




import mpmath  # for roots of unity
import numpy as np




class       a__	:







     def __init__(     self  ,		A=None  ,		A=None			)  ->   Any:



          '''simple docstring'''



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

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

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

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

          # The product
          a     =				self.__multiply()







     def       lowerCAmelCase_    (     self  ,		A			)  ->   Dict:



          '''simple docstring'''



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

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







     def       lowerCAmelCase_    (     self			)  ->   Optional[int]:



          '''simple docstring'''



          a     =				self.__dft("A"			)
          a     =				self.__dft("B"			)
          a     =				[[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
          a     =				2
          while next_ncol <= self.c_max_length:
               a     =				[[] for i in range(__UpperCamelCase			)]
               a     =				self.root ** (next_ncol // 2)
               a     =				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
               a     =				new_inverse_c
               next_ncol *= 2
          # Unpack
          a     =				[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[Any]:



          '''simple docstring'''



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







import math
import unittest

def  SCREAMING_SNAKE_CASE							(       __UpperCamelCase)      ->					bool:
     assert isinstance(__UpperCamelCase		, __UpperCamelCase) and (
         number >= 0
     ), "'number' must been an int and positive"

     if 1 < number < 4:
          # 2 and 3 are primes
          return True
     elif number < 2 or number % 2 == 0 or number % 3 == 0:
          # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
          return False

     # All primes number are in format of 6k +/- 1
     for i in range(5		, int(math.sqrt(__UpperCamelCase) + 1)		, 6):
          if number % i == 0 or number % (i + 2) == 0:
               return False
     return True




class       a__	( unittest.TestCase				):







     def       lowerCAmelCase_    (     self			)  ->   List[str]:



          '''simple docstring'''



          self.assertTrue(is_prime(2			)			)
          self.assertTrue(is_prime(3			)			)
          self.assertTrue(is_prime(5			)			)
          self.assertTrue(is_prime(7			)			)
          self.assertTrue(is_prime(11			)			)
          self.assertTrue(is_prime(13			)			)
          self.assertTrue(is_prime(17			)			)
          self.assertTrue(is_prime(19			)			)
          self.assertTrue(is_prime(23			)			)
          self.assertTrue(is_prime(29			)			)







     def       lowerCAmelCase_    (     self			)  ->   Tuple:



          '''simple docstring'''



          with self.assertRaises(A			):
               is_prime(-19			)
          self.assertFalse(
              is_prime(0			)  ,		"Zero doesn't have any positive factors, primes must have exactly two."  ,		)
          self.assertFalse(
              is_prime(1			)  ,		"One only has 1 positive factor, primes must have exactly two."  ,		)
          self.assertFalse(is_prime(2 * 2			)			)
          self.assertFalse(is_prime(2 * 3			)			)
          self.assertFalse(is_prime(3 * 3			)			)
          self.assertFalse(is_prime(3 * 5			)			)
          self.assertFalse(is_prime(3 * 5 * 7			)			)


if __name__ == "__main__":
   unittest.main()

