

"""simple docstring"""



from __future__ import annotations

from collections.abc import Callable

SCREAMING_SNAKE_CASE_					:	Optional[int]							  =						list[list[float | int]]





def  _snake_case     (    UpperCAmelCase_  :					Matrix					,   UpperCAmelCase_  :					Matrix			):
     A__										=  len(UpperCAmelCase_			)
     A__										=  [[0 for _ in range(size + 1			)] for _ in range(UpperCAmelCase_			)]
     A__										=  42
     A__										=  42
     A__										=  42
     A__										=  42
     A__										=  42
     A__										=  42

     for row in range(UpperCAmelCase_			):
          for col in range(UpperCAmelCase_			):
               A__										=  matrix[row][col]

          A__										=  vector[row][0]

     A__										=  0
     A__										=  0
     while row < size and col < size:
          # pivoting
          A__										=  max((abs(augmented[rowa][col]			), rowa) for rowa in range(UpperCAmelCase_					,   UpperCAmelCase_			)			)[
              1
          ]
          if augmented[pivot_row][col] == 0:
               col += 1
               continue
          else:
               A__			,				A__										=  augmented[pivot_row], augmented[row]

          for rowa in range(row + 1					,   UpperCAmelCase_			):
               A__										=  augmented[rowa][col] / augmented[row][col]
               A__										=  0
               for cola in range(col + 1					,   size + 1			):
                    augmented[rowa][cola] -= augmented[row][cola] * ratio

          row += 1
          col += 1

     # back substitution
     for col in range(1					,   UpperCAmelCase_			):
          for row in range(UpperCAmelCase_			):
               A__										=  augmented[row][col] / augmented[col][col]
               for cola in range(UpperCAmelCase_					,   size + 1			):
                    augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
     return [
         [round(augmented[row][size] / augmented[row][row]					,   10			)] for row in range(UpperCAmelCase_			)
     ]





def  _snake_case     (    UpperCAmelCase_  :					list[int]			):
     A__										=  len(UpperCAmelCase_			)
     A__										=  [[0 for _ in range(UpperCAmelCase_			)] for _ in range(UpperCAmelCase_			)]
     A__										=  [[0] for _ in range(UpperCAmelCase_			)]
     A__										=  42
     A__										=  42
     A__										=  42
     A__										=  42

     for x_val, y_val in enumerate(UpperCAmelCase_			):
          for col in range(UpperCAmelCase_			):
               A__										=  (x_val + 1) ** (size - col - 1)
          A__										=  y_val

     A__										=  solve(UpperCAmelCase_					,   UpperCAmelCase_			)

     def interpolated_func(UpperCAmelCase_  :					int			) -> int:
          return sum(
              round(coeffs[x_val][0]			) * (var ** (size - x_val - 1))
              for x_val in range(UpperCAmelCase_			)			)

     return interpolated_func





def  _snake_case     (    UpperCAmelCase_  :					int			):
     return (
         1
         - variable
         + variable**2
         - variable**3
         + variable**4
         - variable**5
         + variable**6
         - variable**7
         + variable**8
         - variable**9
         + variable**10
     )





def  _snake_case     (    UpperCAmelCase_  :					Callable[[int], int] = question_function					,   UpperCAmelCase_  :					int = 10			):
     A__										=  [func(UpperCAmelCase_			) for x_val in range(1					,   order + 1			)]

     A__										=  [
         interpolate(data_points[:max_coeff]			) for max_coeff in range(1					,   order + 1			)
     ]

     A__										=  0
     A__										=  42
     A__										=  42

     for poly in polynomials:
          A__										=  1
          while func(UpperCAmelCase_			) == poly(UpperCAmelCase_			):
               x_val += 1

          ret += poly(UpperCAmelCase_			)

     return ret


if __name__ == "__main__":
 print(f"""{solution() = }""")









"""simple docstring"""



def  _snake_case     (    UpperCAmelCase_  :					int			):
     A__										=  generate_pascal_triangle(UpperCAmelCase_			)
     for row_idx in range(UpperCAmelCase_			):
          # Print left spaces
          for _ in range(num_rows - row_idx - 1			):
               print(end=""" """			)
          # Print row values
          for col_idx in range(row_idx + 1			):
               if col_idx != row_idx:
                    print(triangle[row_idx][col_idx]					,   end=""" """			)
               else:
                    print(triangle[row_idx][col_idx]					,   end=""""""			)
          print()





def  _snake_case     (    UpperCAmelCase_  :					int			):

     if not isinstance(UpperCAmelCase_					,   UpperCAmelCase_			):
          raise TypeError("""The input value of 'num_rows' should be 'int'"""			)

     if num_rows == 0:
          return []
     elif num_rows < 0:
          raise ValueError(
              """The input value of 'num_rows' should be greater than or equal to 0"""			)

     A__										=  []
     for current_row_idx in range(UpperCAmelCase_			):
          A__										=  populate_current_row(UpperCAmelCase_					,   UpperCAmelCase_			)
          triangle.append(UpperCAmelCase_			)
     return triangle





def  _snake_case     (    UpperCAmelCase_  :					list[list[int]]					,   UpperCAmelCase_  :					int			):
     A__										=  [-1] * (current_row_idx + 1)
     # first and last elements of current row are equal to 1
     A__			,				A__										=  1, 1
     for current_col_idx in range(1					,   UpperCAmelCase_			):
          calculate_current_element(
              UpperCAmelCase_					,   UpperCAmelCase_					,   UpperCAmelCase_					,   UpperCAmelCase_			)
     return current_row





def  _snake_case     (    UpperCAmelCase_  :					list[list[int]]					,   UpperCAmelCase_  :					list[int]					,   UpperCAmelCase_  :					int					,   UpperCAmelCase_  :					int					,   ):
     A__										=  triangle[current_row_idx - 1][current_col_idx - 1]
     A__										=  triangle[current_row_idx - 1][current_col_idx]
     A__										=  above_to_left_elt + above_to_right_elt





def  _snake_case     (    UpperCAmelCase_  :					int			):

     if not isinstance(UpperCAmelCase_					,   UpperCAmelCase_			):
          raise TypeError("""The input value of 'num_rows' should be 'int'"""			)

     if num_rows == 0:
          return []
     elif num_rows < 0:
          raise ValueError(
              """The input value of 'num_rows' should be greater than or equal to 0"""			)

     A__										=  [[1]]

     for row_index in range(1					,   UpperCAmelCase_			):
          A__										=  [0] + result[-1] + [0]
          A__										=  row_index + 1
          # Calculate the number of distinct elements in a row
          A__										=  sum(divmod(UpperCAmelCase_					,   2			)			)
          A__										=  [
              temp_row[i - 1] + temp_row[i] for i in range(1					,   distinct_elements + 1			)
          ]
          A__										=  row_first_half[: (row_index + 1) // 2]
          row_second_half.reverse()
          A__										=  row_first_half + row_second_half
          result.append(UpperCAmelCase_			)

     return result





def  _snake_case     (    ):
     from collections.abc import Callable
     from timeit import timeit

     def benchmark_a_function(UpperCAmelCase_  :					Callable					,   UpperCAmelCase_  :					int			) -> None:
          A__										=  F"""{func.__name__}({value})"""
          A__										=  timeit(F"""__main__.{call}"""					,   setup="""import __main__"""			)
          # print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
          print(F"""{call:38} -- {timing:.4f} seconds"""			)

     for value in range(15			):  # (1, 7, 14):
          for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
               benchmark_a_function(UpperCAmelCase_					,   UpperCAmelCase_			)
          print()


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 benchmark()






