'''simple docstring'''




import numpy as np
from numpy import floataa
from numpy.typing import NDArray






def     snake_case__							(      _A: NDArray[floataa]				,       _A: NDArray[floataa]			)   ->     NDArray[floataa]:

		'''simple docstring'''


		lowerCAmelCase ,					lowerCAmelCase        =  np.shape(_A			)

		lowerCAmelCase        =  np.zeros((rows, 1)				,       dtype=_A			)
		for row in reversed(range(_A			)			):
				lowerCAmelCase        =  0
				for col in range(row + 1				,       _A			):
						total += coefficients[row, col] * x[col]

				lowerCAmelCase        =  (vector[row] - total) / coefficients[row, row]

		return x






def     snake_case__							(      _A: NDArray[floataa]				,       _A: NDArray[floataa]			)   ->     NDArray[floataa]:

		'''simple docstring'''


		lowerCAmelCase ,					lowerCAmelCase        =  np.shape(_A			)
		if rows != columns:
				return np.array(()				,       dtype=_A			)

		# augmented matrix
		lowerCAmelCase        =  np.concatenate((coefficients, vector)				,       axis=1			)
		lowerCAmelCase        =  augmented_mat.astype("""float64"""			)

		# scale the matrix leaving it triangular
		for row in range(rows - 1			):
				lowerCAmelCase        =  augmented_mat[row, row]
				for col in range(row + 1				,       _A			):
						lowerCAmelCase        =  augmented_mat[col, row] / pivot
						augmented_mat[col, :] -= factor * augmented_mat[row, :]

		lowerCAmelCase        =  retroactive_resolution(
		    augmented_mat[:, 0:columns]				,       augmented_mat[:, columns : columns + 1]			)

		return x


if __name__ == "__main__":
						import doctest

						doctest.testmod()





'''simple docstring'''




from __future__ import annotations

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






class     a__:


			'''simple docstring'''






			def __init__(       self	,   __lowerCAmelCase):




					"""simple docstring"""


					lowerCAmelCase        =  -1
					lowerCAmelCase        =  -1
					lowerCAmelCase        =  [-1] * length
					lowerCAmelCase        =  None
					lowerCAmelCase        =  None







			def __repr__(       self):




					"""simple docstring"""


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







def     snake_case__							(      _A: list[int]			)   ->     Node | None:

		'''simple docstring'''


		lowerCAmelCase        =  Node(len(_A			)			)
		lowerCAmelCase ,					lowerCAmelCase        =  min(_A			), max(_A			)
		# Leaf node case where the node contains only one unique value
		if root.minn == root.maxx:
				return root
		lowerCAmelCase        =  (root.minn + root.maxx) // 2

		lowerCAmelCase        =  []
		lowerCAmelCase        =  []

		for index, num in enumerate(_A			):
				if num <= pivot:
						left_arr.append(_A			)
				else:
						right_arr.append(_A			)
				lowerCAmelCase        =  len(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		return root






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		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
		lowerCAmelCase        =  (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				,       _A				,       node.map_left[index] - 1			)
		else:
				# go to the right subtree and map index to the right subtree
				return rank_till_index(node.right				,       _A				,       index - node.map_left[index]			)






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if start > end:
				return 0
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       _A			)
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       start - 1			)
		return rank_till_end - rank_before_start






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		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]
		lowerCAmelCase        =  node.map_left[end] - (
		    node.map_left[start - 1] if start else 0
		)
		if num_elements_in_left_tree > index:
				return quantile(
				    node.left				,       _A				,       (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__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		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
		lowerCAmelCase        =  range_counting(
		    node.left				,       (node.map_left[start - 1] if start else 0)				,       node.map_left[end] - 1				,       _A				,       _A				,       )
		lowerCAmelCase        =  range_counting(
		    node.right				,       start - (node.map_left[start - 1] if start else 0)				,       end - node.map_left[end]				,       _A				,       _A				,       )
		return left + right


if __name__ == "__main__":
						import doctest

						doctest.testmod()




