



'''simple docstring'''

from __future__ import annotations






class 						a_		:


							def __init__(				self   :						str      ,						a_   :						list[list[int]]			)			->					int:
										snake_case:    Optional[int]      							=TypeError(
										    'Matrices must be formed from a list of zero or more lists containing at '
										    'least one and the same number of values, each of which must be of type '
										    'int or float.'			)
										if len(a_			) != 0:
													snake_case:    Union[str, Any]      							=len(rows[0]			)
													if cols == 0:
																raise error
													for row in rows:
																if len(a_			) != cols:
																			raise error
																for value in row:
																			if not isinstance(a_      ,						(int, float)			):
																						raise error
													snake_case:    Any      							=rows
										else:
													snake_case:    Union[str, Any]      							=[]


							def    UpperCamelCase     (				self   :						Any			)			->					list[list[int]]:
										return [[row[i] for row in self.rows] for i in range(len(self.rows[0]			)			)]


							@property
							def    UpperCamelCase     (				self   :						Optional[Any]			)			->					int:
										return len(self.rows			)


							@property
							def    UpperCamelCase     (				self   :						List[str]			)			->					int:
										return len(self.rows[0]			)


							@property
							def    UpperCamelCase     (				self   :						Union[str, Any]			)			->					tuple[int, int]:
										return (self.num_rows, self.num_columns)


							@property
							def    UpperCamelCase     (				self   :						Union[str, Any]			)			->					bool:
										return self.order[0] == self.order[1]


							def    UpperCamelCase     (				self   :						Tuple			)			->					Matrix:
										snake_case:    Tuple      							=[
										    [0 if column_num != row_num else 1 for column_num in range(self.num_rows			)]
										    for row_num in range(self.num_rows			)
										]
										return Matrix(a_			)


							def    UpperCamelCase     (				self   :						Tuple			)			->					int:
										if not self.is_square:
													return 0
										if self.order == (0, 0):
													return 1
										if self.order == (1, 1):
													return int(self.rows[0][0]			)
										if self.order == (2, 2):
													return int(
													    (self.rows[0][0] * self.rows[1][1])
													    - (self.rows[0][1] * self.rows[1][0])			)
										else:
													return sum(
													    self.rows[0][column] * self.cofactors().rows[0][column]
													    for column in range(self.num_columns			)			)


							def    UpperCamelCase     (				self   :						str			)			->					bool:
										return bool(self.determinant()			)


							def    UpperCamelCase     (				self   :						Optional[int]      ,						a_   :						int      ,						a_   :						int			)			->					int:
										snake_case:    Optional[Any]      							=[
										    [
										        self.rows[other_row][other_column]
										        for other_column in range(self.num_columns			)
										        if other_column != column
										    ]
										    for other_row in range(self.num_rows			)
										    if other_row != row
										]
										return Matrix(a_			).determinant()


							def    UpperCamelCase     (				self   :						Any      ,						a_   :						int      ,						a_   :						int			)			->					int:
										if (row + column) % 2 == 0:
													return self.get_minor(a_      ,						a_			)
										return -1 * self.get_minor(a_      ,						a_			)


							def    UpperCamelCase     (				self   :						Dict			)			->					Matrix:
										return Matrix(
										    [
										        [self.get_minor(a_      ,						a_			) for column in range(self.num_columns			)]
										        for row in range(self.num_rows			)
										    ]			)


							def    UpperCamelCase     (				self   :						Tuple			)			->					Matrix:
										return Matrix(
										    [
										        [
										            self.minors().rows[row][column]
										            if (row + column) % 2 == 0
										            else self.minors().rows[row][column] * -1
										            for column in range(self.minors().num_columns			)
										        ]
										        for row in range(self.minors().num_rows			)
										    ]			)


							def    UpperCamelCase     (				self   :						Tuple			)			->					Matrix:
										snake_case:    Optional[int]      							=[
										    [self.cofactors().rows[column][row] for column in range(self.num_columns			)]
										    for row in range(self.num_rows			)
										]
										return Matrix(a_			)


							def    UpperCamelCase     (				self   :						Optional[int]			)			->					Matrix:
										snake_case:    int      							=self.determinant()
										if not determinant:
													raise TypeError('Only matrices with a non-zero determinant have an inverse'			)
										return self.adjugate() * (1 / determinant)


							def __repr__(				self   :						int			)			->					str:
										return str(self.rows			)


							def __str__(				self   :						List[Any]			)			->					str:
										if self.num_rows == 0:
													return "[]"
										if self.num_rows == 1:
													return "[[" + ". ".join(str(self.rows[0]			)			) + "]]"
										return (
										    "["
										    + "\n ".join(
										        [
										            '[' + '. '.join([str(a_			) for value in row]			) + '.]'
										            for row in self.rows
										        ]			)
										    + "]"
										)


