

'''simple docstring'''




from __future__ import annotations

from typing import Any


def       lowerCamelCase__		(						*SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  if all(_check_not_integer(SCREAMING_SNAKE_CASE						) for m in matrix_s						):
    for i in matrix_s[1:]:
      _verify_matrix_sizes(matrix_s[0]       ,   SCREAMING_SNAKE_CASE						)
    return [[sum(SCREAMING_SNAKE_CASE						) for t in zip(*SCREAMING_SNAKE_CASE						)] for m in zip(*SCREAMING_SNAKE_CASE						)]
  raise TypeError('Expected a matrix, got int/list instead'						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]       ,   SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  if (
      _check_not_integer(SCREAMING_SNAKE_CASE						)
      and _check_not_integer(SCREAMING_SNAKE_CASE						)
      and _verify_matrix_sizes(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						)
  ):
    return [[i - j for i, j in zip(*SCREAMING_SNAKE_CASE						)] for m in zip(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						)]
  raise TypeError('Expected a matrix, got int/list instead'						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]       ,   SCREAMING_SNAKE_CASE      :   int | float						):
  return [[x * n for x in row] for row in matrix]


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]       ,   SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  if _check_not_integer(SCREAMING_SNAKE_CASE						) and _check_not_integer(SCREAMING_SNAKE_CASE						):
    UpperCAmelCase							,      UpperCAmelCase													=	_verify_matrix_sizes(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						)

  if cols[0] != rows[1]:
    UpperCAmelCase													=	(
        'Cannot multiply matrix of dimensions '
        f'''({rows[0]},{cols[0]}) and ({rows[1]},{cols[1]})'''
    )
    raise ValueError(SCREAMING_SNAKE_CASE						)
  return [
      [sum(m * n for m, n in zip(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						)						) for j in zip(*SCREAMING_SNAKE_CASE						)] for i in matrix_a
  ]


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   int						):
  UpperCAmelCase													=	int(SCREAMING_SNAKE_CASE						)
  return [[int(row == column						) for column in range(SCREAMING_SNAKE_CASE						)] for row in range(SCREAMING_SNAKE_CASE						)]


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]       ,   SCREAMING_SNAKE_CASE      :   bool = True						):
  if _check_not_integer(SCREAMING_SNAKE_CASE						):
    if return_map:
      return map(SCREAMING_SNAKE_CASE       ,   zip(*SCREAMING_SNAKE_CASE						)						)
    else:
      return list(map(SCREAMING_SNAKE_CASE       ,   zip(*SCREAMING_SNAKE_CASE						)						)						)
  raise TypeError('Expected a matrix, got int/list instead'						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]       ,   SCREAMING_SNAKE_CASE      :   int       ,   SCREAMING_SNAKE_CASE      :   int						):
  UpperCAmelCase													=	matrix[:row] + matrix[row + 1 :]
  return [row[:column] + row[column + 1 :] for row in minor]


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  if len(SCREAMING_SNAKE_CASE						) == 1:
    return matrix[0][0]

  return sum(
      x * determinant(minor(SCREAMING_SNAKE_CASE       ,   0       ,   SCREAMING_SNAKE_CASE						)						) * (-1) ** i
      for i, x in enumerate(matrix[0]						)						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  UpperCAmelCase													=	determinant(SCREAMING_SNAKE_CASE						)
  if det == 0:
    return None

  UpperCAmelCase													=	[
      [determinant(minor(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						)						) for j in range(len(SCREAMING_SNAKE_CASE						)						)]
      for i in range(len(SCREAMING_SNAKE_CASE						)						)
  ]

  UpperCAmelCase													=	[
      [x * (-1) ** (row + col) for col, x in enumerate(matrix_minor[row]						)]
      for row in range(len(SCREAMING_SNAKE_CASE						)						)
  ]
  UpperCAmelCase													=	list(transpose(SCREAMING_SNAKE_CASE						)						)
  return scalar_multiply(SCREAMING_SNAKE_CASE       ,   1 / det						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  return not isinstance(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						) and not isinstance(matrix[0]       ,   SCREAMING_SNAKE_CASE						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  return len(SCREAMING_SNAKE_CASE						), len(matrix[0]						)


def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]       ,   SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  UpperCAmelCase													=	_shape(SCREAMING_SNAKE_CASE						) + _shape(SCREAMING_SNAKE_CASE						)
  if shape[0] != shape[3] or shape[1] != shape[2]:
    UpperCAmelCase													=	(
        'operands could not be broadcast together with shape '
        f'''({shape[0], shape[1]}), ({shape[2], shape[3]})'''
    )
    raise ValueError(SCREAMING_SNAKE_CASE						)
  return (shape[0], shape[2]), (shape[1], shape[3])


