




"""simple docstring"""


from __future__ import annotations

from collections.abc import Callable

_A													=      list[list[float | int]]



def 	lowercase_				(			__UpperCAmelCase    ,		__UpperCAmelCase				)						->    Matrix:
      lowerCAmelCase__      :			int   =	len(__UpperCAmelCase				)
      lowerCAmelCase__      :			Matrix   =	[[0 for _ in range(size + 1				)] for _ in range(__UpperCAmelCase				)]
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			float

      for row in range(__UpperCAmelCase				):
            for col in range(__UpperCAmelCase				):
                  lowerCAmelCase__      :			Tuple   =	matrix[row][col]

            lowerCAmelCase__      :			List[Any]   =	vector[row][0]

      lowerCAmelCase__      :			List[Any]   =	0
      lowerCAmelCase__      :			Dict   =	0
      while row < size and col < size:
            # pivoting
            lowerCAmelCase__      :			Tuple   =	max((abs(augmented[rowa][col]				), rowa) for rowa in range(__UpperCAmelCase    ,		__UpperCAmelCase				)				)[
                1
            ]
            if augmented[pivot_row][col] == 0:
                  col += 1
                  continue
            else:
                  lowerCAmelCase__ ,   lowerCAmelCase__      :			Dict   =	augmented[pivot_row], augmented[row]

            for rowa in range(row + 1    ,		__UpperCAmelCase				):
                  lowerCAmelCase__      :			Union[str, Any]   =	augmented[rowa][col] / augmented[row][col]
                  lowerCAmelCase__      :			List[Any]   =	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				):
                  lowerCAmelCase__      :			Any   =	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 	lowercase_				(			__UpperCAmelCase				)						->    Callable[[int], int]:
      lowerCAmelCase__      :			int   =	len(__UpperCAmelCase				)
      lowerCAmelCase__      :			Matrix   =	[[0 for _ in range(__UpperCAmelCase				)] for _ in range(__UpperCAmelCase				)]
      lowerCAmelCase__      :			Matrix   =	[[0] for _ in range(__UpperCAmelCase				)]
      lowerCAmelCase__      :			Matrix
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			int
      lowerCAmelCase__      :			int

      for x_val, y_val in enumerate(__UpperCAmelCase				):
            for col in range(__UpperCAmelCase				):
                  lowerCAmelCase__      :			List[Any]   =	(x_val + 1) ** (size - col - 1)
            lowerCAmelCase__      :			int   =	y_val

      lowerCAmelCase__      :			str   =	solve(__UpperCAmelCase    ,		__UpperCAmelCase				)

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

      return interpolated_func



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



def 	lowercase_				(			__UpperCAmelCase = question_function    ,		__UpperCAmelCase = 10				)						->    int:
      lowerCAmelCase__      :			list[int]   =	[func(__UpperCAmelCase				) for x_val in range(1    ,		order + 1				)]

      lowerCAmelCase__      :			list[Callable[[int], int]]   =	[
          interpolate(data_points[:max_coeff]				) for max_coeff in range(1    ,		order + 1				)
      ]

      lowerCAmelCase__      :			int   =	0
      lowerCAmelCase__      :			Callable[[int], int]
      lowerCAmelCase__      :			int

      for poly in polynomials:
            lowerCAmelCase__      :			Dict   =	1
            while func(__UpperCAmelCase				) == poly(__UpperCAmelCase				):
                  x_val += 1

            ret += poly(__UpperCAmelCase				)

      return ret


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











"""simple docstring"""


from __future__ import annotations

import math
from collections import Counter
from string import ascii_lowercase



def 	lowercase_				(			__UpperCAmelCase				)						->    None:
      lowerCAmelCase__ ,   lowerCAmelCase__      :			int   =	analyze_text(__UpperCAmelCase				)
      lowerCAmelCase__      :			Optional[Any]   =	list(""" """ + ascii_lowercase				)
      # what is our total sum of probabilities.
      lowerCAmelCase__      :			List[str]   =	sum(single_char_strings.values()				)

      # one length string
      lowerCAmelCase__      :			List[str]   =	0
      # for each alpha we go in our dict and if it is in it we calculate entropy
      for ch in my_alphas:
            if ch in single_char_strings:
                  lowerCAmelCase__      :			List[Any]   =	single_char_strings[ch]
                  lowerCAmelCase__      :			int   =	my_str / all_sum
                  my_fir_sum += prob * math.loga(__UpperCAmelCase				)  # entropy formula.

    # print entropy
      print(f"""{round(-1 * my_fir_sum				):.1f}"""				)

      # two len string
      lowerCAmelCase__      :			Tuple   =	sum(two_char_strings.values()				)
      lowerCAmelCase__      :			str   =	0
      # for each alpha (two in size) calculate entropy.
      for cha in my_alphas:
            for cha in my_alphas:
                  lowerCAmelCase__      :			Optional[int]   =	cha + cha
                  if sequence in two_char_strings:
                        lowerCAmelCase__      :			int   =	two_char_strings[sequence]
                        lowerCAmelCase__      :			str   =	int(__UpperCAmelCase				) / all_sum
                        my_sec_sum += prob * math.loga(__UpperCAmelCase				)

    # print second entropy
      print(f"""{round(-1 * my_sec_sum				):.1f}"""				)

      # print the difference between them
      print(f"""{round((-1 * my_sec_sum) - (-1 * my_fir_sum)				):.1f}"""				)



def 	lowercase_				(			__UpperCAmelCase				)						->    tuple[dict, dict]:
      lowerCAmelCase__      :			Any   =	Counter()  # type: ignore
      lowerCAmelCase__      :			Tuple   =	Counter()  # type: ignore
      single_char_strings[text[-1]] += 1

      # first case when we have space at start.
      two_char_strings[" " + text[0]] += 1
      for i in range(0    ,		len(__UpperCAmelCase				) - 1				):
            single_char_strings[text[i]] += 1
            two_char_strings[text[i : i + 2]] += 1
      return single_char_strings, two_char_strings



def 	lowercase_				(			)						->    Any:
      import doctest

      doctest.testmod()
      # text = (
      #     "Had repulsive dashwoods suspicion sincerity but advantage now him. Remark "
      #     "easily garret nor nay. Civil those mrs enjoy shy fat merry. You greatest "
      #     "jointure saw horrible. He private he on be imagine suppose. Fertile "
      #     "beloved evident through no service elderly is. Blind there if every no so "
      #     "at. Own neglected you preferred way sincerity delivered his attempted. To "
      #     "of message cottage windows do besides against uncivil.  Delightful "
      #     "unreserved impossible few estimating men favourable see entreaties. She "
      #     "propriety immediate was improving. He or entrance humoured likewise "
      #     "moderate. Much nor game son say feel. Fat make met can must form into "
      #     "gate. Me we offending prevailed discovery. "
      # )

      # calculate_prob(text)


if __name__ == "__main__":
  main()





