
def        __UpperCamelCase					(    ):
 return [list(range(1_0_0_0 - i  ,       -1_0_0_0 - i  ,       -1  )  ) for i in range(1_0_0_0  )]


lowercase__          =generate_large_matrix()
lowercase__          =(
    [[4, 3, 2, -1], [3, 2, 1, -1], [1, 1, -1, -2], [-1, -1, -2, -3]],
    [[3, 2], [1, 0]],
    [[7, 7, 6]],
    [[7, 7, 6], [-1, -2, -3]],
    grid,
)




def        __UpperCamelCase					(    lowerCAmelCase__							:    Any  ):
 assert all(row == sorted(a__  ,       reverse=a__  ) for row in grid  )
 assert all(list(a__  ) == sorted(a__  ,       reverse=a__  ) for col in zip(*a__  )  )




def        __UpperCamelCase					(    lowerCAmelCase__							:    Union[str, Any]  ):
 __a			:  str		    =  0
 __a			:  List[Any]		    =  len(a__  ) - 1

 # Edge cases such as no values or all numbers are negative.
 if not array or array[0] < 0:
  return 0

 while right + 1 > left:
  __a			:  Optional[Any]		    =  (left + right) // 2
  __a			:  Optional[Any]		    =  array[mid]

  # Num must be negative and the index must be greater than or equal to 0.
  if num < 0 and array[mid - 1] >= 0:
   return mid

  if num >= 0:
   __a			:  List[str]		    =  mid + 1
  else:
   __a			:  str		    =  mid - 1
  # No negative numbers so return the last index of the array + 1 which is the length.
 return len(a__  )




def        __UpperCamelCase					(    lowerCAmelCase__							:    Tuple  ):
 __a			:  Dict		    =  0
 __a			:  List[Any]		    =  len(grid[0]  )

 for i in range(len(a__  )  ):
  __a			:  Optional[int]		    =  find_negative_index(grid[i][:bound]  )
  total += bound
 return (len(a__  ) * len(grid[0]  )) - total




def        __UpperCamelCase					(    lowerCAmelCase__							:    Any  ):
 return len([number for row in grid for number in row if number < 0]  )




def        __UpperCamelCase					(    lowerCAmelCase__							:    Union[str, Any]  ):
 __a			:  Any		    =  0
 for row in grid:
  for i, number in enumerate(a__  ):
   if number < 0:
    total += len(a__  ) - i
    break
 return total




def        __UpperCamelCase					(    ):
 from timeit import timeit

 print('''Running benchmarks'''  )
 __a			:  Any		    =  (
     '''from __main__ import count_negatives_binary_search, '''
     '''count_negatives_brute_force, count_negatives_brute_force_with_break, grid'''
 )
 for func in (
     "count_negatives_binary_search",  # took 0.7727 seconds
     "count_negatives_brute_force_with_break",  # took 4.6505 seconds
     "count_negatives_brute_force",  # took 12.8160 seconds
 ):
  __a			:  Dict		    =  timeit(f"{func}(grid=grid)"  ,       setup=a__  ,       number=5_0_0  )
  print(f"{func}() took {time:0.4f} seconds"  )


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      benchmark()


def        __UpperCamelCase					(    lowerCAmelCase__							:    list[list[int | float]]  ):
 __a			:  int		    =  len(lowerCAmelCase__  )
 __a			:  Dict		    =  len(matrix[0]  )
 __a			:  Union[str, Any]		    =  min(lowerCAmelCase__  ,       lowerCAmelCase__  )

 for row in range(lowerCAmelCase__  ):
  # Check if diagonal element is not zero
  if matrix[row][row] != 0:
   # Eliminate all the elements below the diagonal
   for col in range(row + 1  ,       lowerCAmelCase__  ):
    __a			:  Dict		    =  matrix[col][row] / matrix[row][row]
    for i in range(lowerCAmelCase__  ,       lowerCAmelCase__  ):
     matrix[col][i] -= multiplier * matrix[row][i]
  else:
   # Find a non-zero diagonal element to swap rows
   __a			:  Optional[int]		    =  True
   for i in range(row + 1  ,       lowerCAmelCase__  ):
    if matrix[i][row] != 0:
     __a    ,	__a			:  Any		    =  matrix[i], matrix[row]
     __a			:  Union[str, Any]		    =  False
     break
   if reduce:
    rank -= 1
    for i in range(lowerCAmelCase__  ):
     __a			:  Optional[Any]		    =  matrix[i][rank]

            # Reduce the row pointer by one to stay on the same row
   row -= 1

 return rank


if __name__ == "__main__":
      import doctest

      doctest.testmod()
