






"""simple docstring"""




import math






def        lowerCAmelCase (      UpperCamelCase_:							int					)       ->							bool:







  '''simple docstring'''







  if 1 < number < 4:
    # 2 and 3 are primes
    return True
  elif number < 2 or number % 2 == 0 or number % 3 == 0:
    # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
    return False

  # All primes number are in format of 6k +/- 1
  for i in range(5     , int(math.sqrt(UpperCamelCase_					) + 1					)     , 6					):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True






def        lowerCAmelCase (      UpperCamelCase_:							int = 10001					)       ->							int:







  '''simple docstring'''







  try:
    _a							=	int(UpperCamelCase_					)
  except (TypeError, ValueError):
    raise TypeError("Parameter nth must be int or castable to int."					) from None
  if nth <= 0:
    raise ValueError("Parameter nth must be greater than or equal to one."					)
  _a							=	[]
  _a							=	2
  while len(UpperCamelCase_					) < nth:
    if is_prime(UpperCamelCase_					):
      primes.append(UpperCamelCase_					)
      num += 1
    else:
      num += 1
  return primes[len(UpperCamelCase_					) - 1]


if __name__ == "__main__":
     print(f"{solution() = }")










"""simple docstring"""




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






def        lowerCAmelCase (      UpperCamelCase_:							ndarray					)       ->							float:







  '''simple docstring'''







  return np.dot(UpperCamelCase_     , UpperCamelCase_					)




class    lowercase_    :






      def __init__(    self     ,     *,
          a_ = np.inf     ,     a_ = "linear"     ,     a_ = 0.0     ,     )	->None:







        '''simple docstring'''


        _a							=	regularization
        _a							=	gamma
        if kernel == "linear":
          _a							=	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"	)
          _a							=	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:
          _a							=	f'''Unknown kernel: {kernel}'''
          raise ValueError(a_	)






      def 			lowerCamelCase__    (    self     ,     a_     ,     a_	)	->float:







        '''simple docstring'''


        return np.dot(a_     ,     a_	)






      def 			lowerCamelCase__    (    self     ,     a_     ,     a_	)	->float:







        '''simple docstring'''


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






      def 			lowerCamelCase__    (    self     ,     a_     ,     a_	)	->None:







        '''simple docstring'''


        _a							=	observations
        _a							=	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

        ((_a)			,	)							=	np.shape(a_	)

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

        _a							=	LinearConstraint(a_     ,     0     ,     0	)
        _a							=	Bounds(0     ,     self.regularization	)

        _a							=	minimize(
            a_     ,     np.ones(a_	)     ,     bounds=a_     ,     constraints=[ly_contraint]	).x
        _a							=	l_star

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






      def 			lowerCamelCase__    (    self     ,     a_	)	->int:







        '''simple docstring'''


        _a							=	sum(
            self.optimum[n]
            * self.classes[n]
            * self.kernel(self.observations[n]     ,     a_	)
            for n in range(len(self.classes	)	)	)
        return 1 if s + self.offset >= 0 else -1


if __name__ == "__main__":
     import doctest

     doctest.testmod()


