
"""simple docstring"""

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




def      lowercase__    (       lowerCAmelCase					:       ndarray			)			->   float:






    """simple docstring"""




    return np.dot(lowerCAmelCase			,					lowerCAmelCase			)
class   _UpperCAmelCase				:




      def __init__(				self ,							*,
          lowercase_ = np.inf ,							lowercase_ = "linear" ,							lowercase_ = 0.0 ,							)    ->       None:
          UpperCAmelCase          =       regularization
          UpperCAmelCase          =       gamma
          if kernel == "linear":
              UpperCAmelCase          =       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'				)
              UpperCAmelCase          =       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:
              UpperCAmelCase          =       F"Unknown kernel: {kernel}"
              raise ValueError(lowercase_				)




      def      a_   (				self ,							lowercase_ ,							lowercase_				)    ->       float:
          return np.dot(lowercase_ ,							lowercase_				)




      def      a_   (				self ,							lowercase_ ,							lowercase_				)    ->       float:
          return np.exp(-(self.gamma * norm_squared(vectora - vectora				))				)




      def      a_   (				self ,							lowercase_ ,							lowercase_				)    ->       None:
          UpperCAmelCase          =       observations
          UpperCAmelCase          =       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

          ((UpperCAmelCase)      ,				)          =       np.shape(lowercase_				)

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

          UpperCAmelCase          =       LinearConstraint(lowercase_ ,							0 ,							0				)
          UpperCAmelCase          =       Bounds(0 ,							self.regularization				)

          UpperCAmelCase          =       minimize(
              lowercase_ ,							np.ones(lowercase_				) ,							bounds=lowercase_ ,							constraints=[ly_contraint]				).x
          UpperCAmelCase          =       l_star

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




      def      a_   (				self ,							lowercase_				)    ->       int:
          UpperCAmelCase          =       sum(
              self.optimum[n]
              * self.classes[n]
              * self.kernel(self.observations[n] ,							lowercase_				)
              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"""

from math import ceil, sqrt




def      lowercase__    (       lowerCAmelCase					:       int = 1_000_000			)			->   int:






    """simple docstring"""




    UpperCAmelCase          =       0

    for outer_width in range(3			,					(limit // 4) + 2			):
        if outer_width**2 > limit:
            UpperCAmelCase          =       max(ceil(sqrt(outer_width**2 - limit			)			)			,					1			)
        else:
            UpperCAmelCase          =       1
        if (outer_width - hole_width_lower_bound) % 2:
            hole_width_lower_bound += 1

        answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

    return answer


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



