

"""simple docstring"""




from __future__ import annotations

SCREAMING_SNAKE_CASE_  :      Union[str, Any]							=      [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]


class 		_A							:





      def __init__(	self     ,    SCREAMING_SNAKE_CASE__		) ->       None:
         lowerCamelCase__						       =      -1
         lowerCamelCase__						       =      -1
         lowerCamelCase__						       =      [-1] * length
         lowerCamelCase__						       =      None
         lowerCamelCase__						       =      None





      def __repr__(	self		) ->       str:
         return f'Node(min_value={self.minn} max_value={self.maxx})'






def 						UpperCAmelCase__ (					A__ )   ->   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 						UpperCAmelCase__ (					A__						,     A__						,     A__ )   ->   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 						UpperCAmelCase__ (					A__						,     A__						,     A__						,     A__ )   ->   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 						UpperCAmelCase__ (					A__						,     A__						,     A__						,     A__ )   ->   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 						UpperCAmelCase__ (					A__						,     A__						,     A__						,     A__						,     A__ )   ->   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()







"""simple docstring"""




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



def 						UpperCAmelCase__ (					A__						,     A__ )   ->   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 						UpperCAmelCase__ (					A__						,     A__ )   ->   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()




