





import mpmath  # for roots of unity
import numpy as np
class SCREAMING_SNAKE_CASE_  :






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







							"""simple docstring"""


							snake_case_						:     Optional[Any]     =		list(poly_a or [0]		)[:]
							snake_case_						:     Any     =		list(poly_b or [0]		)[:]

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

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

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

							# The product
							snake_case_						:     int     =		self.__multiply()






						def UpperCAmelCase_			( self : Optional[Any]						,				_A : List[str]		)							->      int:







							"""simple docstring"""


							snake_case_						:     List[Any]     =		[[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]
							#
							snake_case_						:     Union[str, Any]     =		self.c_max_length // 2
							while next_ncol > 0:
								snake_case_						:     List[Any]     =		[[] for i in range(_A		)]
								snake_case_						:     Any     =		self.root**next_ncol

								# First half of next step
								snake_case_						:     Tuple     =		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
								snake_case_						:     Optional[Any]     =		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
								snake_case_						:     List[str]     =		new_dft
								snake_case_						:     Any     =		next_ncol // 2
							return dft[0]






						def UpperCAmelCase_			( self : List[str]		)							->      Tuple:







							"""simple docstring"""


							snake_case_						:     Any     =		self.__dft('A'		)
							snake_case_						:     Optional[Any]     =		self.__dft('B'		)
							snake_case_						:     Tuple     =		[[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
							snake_case_						:     str     =		2
							while next_ncol <= self.c_max_length:
								snake_case_						:     Union[str, Any]     =		[[] for i in range(_A		)]
								snake_case_						:     List[str]     =		self.root ** (next_ncol // 2)
								snake_case_						:     Optional[int]     =		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
								snake_case_						:     List[str]     =		new_inverse_c
								next_ncol *= 2
							# Unpack
							snake_case_						:     List[str]     =		[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 : Union[str, Any]		)							->      Union[str, Any]:







							"""simple docstring"""


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

import math


def 							SCREAMING_SNAKE_CASE__  (      __a				,	__a				):
	snake_case_						:     Optional[int]     =		u
	for i in range(1				,	__a				):
		snake_case_						:     Optional[Any]     =		temp * (u - i)
	return temp


def 							SCREAMING_SNAKE_CASE__  (      ):
	snake_case_						:     Dict     =		int(input('enter the numbers of values: '				)				)
	snake_case_						:     list[list[float]]     =		[]
	for _ in range(__a				):
		y.append([]				)
	for i in range(__a				):
		for j in range(__a				):
			y[i].append(__a				)
			snake_case_						:     str     =		0

	print('enter the values of parameters in a list: '				)
	snake_case_						:     int     =		list(map(__a				,	input().split()				)				)

	print('enter the values of corresponding parameters: '				)
	for i in range(__a				):
		snake_case_						:     Union[str, Any]     =		float(input()				)

	snake_case_						:     int     =		int(input('enter the value to interpolate: '				)				)
	snake_case_						:     List[Any]     =		(value - x[0]) / (x[1] - x[0])

	# for calculating forward difference table

	for i in range(1				,	__a				):
		for j in range(n - i				):
			snake_case_						:     int     =		y[j + 1][i - 1] - y[j][i - 1]

	snake_case_						:     str     =		y[0][0]
	for i in range(1				,	__a				):
		summ += (ucal(__a				,	__a				) * y[0][i]) / math.factorial(__a				)

	print(f"""the value at {value} is {summ}"""				)


if __name__ == "__main__":
						main()

