

"""simple docstring"""






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



def   lowercase_	(    _lowercase       :		Optional[int]							,      _lowercase       :		str							,      _lowercase       :		List[str] = "x"							,      _lowercase       :		List[Any] = 10**-10							,      _lowercase       :		Optional[int] = 1							,      ):




					'''simple docstring'''






					UpperCAmelCase			:   str      =				symbols(_lowerCamelCase	)
					UpperCAmelCase			:   int      =				lambdify(_lowerCamelCase							,      _lowerCamelCase	)
					UpperCAmelCase			:   List[str]      =				lambdify(_lowerCamelCase							,      diff(_lowerCamelCase							,      _lowerCamelCase	)	)

					UpperCAmelCase			:   Optional[int]      =				starting_point

					while True:
										if diff_function(_lowerCamelCase	) != 0:
															UpperCAmelCase			:   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

										UpperCAmelCase			:   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", 1_0, precision=0.0_05)}''',
	)

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





"""simple docstring"""






def   lowercase_	(    _lowercase       :		List[str]							,      _lowercase       :		Tuple							,      _lowercase       :		int							,      _lowercase       :		Optional[Any]	):




					'''simple docstring'''






					UpperCAmelCase			:   int      =				[False] * len(_lowercase	)
					UpperCAmelCase			:   List[str]      =				[]
					queue.append(_lowercase	)
					UpperCAmelCase			:   List[Any]      =				True

					while queue:
										UpperCAmelCase			:   Any      =				queue.pop(0	)
										for ind in range(len(graph[u]	)	):
															if visited[ind] is False and graph[u][ind] > 0:
																				queue.append(_lowercase	)
																				UpperCAmelCase			:   Optional[int]      =				True
																				UpperCAmelCase			:   int      =				u

					return visited[t]



def   lowercase_	(    _lowercase       :		Tuple							,      _lowercase       :		Dict							,      _lowercase       :		Optional[int]	):




					'''simple docstring'''






					UpperCAmelCase			:   List[Any]      =				[-1] * (len(_lowercase	))
					UpperCAmelCase			:   List[Any]      =				0
					while bfs(_lowercase							,      _lowercase							,      _lowercase							,      _lowercase	):
										UpperCAmelCase			:   List[str]      =				float("Inf"	)
										UpperCAmelCase			:   str      =				sink

										while s != source:
															# Find the minimum value in select path
															UpperCAmelCase			:   Tuple      =				min(_lowercase							,      graph[parent[s]][s]	)
															UpperCAmelCase			:   Optional[Any]      =				parent[s]

										max_flow += path_flow
										UpperCAmelCase			:   List[str]      =				sink

										while v != source:
															UpperCAmelCase			:   Dict      =				parent[v]
															graph[u][v] -= path_flow
															graph[v][u] += path_flow
															UpperCAmelCase			:   Dict      =				parent[v]
					return max_flow


snake_case_   :   Optional[Any]        =       [
    [0, 1_6, 1_3, 0, 0, 0],
    [0, 0, 1_0, 1_2, 0, 0],
    [0, 4, 0, 0, 1_4, 0],
    [0, 0, 9, 0, 0, 2_0],
    [0, 0, 0, 7, 0, 4],
    [0, 0, 0, 0, 0, 0],
]

snake_case_     ,      snake_case_   :   str        =       0, 5




print(ford_fulkerson(graph, source, sink))


