




"""simple docstring"""






import mpmath  # for roots of unity
import numpy as np






class lowerCAmelCase__	:






			'''simple docstring'''



			def __init__(  self  ,      lowercase=None  ,      lowercase=None  ):
							# Input as list
							_lowerCamelCase     :			Optional[int]    =    list(poly_a or [0]  )[:]
							_lowerCamelCase     :			Tuple    =    list(poly_b or [0]  )[:]

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

							while self.polyB[-1] == 0:
											self.polyB.pop()
							_lowerCamelCase     :			Optional[Any]    =    len(self.polyB  )

							# Add 0 to make lengths equal a power of 2
							_lowerCamelCase     :			List[str]    =    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     :			List[str]    =    complex(mpmath.root(x=1  ,      n=self.c_max_length  ,      k=1  )  )

							# The product
							_lowerCamelCase     :			Optional[int]    =    self.__multiply()
			def 					A_    (  self  ,      lowercase  ):
							_lowerCamelCase     :			Any    =    [[x] for x in self.polyA] if which == 'A' else [[x] for x in self.polyB]
							# Corner case
							if len(lowercase  ) <= 1:
											return dft[0]
							#
							_lowerCamelCase     :			Tuple    =    self.c_max_length // 2
							while next_ncol > 0:
											_lowerCamelCase     :			Optional[Any]    =    [[] for i in range(lowercase  )]
											_lowerCamelCase     :			List[Any]    =    self.root**next_ncol

											# First half of next step
											_lowerCamelCase     :			Optional[int]    =    1
											for j in range(self.c_max_length // (next_ncol * 2)  ):
															for i in range(lowercase  ):
																			new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]  )
															current_root *= root
											# Second half of next step
											_lowerCamelCase     :			List[Any]    =    1
											for j in range(self.c_max_length // (next_ncol * 2)  ):
															for i in range(lowercase  ):
																			new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]  )
															current_root *= root
											# Update
											_lowerCamelCase     :			Optional[int]    =    new_dft
											_lowerCamelCase     :			int    =    next_ncol // 2
							return dft[0]
			def 					A_    (  self  ):
							_lowerCamelCase     :			Optional[Any]    =    self.__dft('A'  )
							_lowerCamelCase     :			Optional[Any]    =    self.__dft('B'  )
							_lowerCamelCase     :			Union[str, Any]    =    [[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     :			Any    =    2
							while next_ncol <= self.c_max_length:
											_lowerCamelCase     :			Union[str, Any]    =    [[] for i in range(lowercase  )]
											_lowerCamelCase     :			Tuple    =    self.root ** (next_ncol // 2)
											_lowerCamelCase     :			str    =    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     :			Union[str, Any]    =    new_inverse_c
											next_ncol *= 2
							# Unpack
							_lowerCamelCase     :			Dict    =    [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  ):
							_lowerCamelCase     :			Optional[int]    =    'A = ' + ' + '.join(
							    F'''{coef}*x^{i}''' for coef, i in enumerate(self.polyA[: self.len_A]  )  )
							_lowerCamelCase     :			Any    =    'B = ' + ' + '.join(
							    F'''{coef}*x^{i}''' for coef, i in enumerate(self.polyB[: self.len_B]  )  )
							_lowerCamelCase     :			Optional[Any]    =    '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()







from __future__ import annotations





def 						__UpperCAmelCase			(     a_							,     a_							,     a_							,     a_):  # noqa: E741
							while r - l > 1:
														snake_case_				=							(l + r) // 2
														if v[m] >= key:
																					snake_case_				=							m
														else:
																					snake_case_				=							m  # noqa: E741
							return r





def 						__UpperCAmelCase			(     a_):
							if len(a_) == 0:
														return 0

							snake_case_				=							[0] * len(a_)
							snake_case_				=							1

							snake_case_				=							v[0]

							for i in range(1							,     len(a_)):
														if v[i] < tail[0]:
																					snake_case_				=							v[i]
														elif v[i] > tail[length - 1]:
																					snake_case_				=							v[i]
																					length += 1
														else:
																					snake_case_				=							v[i]

							return length


if __name__ == "__main__":
	import doctest

	doctest.testmod()





