

def 					__lowerCamelCase	( UpperCamelCase__      ):







							'''simple docstring'''






							return 10 - x * x





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							if equation(UpperCamelCase__      ) * equation(UpperCamelCase__      ) >= 0:
														raise ValueError('Wrong space!'      )

							snake_case_							=    a
							while (b - a) >= 0.01:
														# Find middle point
														snake_case_							=    (a + b) / 2
														# Check if middle point is root
														if equation(UpperCamelCase__      ) == 0.0:
																					break
														# Decide the side to repeat the steps
														if equation(UpperCamelCase__      ) * equation(UpperCamelCase__      ) < 0:
																					snake_case_							=    c
														else:
																					snake_case_							=    c
							return c


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					print(bisection(-2, 5))
					print(bisection(0, 6))







def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							global f  # a global dp table for knapsack
							if f[i][j] < 0:
														if j < wt[i - 1]:
																					snake_case_							=    mf_knapsack(i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )
														else:
																					snake_case_							=    max(
																					    mf_knapsack(i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )   ,							mf_knapsack(i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							j - wt[i - 1]      ) + val[i - 1]   ,							)
														snake_case_							=    val
							return f[i][j]





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							snake_case_							=    [[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_:
																												snake_case_							=    max(val[i - 1] + dp[i - 1][w_ - wt[i - 1]]   ,							dp[i - 1][w_]      )
																					else:
																												snake_case_							=    dp[i - 1][w_]

							return dp[n][w_], dp





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							if not (isinstance(UpperCamelCase__   ,							(list, tuple)      ) and isinstance(UpperCamelCase__   ,							(list, tuple)      )):
														raise ValueError(
														    'Both the weights and values vectors must be either lists or tuples'      )

							snake_case_							=    len(UpperCamelCase__      )
							if num_items != len(UpperCamelCase__      ):
														snake_case_							=    (
														    'The number of weights must be the same as the number of values.\n'
														    F'''But got {num_items} weights and {len(UpperCamelCase__      )} values'''
														)
														raise ValueError(UpperCamelCase__      )
							for i in range(UpperCamelCase__      ):
														if not isinstance(wt[i]   ,							UpperCamelCase__      ):
																					snake_case_							=    (
																					    'All weights must be integers but got weight of '
																					    F'''type {type(wt[i]      )} at index {i}'''
																					)
																					raise TypeError(UpperCamelCase__      )

							snake_case_				,    snake_case_							=    knapsack(UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )
							snake_case_							=    set()
							_construct_solution(UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      )

							return optimal_val, example_optional_set





def 					__lowerCamelCase	( UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__   ,							UpperCamelCase__      ):







							'''simple docstring'''






							if i > 0 and j > 0:
														if dp[i - 1][j] == dp[i][j]:
																					_construct_solution(UpperCamelCase__   ,							UpperCamelCase__   ,							i - 1   ,							UpperCamelCase__   ,							UpperCamelCase__      )
														else:
																					optimal_set.add(UpperCamelCase__      )
																					_construct_solution(UpperCamelCase__   ,							UpperCamelCase__   ,							i - 1   ,							j - wt[i - 1]   ,							UpperCamelCase__      )


if __name__ == "__main__":
					_UpperCAmelCase		:   Tuple										=       [3, 2, 4, 4]
					_UpperCAmelCase		:   Optional[Any]										=       [4, 3, 2, 3]
					_UpperCAmelCase		:   List[str]										=       4
					_UpperCAmelCase		:   str										=       6
					_UpperCAmelCase		:   Tuple										=       [[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		:   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)




