


"""simple docstring"""







import mpmath  # for roots of unity
import numpy as np




class  UpperCamelCase_	:

	"""simple docstring"""





	def __init__( self			:				List[Any]			,			UpperCAmelCase__			:				Optional[Any]=None			,			UpperCAmelCase__			:				int=None   )       ->	str:
						# 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   UpperCAmelCase_      ( self			:				Optional[int]			,			UpperCAmelCase__			:				Optional[Any]   )       ->	str:
						__SCREAMING_SNAKE_CASE						   =				[[x] for x in self.polyA] if which == "A" else [[x] for x in self.polyB]
						# Corner case
						if len(UpperCAmelCase__   ) <= 1:
											return dft[0]
						#
						__SCREAMING_SNAKE_CASE						   =				self.c_max_length // 2
						while next_ncol > 0:
											__SCREAMING_SNAKE_CASE						   =				[[] for i in range(UpperCAmelCase__   )]
											__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(UpperCAmelCase__   ):
																					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(UpperCAmelCase__   ):
																					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   UpperCAmelCase_      ( self			:				Union[str, Any]   )       ->	Any:
						__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(UpperCAmelCase__   )]
											__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			:				Optional[int]   )       ->	Any:
						__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()







'''simple docstring'''




import random






def        _lowerCAmelCase			(  __snake_case  : int ,							__snake_case  : float ,							__snake_case  : bool = False			)     ->     dict:
     __A						: dict      =       {i: [] for i in range(__snake_case			)}

     # if probability is greater or equal than 1, then generate a complete graph
     if probability >= 1:
          return complete_graph(__snake_case			)
     # if probability is lower or equal than 0, then return a graph without edges
     if probability <= 0:
          return graph

     # for each couple of nodes, add an edge from u to v
     # if the number randomly generated is greater than probability probability
     for i in range(__snake_case			):
          for j in range(i + 1 ,							__snake_case			):
               if random.random() < probability:
                    graph[i].append(__snake_case			)
                    if not directed:
                         # if the graph is undirected, add an edge in from j to i, either
                         graph[j].append(__snake_case			)
     return graph






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     dict:
     return {
         i: [j for j in range(__snake_case			) if i != j] for i in range(__snake_case			)
     }


if __name__ == "__main__":
  import doctest

  doctest.testmod()