							def    UpperCamelCase     (				self   :						Any      ,						a_   :						list[int]      ,						a_   :						int | None = None			)			->					None:
										snake_case:    List[Any]      							=TypeError('Row must be a list containing all ints and/or floats'			)
										if not isinstance(a_      ,						a_			):
													raise type_error
										for value in row:
													if not isinstance(a_      ,						(int, float)			):
																raise type_error
										if len(a_			) != self.num_columns:
													raise ValueError(
													    'Row must be equal in length to the other rows in the matrix'			)
										if position is None:
													self.rows.append(a_			)
										else:
													snake_case:    Any      							=self.rows[0:position] + [row] + self.rows[position:]


							def    UpperCamelCase     (				self   :						Optional[int]      ,						a_   :						list[int]      ,						a_   :						int | None = None			)			->					None:
										snake_case:    str      							=TypeError(
										    'Column must be a list containing all ints and/or floats'			)
										if not isinstance(a_      ,						a_			):
													raise type_error
										for value in column:
													if not isinstance(a_      ,						(int, float)			):
																raise type_error
										if len(a_			) != self.num_rows:
													raise ValueError(
													    'Column must be equal in length to the other columns in the matrix'			)
										if position is None:
													snake_case:    Any      							=[self.rows[i] + [column[i]] for i in range(self.num_rows			)]
										else:
													snake_case:    Optional[Any]      							=[
													    self.rows[i][0:position] + [column[i]] + self.rows[i][position:]
													    for i in range(self.num_rows			)
													]


							def __eq__(				self   :						Dict      ,						a_   :						object			)			->					bool:
										if not isinstance(a_      ,						a_			):
													return NotImplemented
										return self.rows == other.rows


							def __ne__(				self   :						Any      ,						a_   :						object			)			->					bool:
										return not self == other


							def __neg__(				self   :						Tuple			)			->					Matrix:
										return self * -1


							def __add__(				self   :						Optional[int]      ,						a_   :						Matrix			)			->					Matrix:
										if self.order != other.order:
													raise ValueError('Addition requires matrices of the same order'			)
										return Matrix(
										    [
										        [self.rows[i][j] + other.rows[i][j] for j in range(self.num_columns			)]
										        for i in range(self.num_rows			)
										    ]			)


							def __sub__(				self   :						Dict      ,						a_   :						Matrix			)			->					Matrix:
										if self.order != other.order:
													raise ValueError('Subtraction requires matrices of the same order'			)
										return Matrix(
										    [
										        [self.rows[i][j] - other.rows[i][j] for j in range(self.num_columns			)]
										        for i in range(self.num_rows			)
										    ]			)


							def __mul__(				self   :						Union[str, Any]      ,						a_   :						Matrix | int | float			)			->					Matrix:
										if isinstance(a_      ,						(int, float)			):
													return Matrix(
													    [[int(element * other			) for element in row] for row in self.rows]			)
										elif isinstance(a_      ,						a_			):
													if self.num_columns != other.num_rows:
																raise ValueError(
																    'The number of columns in the first matrix must '
																    'be equal to the number of rows in the second'			)
													return Matrix(
													    [
													        [Matrix.dot_product(a_      ,						a_			) for column in other.columns()]
													        for row in self.rows
													    ]			)
										else:
													raise TypeError(
													    'A Matrix can only be multiplied by an int, float, or another matrix'			)


							def __pow__(				self   :						Tuple      ,						a_   :						int			)			->					Matrix:
										if not isinstance(a_      ,						a_			):
													raise TypeError('A Matrix can only be raised to the power of an int'			)
										if not self.is_square:
													raise ValueError('Only square matrices can be raised to a power'			)
										if other == 0:
													return self.identity()
										if other < 0:
													if self.is_invertable():
																return self.inverse() ** (-other)
													raise ValueError(
													    'Only invertable matrices can be raised to a negative power'			)
										snake_case:    Dict      							=self
										for _ in range(other - 1			):
													result *= self
										return result




							@classmethod
							def    UpperCamelCase     (				cls   :						Any      ,						a_   :						list[int]      ,						a_   :						list[int]			)			->					int:
										return sum(row[i] * column[i] for i in range(len(a_			)			)			)


if __name__ == "__main__":
	import doctest

	doctest.testmod()









'''simple docstring'''

def a_     (    __UpperCAmelCase				,							__UpperCAmelCase				,							__UpperCAmelCase					)				->					list:





			"""simple docstring"""







			snake_case:    Optional[int]      							=len(__UpperCAmelCase					)
			snake_case:    Optional[int]      							=[[0] * n for i in range(__UpperCAmelCase					)]
			for i in range(__UpperCAmelCase					):
						snake_case:    Optional[int]      							=y_points[i]

			for i in range(2				,							__UpperCAmelCase					):
						for j in range(__UpperCAmelCase				,							__UpperCAmelCase					):
									snake_case:    List[str]      							=(
									    (xa - x_points[j - i + 1]) * q[j][i - 1]
									    - (xa - x_points[j]) * q[j - 1][i - 1]
									) / (x_points[j] - x_points[j - i + 1])

			return [q[n - 1][n - 1], q]


if __name__ == "__main__":
	import doctest

	doctest.testmod()




