

import math
class 					A_   :


    '''simple docstring'''






    def __init__(self     ,      lowercase__  )     ->      Optional[Any]:
        __UpperCAmelCase									=			len(lowercase__  )
        __UpperCAmelCase									=			[0] * (
            4 * self.N
        )  # approximate the overall size of segment tree with array N
        self.build(1     ,      0     ,      self.N - 1  )


    def      lowerCAmelCase_						(self     ,      lowercase__  )     ->      Optional[Any]:
        return idx * 2


    def      lowerCAmelCase_						(self     ,      lowercase__  )     ->      int:
        return idx * 2 + 1


    def      lowerCAmelCase_						(self     ,      lowercase__     ,      lowercase__     ,      lowercase__  )     ->      str:  # noqa: E741
        if l == r:
            __UpperCAmelCase									=			A[l]
        else:
            __UpperCAmelCase									=			(l + r) // 2
            self.build(self.left(lowercase__  )     ,      lowercase__     ,      lowercase__  )
            self.build(self.right(lowercase__  )     ,      mid + 1     ,      lowercase__  )
            __UpperCAmelCase									=			max(self.st[self.left(lowercase__  )]     ,      self.st[self.right(lowercase__  )]  )


    def      lowerCAmelCase_						(self     ,      lowercase__     ,      lowercase__     ,      lowercase__  )     ->      Union[str, Any]:
        return self.update_recursive(1     ,      0     ,      self.N - 1     ,      a - 1     ,      b - 1     ,      lowercase__  )


    def      lowerCAmelCase_						(self     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__  )     ->      Optional[Any]:  # noqa: E741
        if r < a or l > b:
            return True
        if l == r:
            __UpperCAmelCase									=			val
            return True
        __UpperCAmelCase									=			(l + r) // 2
        self.update_recursive(self.left(lowercase__  )     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__  )
        self.update_recursive(self.right(lowercase__  )     ,      mid + 1     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__  )
        __UpperCAmelCase									=			max(self.st[self.left(lowercase__  )]     ,      self.st[self.right(lowercase__  )]  )
        return True


    def      lowerCAmelCase_						(self     ,      lowercase__     ,      lowercase__  )     ->      List[Any]:
        return self.query_recursive(1     ,      0     ,      self.N - 1     ,      a - 1     ,      b - 1  )


    def      lowerCAmelCase_						(self     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__  )     ->      Dict:  # noqa: E741
        if r < a or l > b:
            return -math.inf
        if l >= a and r <= b:
            return self.st[idx]
        __UpperCAmelCase									=			(l + r) // 2
        __UpperCAmelCase									=			self.query_recursive(self.left(lowercase__  )     ,      lowercase__     ,      lowercase__     ,      lowercase__     ,      lowercase__  )
        __UpperCAmelCase									=			self.query_recursive(self.right(lowercase__  )     ,      mid + 1     ,      lowercase__     ,      lowercase__     ,      lowercase__  )
        return max(lowercase__     ,      lowercase__  )






    def      lowerCAmelCase_						(self  )     ->      Optional[int]:
        __UpperCAmelCase									=			[]
        for i in range(1     ,      N + 1  ):
            show_list += [self.query(lowercase__     ,      lowercase__  )]
        print(lowercase__  )


if __name__ == "__main__":
       A_		:			Optional[Any]										=   [1, 2, -4, 7, 3, -5, 6, 11, -20, 9, 14, 15, 5, 2, -8]
       A_		:			Optional[Any]										=   15
       A_		:			int										=   SegmentTree(A)
       print(segt.query(4, 6))
       print(segt.query(7, 11))
       print(segt.query(7, 12))
       segt.update(1, 3, 111)
       print(segt.query(1, 15))
       segt.update(7, 8, 235)
       segt.show_data()






import numpy as np


def       __a       (   SCREAMING_SNAKE_CASE      )      ->     Tuple:






    '''simple docstring'''
    __UpperCAmelCase					,     __UpperCAmelCase									=			a.shape
    __UpperCAmelCase									=			min(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE      )
    __UpperCAmelCase									=			np.eye(SCREAMING_SNAKE_CASE      )
    __UpperCAmelCase									=			a.copy()

    for k in range(t - 1      ):
        # select a column of modified matrix A':
        __UpperCAmelCase									=			r[k:, [k]]
        # construct first basis vector
        __UpperCAmelCase									=			np.zeros_like(SCREAMING_SNAKE_CASE      )
        __UpperCAmelCase									=			1.0
        # determine scaling factor
        __UpperCAmelCase									=			np.linalg.norm(SCREAMING_SNAKE_CASE      )
        # construct vector v for Householder reflection
        __UpperCAmelCase									=			x + np.sign(x[0]      ) * alpha * ea
        v /= np.linalg.norm(SCREAMING_SNAKE_CASE      )

        # construct the Householder matrix
        __UpperCAmelCase									=			np.eye(m - k      ) - 2.0 * v @ v.T
        # pad with ones and zeros as necessary
        __UpperCAmelCase									=			np.block([[np.eye(SCREAMING_SNAKE_CASE      ), np.zeros((k, m - k)      )], [np.zeros((m - k, k)      ), q_k]]      )

        __UpperCAmelCase									=			q @ q_k.T
        __UpperCAmelCase									=			q_k @ r

    return q, r


if __name__ == "__main__":
       import doctest

       doctest.testmod()




