
"""simple docstring"""


from typing import Any

import numpy as np



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

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

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

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



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



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

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

	# 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(__UpperCamelCase					,				__UpperCamelCase )

		# The main algorithm.

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

		# Update variables.
		_lowerCAmelCase			=np.copy(__UpperCamelCase )
		_lowerCAmelCase			=np.copy(__UpperCamelCase )
		_lowerCAmelCase			=np.copy(__UpperCamelCase )

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

	return x



def 					_lowerCamelCase()       ->				None:
	_lowerCAmelCase			=3
	_lowerCAmelCase			=_create_spd_matrix(__UpperCamelCase )
	_lowerCAmelCase			=np.random.randn(__UpperCamelCase					,				1 )
	_lowerCAmelCase			=np.dot(__UpperCamelCase					,				__UpperCamelCase )

	# Numpy solution.
	_lowerCAmelCase			=np.linalg.solve(__UpperCamelCase					,				__UpperCamelCase )

	# Our implementation.
	_lowerCAmelCase			=conjugate_gradient(__UpperCamelCase					,				__UpperCamelCase )

	# 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"""


def 					_lowerCamelCase(__UpperCamelCase					,				__UpperCamelCase					,				__UpperCamelCase )       ->				float:
	if mass < 0:
		raise ValueError("""The mass of the body cannot be negative""" )
	if radius <= 0:
		raise ValueError("""The radius is always a positive non zero integer""" )
	return (mass * (velocity) ** 2) / radius


if __name__ == "__main__":
			import doctest

			doctest.testmod(verbose=True)



