.dot(a__    ,						a__ )

							# Numpy solution.
							_UpperCAmelCase					=    np.linalg.solve(a__    ,						a__ )

							# Our implementation.
							_UpperCAmelCase					=    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()




from __future__ import annotations

def  lowerCAmelCase__				(    a__:	list[int]    ,						a__:	int )						->	list[int]:


							'''simple docstring'''





							_UpperCAmelCase					=    {}
							for index, val in enumerate(a__ ):
														_UpperCAmelCase					=    target - val
														if compl in chk_map:
																					return [chk_map[compl], index]
														_UpperCAmelCase					=    index
							return []


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		print(f'''{two_sum([2, 7, 1_1, 1_5], 9) = }''')
