



"""simple docstring"""






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


def     _lowercase       (						__snake_case			)		->     float:
			return np.dot(__snake_case      ,__snake_case			)



class     A__     :




	'''simple docstring'''





	def __init__(  self:	Dict ,    *,
	    _SCREAMING_SNAKE_CASE:	float = np.inf ,    _SCREAMING_SNAKE_CASE:	str = "linear" ,    _SCREAMING_SNAKE_CASE:	float = 0.0 ,    )       -> None:

				"""simple docstring"""

				__lowerCAmelCase   :						Optional[Any]				= regularization
				__lowerCAmelCase   :						Dict				= gamma
				if kernel == "linear":
							__lowerCAmelCase   :						Optional[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")
							__lowerCAmelCase   :						str				= 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:
							__lowerCAmelCase   :						Dict				= F"""Unknown kernel: {kernel}"""
							raise ValueError(_SCREAMING_SNAKE_CASE)

	def 							_SCREAMING_SNAKE_CASE     (  self:	Any ,    _SCREAMING_SNAKE_CASE:	ndarray ,    _SCREAMING_SNAKE_CASE:	ndarray)       -> float:

				"""simple docstring"""

				return np.dot(_SCREAMING_SNAKE_CASE ,    _SCREAMING_SNAKE_CASE)

	def 							_SCREAMING_SNAKE_CASE     (  self:	List[Any] ,    _SCREAMING_SNAKE_CASE:	ndarray ,    _SCREAMING_SNAKE_CASE:	ndarray)       -> float:

				"""simple docstring"""

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

	def 							_SCREAMING_SNAKE_CASE     (  self:	Any ,    _SCREAMING_SNAKE_CASE:	list[ndarray] ,    _SCREAMING_SNAKE_CASE:	ndarray)       -> None:

				"""simple docstring"""

				__lowerCAmelCase   :						Dict				= observations
				__lowerCAmelCase   :						int				= 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

				((__lowerCAmelCase)		,			)   :						Optional[Any]				= np.shape(_SCREAMING_SNAKE_CASE)

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

				__lowerCAmelCase   :						List[str]				= LinearConstraint(_SCREAMING_SNAKE_CASE ,    0 ,    0)
				__lowerCAmelCase   :						Any				= Bounds(0 ,    self.regularization)

				__lowerCAmelCase   :						Any				= minimize(
				    _SCREAMING_SNAKE_CASE ,    np.ones(_SCREAMING_SNAKE_CASE) ,    bounds=_SCREAMING_SNAKE_CASE ,    constraints=[ly_contraint]).x
				__lowerCAmelCase   :						str				= l_star

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







	def 							_SCREAMING_SNAKE_CASE     (  self:	Optional[int] ,    _SCREAMING_SNAKE_CASE:	ndarray)       -> int:

				"""simple docstring"""

				__lowerCAmelCase   :						Union[str, Any]				= sum(
				    self.optimum[n]
				    * self.classes[n]
				    * self.kernel(self.observations[n] ,    _SCREAMING_SNAKE_CASE)
				    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"""






def     _lowercase       (						__snake_case = 3      ,__snake_case = 7      ,__snake_case = 1_000_000			)		->     int:
			__lowerCAmelCase   :						Optional[Any]				= 0
			__lowerCAmelCase   :						List[str]				= 1

			for current_denominator in range(1      ,limit + 1			):
						__lowerCAmelCase   :						Optional[int]				= current_denominator * numerator // denominator
						if current_denominator % denominator == 0:
									current_numerator -= 1
						if current_numerator * max_denominator > current_denominator * max_numerator:
									__lowerCAmelCase   :						Union[str, Any]				= current_numerator
									__lowerCAmelCase   :						int				= current_denominator
			return max_numerator


if __name__ == "__main__":
	print(solution(numerator=3, denominator=7, limit=1_000_000))