

import mpmath  # for roots of unity
import numpy as np



class       __snake_case  :

  def __init__(					self						,		_A=None						,		_A=None):
      # Input as list
      SCREAMING_SNAKE_CASE_									=						list(poly_a or [0])[:]
      SCREAMING_SNAKE_CASE_									=						list(poly_b or [0])[:]

      # Remove leading zero coefficients
      while self.polyA[-1] == 0:
          self.polyA.pop()
      SCREAMING_SNAKE_CASE_									=						len(self.polyA)

      while self.polyB[-1] == 0:
          self.polyB.pop()
      SCREAMING_SNAKE_CASE_									=						len(self.polyB)

      # Add 0 to make lengths equal a power of 2
      SCREAMING_SNAKE_CASE_									=						int(
          2 ** np.ceil(np.loga(len(self.polyA) + len(self.polyB) - 1)))

      while len(self.polyA) < self.c_max_length:
          self.polyA.append(0)
      while len(self.polyB) < self.c_max_length:
          self.polyB.append(0)
      # A complex root used for the fourier transform
      SCREAMING_SNAKE_CASE_									=						complex(mpmath.root(x=1						,		n=self.c_max_length						,		k=1))

      # The product
      SCREAMING_SNAKE_CASE_									=						self.__multiply()

  def 						lowerCAmelCase__     (					self						,		_A):
      SCREAMING_SNAKE_CASE_									=						[[x] for x in self.polyA] if which == 'A' else [[x] for x in self.polyB]
      # Corner case
      if len(_A) <= 1:
          return dft[0]
      #
      SCREAMING_SNAKE_CASE_									=						self.c_max_length // 2
      while next_ncol > 0:
          SCREAMING_SNAKE_CASE_									=						[[] for i in range(_A)]
          SCREAMING_SNAKE_CASE_									=						self.root**next_ncol

          # First half of next step
          SCREAMING_SNAKE_CASE_									=						1
          for j in range(self.c_max_length // (next_ncol * 2)):
              for i in range(_A):
                  new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j])
              current_root *= root
          # Second half of next step
          SCREAMING_SNAKE_CASE_									=						1
          for j in range(self.c_max_length // (next_ncol * 2)):
              for i in range(_A):
                  new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j])
              current_root *= root
          # Update
          SCREAMING_SNAKE_CASE_									=						new_dft
          SCREAMING_SNAKE_CASE_									=						next_ncol // 2
      return dft[0]

  def 						lowerCAmelCase__     (					self):
      SCREAMING_SNAKE_CASE_									=						self.__dft('A')
      SCREAMING_SNAKE_CASE_									=						self.__dft('B')
      SCREAMING_SNAKE_CASE_									=						[[dft_a[i] * dft_b[i] for i in range(self.c_max_length)]]
      del dft_a
      del dft_b

      # Corner Case
      if len(inverce_c[0]) <= 1:
          return inverce_c[0]
      # Inverse DFT
      SCREAMING_SNAKE_CASE_									=						2
      while next_ncol <= self.c_max_length:
          SCREAMING_SNAKE_CASE_									=						[[] for i in range(_A)]
          SCREAMING_SNAKE_CASE_									=						self.root ** (next_ncol // 2)
          SCREAMING_SNAKE_CASE_									=						1
          # First half of next step
          for j in range(self.c_max_length // next_ncol):
              for i in range(next_ncol // 2):
                  # Even positions
                  new_inverse_c[i].append(
                      (
                          inverce_c[i][j]
                          + inverce_c[i][j + self.c_max_length // next_ncol]
                      )
                      / 2)
                  # Odd positions
                  new_inverse_c[i + next_ncol // 2].append(
                      (
                          inverce_c[i][j]
                          - inverce_c[i][j + self.c_max_length // next_ncol]
                      )
                      / (2 * current_root))
              current_root *= root
          # Update
          SCREAMING_SNAKE_CASE_									=						new_inverse_c
          next_ncol *= 2
      # Unpack
      SCREAMING_SNAKE_CASE_									=						[round(x[0].real						,		8) + round(x[0].imag						,		8) * 1J for x in inverce_c]

      # Remove leading 0's
      while inverce_c[-1] == 0:
          inverce_c.pop()
      return inverce_c


  def __str__(					self):
      SCREAMING_SNAKE_CASE_									=						'A = ' + ' + '.join(
          f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyA[: self.len_A]))
      SCREAMING_SNAKE_CASE_									=						'B = ' + ' + '.join(
          f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyB[: self.len_B]))
      SCREAMING_SNAKE_CASE_									=						'A*B = ' + ' + '.join(
          f"""{coef}*x^{i}""" for coef, i in enumerate(self.product))

      return f"""{a}\n{b}\n{c}"""


# Unit tests
if __name__ == "__main__":
       import doctest

       doctest.testmod()





def 					_UpperCAmelCase       (  _SCREAMING_SNAKE_CASE					:		list ,   _SCREAMING_SNAKE_CASE					:		list ,   _SCREAMING_SNAKE_CASE					:		int ,   _SCREAMING_SNAKE_CASE					:		int ,   _SCREAMING_SNAKE_CASE					:		int       ):





    """simple docstring"""


    if index == number_of_items:
        return 0
    SCREAMING_SNAKE_CASE_									=						0
    SCREAMING_SNAKE_CASE_									=						0
    SCREAMING_SNAKE_CASE_									=						knapsack(_SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE ,   index + 1       )
    if weights[index] <= max_weight:
        SCREAMING_SNAKE_CASE_									=						values[index] + knapsack(
            _SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE ,   max_weight - weights[index] ,   index + 1       )
    return max(_SCREAMING_SNAKE_CASE ,   _SCREAMING_SNAKE_CASE       )


if __name__ == "__main__":
       import doctest

       doctest.testmod()


