"""simple docstring"""







import numpy as np
from numpy import ndarray
from scipy.optimize import Bounds, LinearConstraint, minimize


def      __lowerCAmelCase(		__UpperCAmelCase			):






	"""simple docstring"""





	return np.dot(__UpperCAmelCase     ,__UpperCAmelCase			)



class  _lowerCamelCase			:







				def __init__( self    :       Optional[int]			,					*,
				    lowerCamelCase_    :       float = np.inf			,					lowerCamelCase_    :       str = "linear"			,					lowerCamelCase_    :       float = 0.0			,					):

					"""simple docstring"""






					_lowercase :    Any							=							regularization
					_lowercase :    List[Any]							=							gamma
					if kernel == "linear":
						_lowercase :    int							=							self.__linear
					elif kernel == "rbf":
						if self.gamma == 0:
							raise ValueError('rbf kernel requires gamma'						)
						if not isinstance(self.gamma			,					(float, int)						):
							raise ValueError('gamma must be float or int'						)
						if not self.gamma > 0:
							raise ValueError('gamma must be > 0'						)
						_lowercase :    Optional[int]							=							self.__rbf
						# in the future, there could be a default value like in sklearn
						# sklear: def_gamma = 1/(n_features * X.var()) (wiki)
						# previously it was 1/(n_features)
					else:
						_lowercase :    Any							=							F'''Unknown kernel: {kernel}'''
						raise ValueError(lowerCamelCase_						)







				def 							__UpperCAmelCase					( self    :       Tuple			,					lowerCamelCase_    :       ndarray			,					lowerCamelCase_    :       ndarray						):

					"""simple docstring"""






					return np.dot(lowerCamelCase_			,					lowerCamelCase_						)







				def 							__UpperCAmelCase					( self    :       Optional[Any]			,					lowerCamelCase_    :       ndarray			,					lowerCamelCase_    :       ndarray						):

					"""simple docstring"""






					return np.exp(-(self.gamma * norm_squared(vectora - vectora						))						)







				def 							__UpperCAmelCase					( self    :       Tuple			,					lowerCamelCase_    :       list[ndarray]			,					lowerCamelCase_    :       ndarray						):

					"""simple docstring"""






					_lowercase :    Optional[Any]							=							observations
					_lowercase :    Any							=							classes

					# using Wolfe's Dual to calculate w.
					# Primal problem: minimize 1/2*norm_squared(w)
					#   constraint: yn(w . xn + b) >= 1
					#
					# With l a vector
					# Dual problem: maximize sum_n(ln) -
					#       1/2 * sum_n(sum_m(ln*lm*yn*ym*xn . xm))
					#   constraint: self.C >= ln >= 0
					#           and sum_n(ln*yn) = 0
					# Then we get w using w = sum_n(ln*yn*xn)
					# At the end we can get b ~= mean(yn - w . xn)
					#
					# Since we use kernels, we only need l_star to calculate b
					# and to classify observations

					(_lowercase      ) :    Optional[Any]							=							np.shape(lowerCamelCase_						)

					def to_minimize(lowerCamelCase_    :       ndarray						) -> float:
						_lowercase :    List[str]							=							0
						(_lowercase      ) :    List[Any]							=							np.shape(lowerCamelCase_						)
						for i in range(lowerCamelCase_						):
							for j in range(lowerCamelCase_						):
								s += (
								    candidate[i]
								    * candidate[j]
								    * classes[i]
								    * classes[j]
								    * self.kernel(observations[i]			,					observations[j]						)
								)
						return 1 / 2 * s - sum(lowerCamelCase_						)

					_lowercase :    List[Any]							=							LinearConstraint(lowerCamelCase_			,					0			,					0						)
					_lowercase :    Optional[Any]							=							Bounds(0			,					self.regularization						)

					_lowercase :    List[str]							=							minimize(
					    lowerCamelCase_			,					np.ones(lowerCamelCase_						)			,					bounds=lowerCamelCase_			,					constraints=[ly_contraint]						).x
					_lowercase :    Optional[Any]							=							l_star

					# calculating mean offset of separation plane to points
					_lowercase :    List[str]							=							0
					for i in range(lowerCamelCase_						):
						for j in range(lowerCamelCase_						):
							s += classes[i] - classes[i] * self.optimum[i] * self.kernel(
							    observations[i]			,					observations[j]						)
					_lowercase :    Union[str, Any]							=							s / n







				def 							__UpperCAmelCase					( self    :       List[str]			,					lowerCamelCase_    :       ndarray						):

					"""simple docstring"""






					_lowercase :    Union[str, Any]							=							sum(
					    self.optimum[n]
					    * self.classes[n]
					    * self.kernel(self.observations[n]			,					lowerCamelCase_						)
					    for n in range(len(self.classes						)						)						)
					return 1 if s + self.offset >= 0 else -1


if __name__ == "__main__":
				import doctest

				doctest.testmod()




"""simple docstring"""







import math


def      __lowerCAmelCase(		__UpperCAmelCase			):






	"""simple docstring"""





	if not isinstance(__UpperCAmelCase     ,__UpperCAmelCase			):
		_lowercase :    List[Any]							=							F'''Input value of [number={number}] must be an integer'''
		raise TypeError(__UpperCAmelCase			)

	if number < 1:
		_lowercase :    List[Any]							=							F'''Input value of [number={number}] must be > 0'''
		raise ValueError(__UpperCAmelCase			)
	elif number == 1:
		return 3
	elif number == 2:
		return 5
	else:
		_lowercase :    str							=							int(math.log(number // 3     ,2			)			) + 2

		_lowercase :    Union[str, Any]							=							[3, 5]
		_lowercase :    Optional[int]							=							2
		_lowercase :    List[Any]							=							3
		for block in range(1     ,__UpperCAmelCase			):
			for _ in range(__UpperCAmelCase			):
				proth_list.append(2 ** (block + 1) + proth_list[proth_index - 1]			)
				proth_index += 1
			increment *= 2

	return proth_list[number - 1]


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						for number in range(11):
												SCREAMING_SNAKE_CASE									=						0
												try:
																		SCREAMING_SNAKE_CASE									=						proth(number)
												except ValueError:
																		print(f"""ValueError: there is no {number}th Proth number""")
																		continue

												print(f"""The {number}th Proth number: {value}""")



