
"""simple docstring"""




import mpmath  # for roots of unity
import numpy as np

class 		a__						:
   def __init__(			self	,    _a=None	,    _a=None				):
          lowercase    :				int       		=							list(poly_a or [0]				)[:]
          lowercase    :				List[str]       		=							list(poly_b or [0]				)[:]

          # Remove leading zero coefficients
          while self.polyA[-1] == 0:
                 self.polyA.pop()
          lowercase    :				str       		=							len(self.polyA				)

          while self.polyB[-1] == 0:
                 self.polyB.pop()
          lowercase    :				List[str]       		=							len(self.polyB				)

          # Add 0 to make lengths equal a power of 2
          lowercase    :				int       		=							int(
              2 ** np.ceil(np.loga(len(self.polyA				) + len(self.polyB				) - 1				)				)				)

          while len(self.polyA				) < self.c_max_length:
                 self.polyA.append(0				)
          while len(self.polyB				) < self.c_max_length:
                 self.polyB.append(0				)
          # A complex root used for the fourier transform
          lowercase    :				Optional[Any]       		=							complex(mpmath.root(x=1	,    n=self.c_max_length	,    k=1				)				)

          # The product
          lowercase    :				List[Any]       		=							self.__multiply()
   def 			__magic_name__			(			self	,    _a				):
          lowercase    :				str       		=							[[x] for x in self.polyA] if which == "A" else [[x] for x in self.polyB]
          # Corner case
          if len(_a				) <= 1:
                 return dft[0]
          #
          lowercase    :				List[str]       		=							self.c_max_length // 2
          while next_ncol > 0:
                 lowercase    :				Tuple       		=							[[] for i in range(_a				)]
                 lowercase    :				str       		=							self.root**next_ncol

                 # First half of next step
                 lowercase    :				List[str]       		=							1
                 for j in range(self.c_max_length // (next_ncol * 2)				):
                        for i in range(_a				):
                               new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]				)
                        current_root *= root
                 # Second half of next step
                 lowercase    :				Any       		=							1
                 for j in range(self.c_max_length // (next_ncol * 2)				):
                        for i in range(_a				):
                               new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]				)
                        current_root *= root
                 # Update
                 lowercase    :				Any       		=							new_dft
                 lowercase    :				Union[str, Any]       		=							next_ncol // 2
          return dft[0]
   def 			__magic_name__			(			self				):
          lowercase    :				Any       		=							self.__dft("A"				)
          lowercase    :				Union[str, Any]       		=							self.__dft("B"				)
          lowercase    :				Tuple       		=							[[dft_a[i] * dft_b[i] for i in range(self.c_max_length				)]]
          del dft_a
          del dft_b

          # Corner Case
          if len(inverce_c[0]				) <= 1:
                 return inverce_c[0]
          # Inverse DFT
          lowercase    :				List[Any]       		=							2
          while next_ncol <= self.c_max_length:
                 lowercase    :				Dict       		=							[[] for i in range(_a				)]
                 lowercase    :				Optional[Any]       		=							self.root ** (next_ncol // 2)
                 lowercase    :				List[Any]       		=							1
                 # First half of next step
                 for j in range(self.c_max_length // next_ncol				):
                        for i in range(next_ncol // 2				):
                               # Even positions
                               new_inverse_c[i].append(
                                   (
                                       inverce_c[i][j]
                                       + inverce_c[i][j + self.c_max_length // next_ncol]
                                   )
                                   / 2				)
                               # Odd positions
                               new_inverse_c[i + next_ncol // 2].append(
                                   (
                                       inverce_c[i][j]
                                       - inverce_c[i][j + self.c_max_length // next_ncol]
                                   )
                                   / (2 * current_root)				)
                        current_root *= root
                 # Update
                 lowercase    :				Optional[Any]       		=							new_inverse_c
                 next_ncol *= 2
          # Unpack
          lowercase    :				Optional[Any]       		=							[round(x[0].real	,    8				) + round(x[0].imag	,    8				) * 1j for x in inverce_c]

          # Remove leading 0's
          while inverce_c[-1] == 0:
                 inverce_c.pop()
          return inverce_c

   def __str__(			self				):
          lowercase    :				int       		=							"A = " + " + ".join(
              f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyA[: self.len_A]				)				)
          lowercase    :				List[str]       		=							"B = " + " + ".join(
              f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyB[: self.len_B]				)				)
          lowercase    :				Optional[Any]       		=							"A*B = " + " + ".join(
              f"""{coef}*x^{i}""" for coef, i in enumerate(self.product				)				)

          return f"""{a}\n{b}\n{c}"""


# Unit tests
if __name__ == "__main__":
   import doctest

   doctest.testmod()




from __future__ import annotations

from typing import Any




class    SCREAMING_SNAKE_CASE__     :





	'''simple docstring'''



	def __init__(  self	:    Tuple			,  lowercase	:    int			,  lowercase	:    int			,  lowercase	:    float = 0	):


				'''simple docstring'''


				_snake_case	,					_snake_case		=			row, column
				_snake_case		=			[[default_value for c in range(lowercase	)] for r in range(lowercase	)]



	def __str__(  self	:    int	):


