



UpperCAmelCase :				dict[str, float]							={
    "kilogram": 1,
    "gram": pow(10, 3),
    "milligram": pow(10, 6),
    "metric-ton": pow(10, -3),
    "long-ton": 0.0_009_842_073,
    "short-ton": 0.0_011_023_122,
    "pound": 2.2_046_244_202,
    "stone": 0.1_574_731_728,
    "ounce": 35.273_990_723,
    "carrat": 5000,
    "atomic-mass-unit": 6.022_136_652e26,
}

UpperCAmelCase :				dict[str, float]							={
    "kilogram": 1,
    "gram": pow(10, -3),
    "milligram": pow(10, -6),
    "metric-ton": pow(10, 3),
    "long-ton": 1_016.04_608,
    "short-ton": 907.184,
    "pound": 0.453_592,
    "stone": 6.35_029,
    "ounce": 0.0_283_495,
    "carrat": 0.0_002,
    "atomic-mass-unit": 1.660_540_199e-27,
}



def  _lowerCAmelCase     (_lowerCAmelCase ,						_lowerCAmelCase ,						_lowerCAmelCase):
					if to_type not in KILOGRAM_CHART or from_type not in WEIGHT_TYPE_CHART:
										UpperCamelCase_			= (
										    f"""Invalid 'from_type' or 'to_type' value: {from_type!r}, {to_type!r}\n"""
										    f"""Supported values are: {', '.join(_lowerCAmelCase)}"""
										)
										raise ValueError(_lowerCAmelCase)
					return value * KILOGRAM_CHART[to_type] * WEIGHT_TYPE_CHART[from_type]


if __name__ == "__main__":
	import doctest

	doctest.testmod()







from typing import Any

import numpy as np





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





def  __UpperCamelCase   (     a,  a,  a = 1000,  a = 1E-8,  )			->Any:
					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.
					lowerCamelCase__      =							np.zeros((np.shape(a)[0], 1))
					lowerCamelCase__      =							np.copy(a)
					lowerCamelCase__      =							np.copy(a)

					# 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(a,  a)

										# The main algorithm.

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

										# Update variables.
										lowerCamelCase__      =							np.copy(a)
										lowerCamelCase__      =							np.copy(a)
										lowerCamelCase__      =							np.copy(a)

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

					return x





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

					# Numpy solution.
					lowerCamelCase__      =							np.linalg.solve(a,  a)

					# Our implementation.
					lowerCamelCase__      =							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()

