

"""simple docstring"""


def    a__   (   __SCREAMING_SNAKE_CASE			)     ->     list:
		__lowerCAmelCase:	List[str]   				=      [0] * len(__SCREAMING_SNAKE_CASE			)

		for i in range(1 ,			len(__SCREAMING_SNAKE_CASE			)			):
				# use last results for better performance - dynamic programming
				__lowerCAmelCase:	Optional[Any]   				=      prefix_result[i - 1]
				while j > 0 and input_string[i] != input_string[j]:
						__lowerCAmelCase:	List[Any]   				=      prefix_result[j - 1]

				if input_string[i] == input_string[j]:
						j += 1
				__lowerCAmelCase:	Union[str, Any]   				=      j

		return prefix_result






def    a__   (   __SCREAMING_SNAKE_CASE			)     ->     int:
		return max(prefix_function(__SCREAMING_SNAKE_CASE			)			)


if __name__ == "__main__":
			import doctest

			doctest.testmod()





"""simple docstring"""


from __future__ import annotations

from decimal import Decimal
from math import *  # noqa: F403

from sympy import diff






def    a__   (   __SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE ,			__SCREAMING_SNAKE_CASE = 1_0**-1_0			)     ->     float:
		__lowerCAmelCase:	Union[str, Any]   				=      a
		while True:
				__lowerCAmelCase:	Optional[int]   				=      Decimal(__SCREAMING_SNAKE_CASE			) - (
				    Decimal(eval(__SCREAMING_SNAKE_CASE			)			) / Decimal(eval(str(diff(__SCREAMING_SNAKE_CASE			)			)			)			)  # noqa: S307
				)
				# This number dictates the accuracy of the answer
				if abs(eval(__SCREAMING_SNAKE_CASE			)			) < precision:  # noqa: S307
						return float(__SCREAMING_SNAKE_CASE			)


# Let's Execute
if __name__ == "__main__":
			# Find root of trigonometric function
			# Find value of pi
			print(F'''The root of sin(x) = 0 is {newton_raphson("sin(x)", 2)}''')
			# Find root of polynomial
			print(F'''The root of x**2 - 5*x + 2 = 0 is {newton_raphson("x**2 - 5*x + 2", 0.4)}''')
			# Find Square Root of 5
			print(F'''The root of log(x) - 1 = 0 is {newton_raphson("log(x) - 1", 2)}''')
			# Exponential Roots
			print(F'''The root of exp(x) - 1 = 0 is {newton_raphson("exp(x) - 1", 0)}''')


