
def        A__					(						lowerCamelCase						)  ->	list[int]:
				UpperCamelCase_:							List[Any]	=						[0 for i in range(len(lowerCamelCase						)						)]

				# initialize interval's left pointer and right pointer
				UpperCamelCase_,   UpperCamelCase_:							str	=						0, 0

				for i in range(1     ,    len(lowerCamelCase						)						):
								# case when current index is inside the interval
								if i <= right_pointer:
												UpperCamelCase_:							Optional[int]	=						min(right_pointer - i + 1     ,    z_result[i - left_pointer]						)
												UpperCamelCase_:							Dict	=						min_edge

								while go_next(lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase						):
												z_result[i] += 1

								# if new index's result gives us more right interval,
								# we've to update left_pointer and right_pointer
								if i + z_result[i] - 1 > right_pointer:
												UpperCamelCase_,   UpperCamelCase_:							List[Any]	=						i, i + z_result[i] - 1

				return z_result







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase						)  ->	bool:
				return i + z_result[i] < len(lowerCamelCase						) and s[z_result[i]] == s[i + z_result[i]]







def        A__					(						lowerCamelCase     ,    lowerCamelCase						)  ->	int:
				UpperCamelCase_:							List[Any]	=						0
				# concatenate 'pattern' and 'input_str' and call z_function
				# with concatenated string
				UpperCamelCase_:							Dict	=						z_function(pattern + input_str						)

				for val in z_result:
								# if value is greater then length of the pattern string
								# that means this index is starting position of substring
								# which is equal to pattern string
								if val >= len(lowerCamelCase						):
												answer += 1

				return answer


if __name__ == "__main__":
							import doctest

							doctest.testmod()








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







def        A__					(						lowerCamelCase     ,    lowerCamelCase     ,    lowerCamelCase = "x"     ,    lowerCamelCase = 10**-10     ,    lowerCamelCase = 1     ,    )  ->	complex:
				UpperCamelCase_:							Optional[Any]	=						symbols(lowerCamelCase						)
				UpperCamelCase_:							int	=						lambdify(lowerCamelCase     ,    lowerCamelCase						)
				UpperCamelCase_:							Optional[Any]	=						lambdify(lowerCamelCase     ,    diff(lowerCamelCase     ,    lowerCamelCase						)						)

				UpperCamelCase_:							Tuple	=						starting_point

				while True:
								if diff_function(lowerCamelCase						) != 0:
												UpperCamelCase_:							List[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

								UpperCamelCase_:							Any	=						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)}""")






