

def 							__SCREAMING_SNAKE_CASE     (   a__    :   List[Any]			,a__    :   Tuple			,a__    :   Any			,a__    :   Any       )					->				Tuple:
	global f  # a global dp table for knapsack
	if f[i][j] < 0:
		if j < wt[i - 1]:
			__A :			Any		    =			mf_knapsack(i - 1			,a__			,a__			,a__       )
		else:
			__A :			str		    =			max(
			    mf_knapsack(i - 1			,a__			,a__			,a__       )			,mf_knapsack(i - 1			,a__			,a__			,j - wt[i - 1]       ) + val[i - 1]			,)
		__A :			Optional[Any]		    =			val
	return f[i][j]
def 							__SCREAMING_SNAKE_CASE     (   a__    :   int			,a__    :   List[str]			,a__    :   Optional[int]			,a__    :   int       )					->				Tuple:
	__A :			int		    =			[[0] * (w + 1) for _ in range(n + 1       )]

	for i in range(1			,n + 1       ):
		for w_ in range(1			,w + 1       ):
			if wt[i - 1] <= w_:
				__A :			Optional[int]		    =			max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]			,dp[i - 1][w_]       )
			else:
				__A :			int		    =			dp[i - 1][w_]

	return dp[n][w_], dp
def 							__SCREAMING_SNAKE_CASE     (   a__    :   int			,a__    :   list			,a__    :   list       )					->				Optional[Any]:
	if not (isinstance(a__			,(list, tuple)       ) and isinstance(a__			,(list, tuple)       )):
		raise ValueError(
		    """Both the weights and values vectors must be either lists or tuples"""       )

	__A :			Tuple		    =			len(a__       )
	if num_items != len(a__       ):
		__A :			Dict		    =			(
		    """The number of weights must be the same as the number of values.\n"""
		    f"""But got {num_items} weights and {len(a__       )} values"""
		)
		raise ValueError(a__       )
	for i in range(a__       ):
		if not isinstance(wt[i]			,a__       ):
			__A :			Optional[int]		    =			(
			    """All weights must be integers but got weight of """
			    f"""type {type(wt[i]       )} at index {i}"""
			)
			raise TypeError(a__       )

	__A    ,				__A :			Union[str, Any]		    =			knapsack(a__			,a__			,a__			,a__       )
	__A :			set		    =			set()
	_construct_solution(a__			,a__			,a__			,a__			,a__       )

	return optimal_val, example_optional_set
def 							__SCREAMING_SNAKE_CASE     (   a__    :   list			,a__    :   list			,a__    :   int			,a__    :   int			,a__    :   set       )					->				Any:
	# for the current item i at a maximum weight j to be part of an optimal subset,
	# the optimal value at (i, j) must be greater than the optimal value at (i-1, j).
	# where i - 1 means considering only the previous items at the given maximum weight
	if i > 0 and j > 0:
		if dp[i - 1][j] == dp[i][j]:
			_construct_solution(a__			,a__			,i - 1			,a__			,a__       )
		else:
			optimal_set.add(a__       )
			_construct_solution(a__			,a__			,i - 1			,j - wt[i - 1]			,a__       )


if __name__ == "__main__":
			UpperCAmelCase_							:					Optional[int]		 =   [3, 2, 4, 4]
			UpperCAmelCase_							:					Tuple		 =   [4, 3, 2, 3]
			UpperCAmelCase_							:					List[Any]		 =   4
			UpperCAmelCase_							:					List[str]		 =   6
			UpperCAmelCase_							:					Optional[int]		 =   [[0] * (w + 1)] + [[0] + [-1] * (w + 1) for _ in range(n + 1)]
			UpperCAmelCase_ ,		UpperCAmelCase_							:					List[Any]		 =   knapsack(w, wt, val, n)
			print(optimal_solution)
			print(mf_knapsack(n, wt, val, w))  # switched the n and w

			# testing the dynamic programming problem with example
			# the optimal subset for the above example are items 3 and 4
			UpperCAmelCase_ ,		UpperCAmelCase_							:					Union[str, Any]		 =   knapsack_with_example_solution(w, wt, val)
			assert optimal_solution == 8
			assert optimal_subset == {3, 4}
			print('''optimal_value = ''', optimal_solution)
			print('''An optimal subset corresponding to the optimal value''', optimal_subset)








from sympy import diff, lambdify, symbols
from sympy.functions import *  # noqa: F403
def 							__SCREAMING_SNAKE_CASE     (   a__    :   str			,a__    :   complex			,a__    :   str = "x"			,a__    :   float = 10**-10			,a__    :   int = 1			,)					->				complex:
	__A :			Tuple		    =			symbols(a__       )
	__A :			List[str]		    =			lambdify(a__			,a__       )
	__A :			Any		    =			lambdify(a__			,diff(a__			,a__       )       )

	__A :			Dict		    =			starting_point

	while True:
		if diff_function(a__       ) != 0:
			__A :			Optional[int]		    =			prev_guess - multiplicity * func(a__       ) / diff_function(
			    a__       )
		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 :			List[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)}""")





