




'''simple docstring'''




import math as m




def 						snake_case__    ( lowerCamelCase__				:   List[Any]			,		lowerCamelCase__				:   Tuple			,		lowerCamelCase__				:   List[Any]=0.001			)   ->   int:
					return (f(a + h			) - f(a - h			)) / (2 * h)




def 						snake_case__    ( lowerCamelCase__				:   Optional[int]			,		lowerCamelCase__				:   Tuple=0			,		lowerCamelCase__				:   Union[str, Any]=1_0_0			,		lowerCamelCase__				:   List[Any]=0.0001			,		lowerCamelCase__				:   Any=1e-6			,		lowerCamelCase__				:   Dict=False			)   ->   Any:
					A_ :     Union[str, Any] =							xa  # set the initial guess
					A_ :     Tuple =							[a]
					A_ :     Optional[Any] =							abs(f(lowercase__			)			)
					A_ :     Any =							lambda lowerCamelCase__			: calc_derivative(lowercase__			,		lowercase__			,		h=lowercase__			)  # noqa: E731  Derivative of f(x)
					for _ in range(lowercase__			):
										if fa(lowercase__			) == 0:
															raise ValueError('''No converging solution found'''			)
										A_ :     str =							a - f(lowercase__			) / fa(lowercase__			)  # Calculate the next estimate
										if logsteps:
															steps.append(lowercase__			)
										if error < maxerror:
															break
					else:
										raise ValueError('''Iteration limit reached, no converging solution found'''			)
					if logsteps:
										# If logstep is true, then log intermediate steps
										return a, error, steps
					return a, error


if __name__ == "__main__":
							from matplotlib import pyplot as plt

							snake_case__          =						lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
							snake_case__       ,						snake_case__       ,						snake_case__          =						newton_raphson(
							    f, xa=10, maxiter=10_00, step=1e-6, logsteps=True
							)
							plt.plot([abs(f(x)) for x in steps])
							plt.xlabel("""step""")
							plt.ylabel("""error""")
							plt.show()
							print(F'solution = {{{solution:f}}}, error = {{{error:f}}}')









'''simple docstring'''




from __future__ import annotations

snake_case__          =						[2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]






class     UpperCamelCase_     :







						"""simple docstring"""


						def __init__(     self      :       Dict							, _lowerCamelCase      :       int    ):





											"""simple docstring"""







											A_ :     int =							-1
											A_ :     int =							-1
											A_ :     list[int] =							[-1] * length
											A_ :     Node | None =							None
											A_ :     Node | None =							None







						def __repr__(     self      :       List[str]    ):





											"""simple docstring"""







											return f'Node(min_value={self.minn} max_value={self.maxx})'






def 						snake_case__    ( lowerCamelCase__				:   list[int]			)   ->   Node | None:
					A_ :     Optional[Any] =							Node(len(lowerCamelCase__			)			)
					A_ ,A_ :     str =							min(lowerCamelCase__			), max(lowerCamelCase__			)
					# Leaf node case where the node contains only one unique value
					if root.minn == root.maxx:
										return root
					A_ :     Optional[int] =							(root.minn + root.maxx) // 2

					A_ :     list[int] =							[]
					A_ :     list[int] =							[]

					for index, num in enumerate(lowerCamelCase__			):
										if num <= pivot:
															left_arr.append(lowerCamelCase__			)
										else:
															right_arr.append(lowerCamelCase__			)
										A_ :     Union[str, Any] =							len(lowerCamelCase__			)
					A_ :     Optional[int] =							build_tree(lowerCamelCase__			)
					A_ :     Optional[int] =							build_tree(lowerCamelCase__			)
					return root




def 						snake_case__    ( lowerCamelCase__				:   Node | None			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			)   ->   int:
					if index < 0 or node is None:
										return 0
					# Leaf node cases
					if node.minn == node.maxx:
										return index + 1 if node.minn == num else 0
					A_ :     List[Any] =							(node.minn + node.maxx) // 2
					if num <= pivot:
										# go the left subtree and map index to the left subtree
										return rank_till_index(node.left			,		lowerCamelCase__			,		node.map_left[index] - 1			)
					else:
										# go to the right subtree and map index to the right subtree
										return rank_till_index(node.right			,		lowerCamelCase__			,		index - node.map_left[index]			)




def 						snake_case__    ( lowerCamelCase__				:   Node | None			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			)   ->   int:
					if start > end:
										return 0
					A_ :     Union[str, Any] =							rank_till_index(lowerCamelCase__			,		lowerCamelCase__			,		lowerCamelCase__			)
					A_ :     Union[str, Any] =							rank_till_index(lowerCamelCase__			,		lowerCamelCase__			,		start - 1			)
					return rank_till_end - rank_before_start




def 						snake_case__    ( lowerCamelCase__				:   Node | None			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			)   ->   int:
					if index > (end - start) or start > end or node is None:
										return -1
					# Leaf node case
					if node.minn == node.maxx:
										return node.minn
					# Number of elements in the left subtree in interval [start, end]
					A_ :     List[str] =							node.map_left[end] - (
					    node.map_left[start - 1] if start else 0
					)
					if num_elements_in_left_tree > index:
										return quantile(
										    node.left			,		lowerCamelCase__			,		(node.map_left[start - 1] if start else 0)			,		node.map_left[end] - 1			,		)
					else:
										return quantile(
										    node.right			,		index - num_elements_in_left_tree			,		start - (node.map_left[start - 1] if start else 0)			,		end - node.map_left[end]			,		)




def 						snake_case__    ( lowerCamelCase__				:   Node | None			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			,		lowerCamelCase__				:   int			)   ->   int:
					if (
					    start > end
					    or node is None
					    or start_num > end_num
					    or node.minn > end_num
					    or node.maxx < start_num
					):
										return 0
					if start_num <= node.minn and node.maxx <= end_num:
										return end - start + 1
					A_ :     int =							range_counting(
					    node.left			,		(node.map_left[start - 1] if start else 0)			,		node.map_left[end] - 1			,		lowerCamelCase__			,		lowerCamelCase__			,		)
					A_ :     List[str] =							range_counting(
					    node.right			,		start - (node.map_left[start - 1] if start else 0)			,		end - node.map_left[end]			,		lowerCamelCase__			,		lowerCamelCase__			,		)
					return left + right


if __name__ == "__main__":
							import doctest

							doctest.testmod()



