




import mpmath  # for roots of unity
import numpy as np







class   A:



 '''simple docstring'''


 def __init__(    self :  Optional[Any]						,							A_ :  Tuple=None						,							A_ :  str=None  )       ->     Tuple:

       """simple docstring"""






       lowerCamelCase_							=   list(poly_a or [0]  )[:]
       lowerCamelCase_							=   list(poly_b or [0]  )[:]

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

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

       # Add 0 to make lengths equal a power of 2
       lowerCamelCase_							=   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
       lowerCamelCase_							=   complex(mpmath.root(x=1						,							n=self.c_max_length						,							k=1  )  )

       # The product
       lowerCamelCase_							=   self.__multiply()

 def     a__			(    self :  List[str]						,							A_ :  Optional[Any]  )       ->     Optional[Any]:

       """simple docstring"""






       lowerCamelCase_							=   [[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]
       #
       lowerCamelCase_							=   self.c_max_length // 2
       while next_ncol > 0:
             lowerCamelCase_							=   [[] for i in range(A_  )]
             lowerCamelCase_							=   self.root**next_ncol

             # First half of next step
             lowerCamelCase_							=   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
             lowerCamelCase_							=   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
             lowerCamelCase_							=   new_dft
             lowerCamelCase_							=   next_ncol // 2
       return dft[0]

 def     a__			(    self :  str  )       ->     Union[str, Any]:

       """simple docstring"""






       lowerCamelCase_							=   self.__dft('A'  )
       lowerCamelCase_							=   self.__dft('B'  )
       lowerCamelCase_							=   [[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
       lowerCamelCase_							=   2
       while next_ncol <= self.c_max_length:
             lowerCamelCase_							=   [[] for i in range(A_  )]
             lowerCamelCase_							=   self.root ** (next_ncol // 2)
             lowerCamelCase_							=   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
             lowerCamelCase_							=   new_inverse_c
             next_ncol *= 2
       # Unpack
       lowerCamelCase_							=   [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 :  str  )       ->     Dict:

       """simple docstring"""






       lowerCamelCase_							=   'A = ' + ' + '.join(
           f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyA[: self.len_A]  )  )
       lowerCamelCase_							=   'B = ' + ' + '.join(
           f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyB[: self.len_B]  )  )
       lowerCamelCase_							=   '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       _SCREAMING_SNAKE_CASE      (							lowercase     :	str     ,							lowercase     :	int					):


      '''simple docstring'''



      return [sentence[i : i + ngram_size] for i in range(len(lowercase					) - ngram_size + 1					)]


if __name__ == "__main__":
 from doctest import testmod

 testmod()



