


'''simple docstring'''



from __future__ import annotations

import math







def     _UpperCamelCase      (		UpperCamelCase__ ,      UpperCamelCase__					):
			UpperCAmelCase__  :					int        =     u
			for i in range(1 ,      UpperCamelCase__					):
						UpperCAmelCase__  :					Optional[Any]        =     temp * (u - i)
			return temp







def     _UpperCamelCase      (		):
			UpperCAmelCase__  :					Optional[int]        =     int(input("""enter the numbers of values: """					)					)
			UpperCAmelCase__  :					list[list[float]]        =     []
			for _ in range(UpperCamelCase__					):
						y.append([]					)
			for i in range(UpperCamelCase__					):
						for j in range(UpperCamelCase__					):
									y[i].append(UpperCamelCase__					)
									UpperCAmelCase__  :					Union[str, Any]        =     0

			print("""enter the values of parameters in a list: """					)
			UpperCAmelCase__  :					Optional[int]        =     list(map(UpperCamelCase__ ,      input().split()					)					)

			print("""enter the values of corresponding parameters: """					)
			for i in range(UpperCamelCase__					):
						UpperCAmelCase__  :					int        =     float(input()					)

			UpperCAmelCase__  :					List[str]        =     int(input("""enter the value to interpolate: """					)					)
			UpperCAmelCase__  :					Union[str, Any]        =     (value - x[0]) / (x[1] - x[0])

			# for calculating forward difference table

			for i in range(1 ,      UpperCamelCase__					):
						for j in range(n - i					):
									UpperCAmelCase__  :					Optional[int]        =     y[j + 1][i - 1] - y[j][i - 1]

			UpperCAmelCase__  :					Optional[int]        =     y[0][0]
			for i in range(1 ,      UpperCamelCase__					):
						summ += (ucal(UpperCamelCase__ ,      UpperCamelCase__					) * y[0][i]) / math.factorial(UpperCamelCase__					)

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


if __name__ == "__main__":
		main()



'''simple docstring'''



from __future__ import annotations

import matplotlib.pyplot as plt  # type: ignore
import numpy

# initial triangle of Koch snowflake
__A	=numpy.array([0, 0])
__A	=numpy.array([0.5, 0.8_6_6_0_2_5_4])
__A	=numpy.array([1, 0])
__A	=[VECTOR_1, VECTOR_2, VECTOR_3, VECTOR_1]







def     _UpperCamelCase      (		UpperCamelCase__ ,      UpperCamelCase__					):
			UpperCAmelCase__  :					Union[str, Any]        =     initial_vectors
			for _ in range(UpperCamelCase__					):
						UpperCAmelCase__  :					Dict        =     iteration_step(UpperCamelCase__					)
			return vectors







def     _UpperCamelCase      (		UpperCamelCase__					):
			UpperCAmelCase__  :					Optional[Any]        =     []
			for i, start_vector in enumerate(vectors[:-1]					):
						UpperCAmelCase__  :					Tuple        =     vectors[i + 1]
						new_vectors.append(UpperCamelCase__					)
						UpperCAmelCase__  :					Optional[Any]        =     end_vector - start_vector
						new_vectors.append(start_vector + difference_vector / 3					)
						new_vectors.append(
						    start_vector + difference_vector / 3 + rotate(difference_vector / 3 ,      6_0					)					)
						new_vectors.append(start_vector + difference_vector * 2 / 3					)
			new_vectors.append(vectors[-1]					)
			return new_vectors







def     _UpperCamelCase      (		UpperCamelCase__ ,      UpperCamelCase__					):
			UpperCAmelCase__  :					Any        =     numpy.radians(UpperCamelCase__					)
			UpperCAmelCase__   ,       UpperCAmelCase__  :					int        =     numpy.cos(UpperCamelCase__					), numpy.sin(UpperCamelCase__					)
			UpperCAmelCase__  :					Dict        =     numpy.array(((c, -s), (s, c))					)
			return numpy.dot(UpperCamelCase__ ,      UpperCamelCase__					)







def     _UpperCamelCase      (		UpperCamelCase__					):
			UpperCAmelCase__  :					Any        =     plt.gca()
			axes.set_aspect("""equal"""					)

			# matplotlib.pyplot.plot takes a list of all x-coordinates and a list of all
			# y-coordinates as inputs, which are constructed from the vector-list using
			# zip()
			UpperCAmelCase__   ,       UpperCAmelCase__  :					Tuple        =     zip(*UpperCamelCase__					)
			plt.plot(UpperCamelCase__ ,      UpperCamelCase__					)
			plt.show()


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		__A	=iterate(INITIAL_VECTORS, 5)
		plot(processed_vectors)