def       lowerCamelCase__		(						):
  UpperCAmelCase													=	[[12, 10], [3, 9]]
  UpperCAmelCase													=	[[3, 4], [7, 4]]
  UpperCAmelCase													=	[[11, 12, 13, 14], [21, 22, 23, 24], [31, 32, 33, 34], [41, 42, 43, 44]]
  UpperCAmelCase													=	[[3, 0, 2], [2, 0, -2], [0, 1, 1]]
  print(f'''Add Operation, {add(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						) = } \n'''						)
  print(
      f'''Multiply Operation, {multiply(SCREAMING_SNAKE_CASE       ,   SCREAMING_SNAKE_CASE						) = } \n'''       ,   )
  print(f'''Identity: {identity(5						)}\n'''						)
  print(f'''Minor of {matrix_c} = {minor(SCREAMING_SNAKE_CASE       ,   1       ,   2						)} \n'''						)
  print(f'''Determinant of {matrix_b} = {determinant(SCREAMING_SNAKE_CASE						)} \n'''						)
  print(f'''Inverse of {matrix_d} = {inverse(SCREAMING_SNAKE_CASE						)}\n'''						)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  main()





'''simple docstring'''




def       lowerCamelCase__		(						SCREAMING_SNAKE_CASE      :   list[list[int]]						):
  if len(SCREAMING_SNAKE_CASE						) == 0 or not all(len(SCREAMING_SNAKE_CASE						) == 2 for pair in coordinates						):
    raise ValueError('The program cannot work out a fitting polynomial.'						)

  if len({tuple(SCREAMING_SNAKE_CASE						) for pair in coordinates}						) != len(SCREAMING_SNAKE_CASE						):
    raise ValueError('The program cannot work out a fitting polynomial.'						)

  UpperCAmelCase													=	{x for x, _ in coordinates}
  if len(SCREAMING_SNAKE_CASE						) == 1:
    return f'''x={coordinates[0][0]}'''

  if len(SCREAMING_SNAKE_CASE						) != len(SCREAMING_SNAKE_CASE						):
    raise ValueError('The program cannot work out a fitting polynomial.'						)

  UpperCAmelCase													=	len(SCREAMING_SNAKE_CASE						)

  UpperCAmelCase													=	0
  UpperCAmelCase													=	[]
  # put the x and x to the power values in a matrix
  while count_of_line < x:
    UpperCAmelCase													=	0
    UpperCAmelCase													=	coordinates[count_of_line][0]
    UpperCAmelCase													=	[]
    while count_in_line < x:
      count_line.append(a ** (x - (count_in_line + 1))						)
      count_in_line += 1
    matrix.append(SCREAMING_SNAKE_CASE						)
    count_of_line += 1

  UpperCAmelCase													=	0
  # put the y values into a vector
  UpperCAmelCase													=	[]
  while count_of_line < x:
    vector.append(coordinates[count_of_line][1]						)
    count_of_line += 1

  UpperCAmelCase													=	0

  while count < x:
    UpperCAmelCase													=	0
    while zahlen < x:
      if count == zahlen:
        zahlen += 1
      if zahlen == x:
        break
      UpperCAmelCase													=	matrix[zahlen][count] / matrix[count][count]
      for counting_columns, item in enumerate(matrix[count]						):
        # manipulating all the values in the matrix
        matrix[zahlen][counting_columns] -= item * bruch
      # manipulating the values in the vector
      vector[zahlen] -= vector[count] * bruch
      zahlen += 1
    count += 1

  UpperCAmelCase													=	0
  # make solutions
  UpperCAmelCase													=	[]
  while count < x:
    solution.append(str(vector[count] / matrix[count][count]						)						)
    count += 1

  UpperCAmelCase													=	0
  UpperCAmelCase													=	'f(x)='

  while count < x:
    UpperCAmelCase													=	solution[count].split('E'						)
    if len(SCREAMING_SNAKE_CASE						) > 1:
      UpperCAmelCase													=	f'''{remove_e[0]}*10^{remove_e[1]}'''
    solved += f'''x^{x - (count + 1)}*{solution[count]}'''
    if count + 1 != x:
      solved += "+"
    count += 1

  return solved


if __name__ == "__main__":
  print(points_to_polynomial([]))
  print(points_to_polynomial([[]]))
  print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
  print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
  print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
  print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
  print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
  print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
  print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))