				'''simple docstring'''


				_snake_case		=			f'''Matrix consist of {self.row} rows and {self.column} columns\n'''

				# Make string identifier
				_snake_case		=			0
				for row_vector in self.array:
							for obj in row_vector:
										_snake_case		=			max(lowercase			,  len(str(lowercase	)	)	)
				_snake_case		=			f'''%{max_element_length}s'''

				# Make string and return
				def single_line(lowercase	:    list[float]	) -> str:
							nonlocal string_format_identifier
							_snake_case		=			'['
							line += ", ".join(string_format_identifier % (obj,) for obj in row_vector	)
							line += "]"
							return line

				s += "\n".join(single_line(lowercase	) for row_vector in self.array	)
				return s



	def __repr__(  self	:    Dict	):


				'''simple docstring'''


				return str(self	)



	def  A							(  self	:    str			,  lowercase	:    tuple[int, int]	):


				'''simple docstring'''


				if not (isinstance(lowercase			,  (list, tuple)	) and len(lowercase	) == 2):
							return False
				elif not (0 <= loc[0] < self.row and 0 <= loc[1] < self.column):
							return False
				else:
							return True



	def __getitem__(  self	:    Dict			,  lowercase	:    tuple[int, int]	):


				'''simple docstring'''


				assert self.validate_indicies(lowercase	)
				return self.array[loc[0]][loc[1]]



	def __setitem__(  self	:    str			,  lowercase	:    tuple[int, int]			,  lowercase	:    float	):


				'''simple docstring'''


				assert self.validate_indicies(lowercase	)
				_snake_case		=			value



	def __add__(  self	:    str			,  lowercase	:    Matrix	):


				'''simple docstring'''


				assert isinstance(lowercase			,  lowercase	)
				assert self.row == another.row and self.column == another.column

				# Add
				_snake_case		=			Matrix(self.row			,  self.column	)
				for r in range(self.row	):
							for c in range(self.column	):
										_snake_case		=			self[r, c] + another[r, c]
				return result



	def __neg__(  self	:    Tuple	):


				'''simple docstring'''


				_snake_case		=			Matrix(self.row			,  self.column	)
				for r in range(self.row	):
							for c in range(self.column	):
										_snake_case		=			-self[r, c]
				return result



	def __sub__(  self	:    List[str]			,  lowercase	:    Matrix	):


				'''simple docstring'''


				return self + (-another)



	def __mul__(  self	:    Dict			,  lowercase	:    int | float | Matrix	):


				'''simple docstring'''


				if isinstance(lowercase			,  (int, float)	):  # Scalar multiplication
							_snake_case		=			Matrix(self.row			,  self.column	)
							for r in range(self.row	):
										for c in range(self.column	):
													_snake_case		=			self[r, c] * another
							return result
				elif isinstance(lowercase			,  lowercase	):  # Matrix multiplication
							assert self.column == another.row
							_snake_case		=			Matrix(self.row			,  another.column	)
							for r in range(self.row	):
										for c in range(another.column	):
													for i in range(self.column	):
																result[r, c] += self[r, i] * another[i, c]
							return result
				else:
							_snake_case		=			f'''Unsupported type given for another ({type(lowercase	)})'''
							raise TypeError(lowercase	)



	def  A							(  self	:    Optional[Any]	):


				'''simple docstring'''


				_snake_case		=			Matrix(self.column			,  self.row	)
				for r in range(self.row	):
							for c in range(self.column	):
										_snake_case		=			self[r, c]
				return result





	def  A							(  self	:    List[Any]			,  lowercase	:    Matrix			,  lowercase	:    Matrix	):


				'''simple docstring'''


				assert isinstance(lowercase			,  lowercase	) and isinstance(lowercase			,  lowercase	)
				assert self.row == self.column == u.row == v.row  # u, v should be column vector
				assert u.column == v.column == 1  # u, v should be column vector

				# Calculate
				_snake_case		=			v.transpose()
				_snake_case		=			(v_t * self * u)[0, 0] + 1
				if numerator_factor == 0:
							return None  # It's not invertable
				return self - ((self * u) * (v_t * self) * (1.0 / numerator_factor))


# Testing
if __name__ == "__main__":






			def        a_		(       ) ->  None:
						# a^(-1)
						_snake_case		=			Matrix(3       ,					3       ,					0 )
						for i in range(3 ):
									_snake_case		=			1
						print(f'''a^(-1) is {ainv}''' )
						# u, v
						_snake_case		=			Matrix(3       ,					1       ,					0 )
						_snake_case	,					_snake_case	,					_snake_case		=			1, 2, -3
						_snake_case		=			Matrix(3       ,					1       ,					0 )
						_snake_case	,					_snake_case	,					_snake_case		=			4, -2, 5
						print(f'''u is {u}''' )
						print(f'''v is {v}''' )
						print(f'''uv^T is {u * v.transpose()}''' )
						# Sherman Morrison
						print(f'''(a + uv^T)^(-1) is {ainv.sherman_morrison(__lowercase       ,					__lowercase )}''' )






			def        a_		(       ) ->  None:
						import doctest

						doctest.testmod()

			testa()