



def 	UpperCAmelCase   (						_lowerCamelCase    ):
			A							:					Any              =						[0] * len(_lowerCamelCase    )

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

						if input_string[i] == input_string[j]:
									j += 1
						A							:					Optional[int]              =						j

			return prefix_result




def 	UpperCAmelCase   (						_lowerCamelCase    ):
			return max(prefix_function(_lowerCamelCase    )    )


if __name__ == "__main__":
							import doctest

							doctest.testmod()




from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403




def 	UpperCAmelCase   (						_lowerCamelCase ,    _lowerCamelCase ,    _lowerCamelCase = "x" ,    _lowerCamelCase = 10**-10 ,    _lowerCamelCase = 1 ,    ):
			A							:					str              =						symbols(_lowerCamelCase    )
			A							:					int              =						lambdify(_lowerCamelCase ,    _lowerCamelCase    )
			A							:					List[str]              =						lambdify(_lowerCamelCase ,    diff(_lowerCamelCase ,    _lowerCamelCase    )    )

			A							:					Optional[int]              =						starting_point

			while True:
						if diff_function(_lowerCamelCase    ) != 0:
									A							:					Optional[Any]              =						prev_guess - multiplicity * func(_lowerCamelCase    ) / diff_function(
									    _lowerCamelCase    )
						else:
									raise ZeroDivisionError("Could not find root"    ) from None

						# Precision is checked by comparing the difference of consecutive guesses
						if abs(next_guess - prev_guess    ) < precision:
									return next_guess

						A							:					int              =						next_guess


# 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
							# Find fourth Root of 5
							print(F"""The root of x**4 - 5 = 0 is {newton_raphson('x**4 -5', 0.4 +5j)}""")

							# Find value of e
							print(
							    """The root of log(y) - 1 = 0 is """,
							    F"""{newton_raphson('log(y) - 1', 2, variable='y')}""",
							)

							# Exponential Roots
							print(
							    """The root of exp(x) - 1 = 0 is""",
							    F"""{newton_raphson('exp(x) - 1', 10, precision=0.005)}""",
							)

							# Find root of cos(x)
							print(F"""The root of cos(x) = 0 is {newton_raphson('cos(x)', 0)}""")