



from typing import Any

import numpy as np







def 							A     (						__UpperCAmelCase      )			->			bool:



						'''simple docstring'''






						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.
						UpperCAmelCase_				, UpperCAmelCase_					=	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 							A     (						__UpperCAmelCase      )			->			Any:



						'''simple docstring'''






						UpperCAmelCase_					=	np.random.randn(__UpperCAmelCase				,   __UpperCAmelCase      )
						UpperCAmelCase_					=	np.dot(__UpperCAmelCase				,   random_matrix.T      )
						assert _is_matrix_spd(__UpperCAmelCase      )
						return spd_matrix







def 							A     (						__UpperCAmelCase				,   __UpperCAmelCase				,   __UpperCAmelCase = 1000				,   __UpperCAmelCase = 1e-8				,   )			->			Any:



						'''simple docstring'''






						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.
						UpperCAmelCase_					=	np.zeros((np.shape(__UpperCAmelCase      )[0], 1)      )
						UpperCAmelCase_					=	np.copy(__UpperCAmelCase      )
						UpperCAmelCase_					=	np.copy(__UpperCAmelCase      )

						# 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(__UpperCAmelCase				,   __UpperCAmelCase      )

												# The main algorithm.

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

												# Update variables.
												UpperCAmelCase_					=	np.copy(__UpperCAmelCase      )
												UpperCAmelCase_					=	np.copy(__UpperCAmelCase      )
												UpperCAmelCase_					=	np.copy(__UpperCAmelCase      )

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

						return x







def 							A     (						)			->			None:



						'''simple docstring'''






						UpperCAmelCase_					=	3
						UpperCAmelCase_					=	_create_spd_matrix(__UpperCAmelCase      )
						UpperCAmelCase_					=	np.random.randn(__UpperCAmelCase				,   1      )
						UpperCAmelCase_					=	np.dot(__UpperCAmelCase				,   __UpperCAmelCase      )

						# Numpy solution.
						UpperCAmelCase_					=	np.linalg.solve(__UpperCAmelCase				,   __UpperCAmelCase      )

						# Our implementation.
						UpperCAmelCase_					=	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()










import math







def 							A     (						__UpperCAmelCase      )			->			str:



						'''simple docstring'''






						if number == 0:
												UpperCAmelCase_					=	'''Zero'''
												return words
						else:
												UpperCAmelCase_					=	math.logaa(__UpperCAmelCase      )
												UpperCAmelCase_					=	digits + 1
												UpperCAmelCase_					=	{}
												UpperCAmelCase_					=	''''''
												UpperCAmelCase_					=	'''One'''
												UpperCAmelCase_					=	'''Two'''
												UpperCAmelCase_					=	'''Three'''
												UpperCAmelCase_					=	'''Four'''
												UpperCAmelCase_					=	'''Five'''
												UpperCAmelCase_					=	'''Six'''
												UpperCAmelCase_					=	'''Seven'''
												UpperCAmelCase_					=	'''Eight'''
												UpperCAmelCase_					=	'''Nine'''

												UpperCAmelCase_					=	{}
												UpperCAmelCase_					=	''''''
												UpperCAmelCase_					=	'''Twenty'''
												UpperCAmelCase_					=	'''Thirty'''
												UpperCAmelCase_					=	'''Forty'''
												UpperCAmelCase_					=	'''Fifty'''
												UpperCAmelCase_					=	'''Sixty'''
												UpperCAmelCase_					=	'''Seventy'''
												UpperCAmelCase_					=	'''Eighty'''
												UpperCAmelCase_					=	'''Ninety'''

												UpperCAmelCase_					=	{}
												UpperCAmelCase_					=	'''Ten'''
												UpperCAmelCase_					=	'''Eleven'''
												UpperCAmelCase_					=	'''Twelve'''
												UpperCAmelCase_					=	'''Thirteen'''
												UpperCAmelCase_					=	'''Fourteen'''
												UpperCAmelCase_					=	'''Fifteen'''
												UpperCAmelCase_					=	'''Sixteen'''
												UpperCAmelCase_					=	'''Seventeen'''
												UpperCAmelCase_					=	'''Eighteen'''
												UpperCAmelCase_					=	'''Nineteen'''

												UpperCAmelCase_					=	{}
												UpperCAmelCase_					=	'''Hundred,'''
												UpperCAmelCase_					=	'''Thousand,'''
												UpperCAmelCase_					=	'''Lakh,'''
												UpperCAmelCase_					=	'''Crore,'''

												UpperCAmelCase_					=	number
												UpperCAmelCase_					=	''''''
												UpperCAmelCase_					=	0
												UpperCAmelCase_					=	int(__UpperCAmelCase      )
												while counter < digits:
																		UpperCAmelCase_					=	temp_num % 10
																		if counter % 2 == 0:
																								UpperCAmelCase_					=	''''''
																								if counter in placevalue and current != 0:
																														UpperCAmelCase_					=	placevalue[counter]
																								if counter == 2:
																														UpperCAmelCase_					=	singles[current] + addition + words
																								elif counter == 0:
																														if ((temp_num % 100) // 10) == 1:
																																				UpperCAmelCase_					=	teens[current] + addition + words
																																				UpperCAmelCase_					=	temp_num // 10
																																				counter += 1
																														else:
																																				UpperCAmelCase_					=	singles[current] + addition + words

																								else:
																														UpperCAmelCase_					=	doubles[current] + addition + words

																		else:
																								if counter == 1:
																														if current == 1:
																																				UpperCAmelCase_					=	teens[number % 10] + words
																														else:
																																				UpperCAmelCase_					=	''''''
																																				if counter in placevalue:
																																										UpperCAmelCase_					=	placevalue[counter]
																																				UpperCAmelCase_					=	doubles[current] + addition + words
																								else:
																														UpperCAmelCase_					=	''''''
																														if counter in placevalue:
																																				if current == 0 and ((temp_num % 100) // 10) == 0:
																																										UpperCAmelCase_					=	''''''
																																				else:
																																										UpperCAmelCase_					=	placevalue[counter]
																														if ((temp_num % 100) // 10) == 1:
																																				UpperCAmelCase_					=	teens[current] + addition + words
																																				UpperCAmelCase_					=	temp_num // 10
																																				counter += 1
																														else:
																																				UpperCAmelCase_					=	singles[current] + addition + words
																		counter += 1
																		UpperCAmelCase_					=	temp_num // 10
						return words


if __name__ == "__main__":
							import doctest

							doctest.testmod()





