"""simple docstring"""







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



def 			lowercase__(       __SCREAMING_SNAKE_CASE							:     ndarray      ):
     return np.dot(__SCREAMING_SNAKE_CASE     ,			__SCREAMING_SNAKE_CASE      )







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


        '''simple docstring'''


        lowercase_	:				Union[str, Any]													=	regularization
        lowercase_	:				int													=	gamma
        if kernel == "linear":
             lowercase_	:				str													=	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_	:				Union[str, Any]													=	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_	:				Union[str, Any]													=	f'''Unknown kernel: {kernel}'''
             raise ValueError(__UpperCamelCase			)
   def 			_UpperCAmelCase   (			self     ,__UpperCamelCase     ,__UpperCamelCase			)	->      float:


        '''simple docstring'''


        return np.dot(__UpperCamelCase     ,__UpperCamelCase			)
   def 			_UpperCAmelCase   (			self     ,__UpperCamelCase     ,__UpperCamelCase			)	->      float:


        '''simple docstring'''


        return np.exp(-(self.gamma * norm_squared(vectora - vectora			))			)
   def 			_UpperCAmelCase   (			self     ,__UpperCamelCase     ,__UpperCamelCase			)	->      None:


        '''simple docstring'''


        lowercase_	:				Union[str, Any]													=	observations
        lowercase_	:				Optional[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

        ((lowercase_)							,  )	:				Dict													=	np.shape(__UpperCamelCase			)

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

        lowercase_	:				int													=	LinearConstraint(__UpperCamelCase     ,0     ,0			)
        lowercase_	:				List[Any]													=	Bounds(0     ,self.regularization			)

        lowercase_	:				List[Any]													=	minimize(
            __UpperCamelCase     ,np.ones(__UpperCamelCase			)     ,bounds=__UpperCamelCase     ,constraints=[ly_contraint]			).x
        lowercase_	:				List[str]													=	l_star

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







   def 			_UpperCAmelCase   (			self     ,__UpperCamelCase			)	->      int:


        '''simple docstring'''


        lowercase_	:				Union[str, Any]													=	sum(
            self.optimum[n]
            * self.classes[n]
            * self.kernel(self.observations[n]     ,__UpperCamelCase			)
            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 __future__ import annotations

import math
from collections.abc import Callable



def 			lowercase__(       __SCREAMING_SNAKE_CASE							:     Callable[[int | float], int | float]     ,			__SCREAMING_SNAKE_CASE							:     int | float     ,			__SCREAMING_SNAKE_CASE							:     int | float     ,			__SCREAMING_SNAKE_CASE							:     int = 1_00     ,			):
     lowercase_	:				Dict													=	x_start
     lowercase_	:				Optional[Any]													=	fnc(__SCREAMING_SNAKE_CASE      )
     lowercase_	:				Optional[Any]													=	0.0

     for _ in range(__SCREAMING_SNAKE_CASE      ):
          # Approximates curve as a sequence of linear lines and sums their length
          lowercase_	:				Union[str, Any]													=	(x_end - x_start) / steps + xa
          lowercase_	:				str													=	fnc(__SCREAMING_SNAKE_CASE      )
          length += math.hypot(xa - xa     ,			fxa - fxa      )

          # Increment step
          lowercase_	:				List[Any]													=	xa
          lowercase_	:				Optional[Any]													=	fxa

     return length


if __name__ == "__main__":



       def 			lowercase__(       __SCREAMING_SNAKE_CASE							:     Any      ):
            return math.sin(10 * x      )

       print("f(x) = sin(10 * x)")
       print("The length of the curve from x = -10 to x = 10 is:")
       __SCREAMING_SNAKE_CASE    	=10
       while i <= 10_0000:
              print(F"With {i} steps: {line_length(f, -10, 10, i)}")
              i *= 10

