

"""simple docstring"""




from __future__ import annotations

from collections.abc import MutableSequence







class      __lowercase			:

  def __init__(  self		:			Tuple					,A		:			int					,A		:			MutableSequence[float]    ):






   '''simple docstring'''
   if len(A    ) != degree + 1:
    raise ValueError(
        """The number of coefficients should be equal to the degree + 1."""    )

   UpperCAmelCase__					:  list[float]					   =					list(A    )
   UpperCAmelCase__					:  Any					   =					degree

  def __add__(  self		:			Dict					,A		:			Polynomial    ):






   '''simple docstring'''

   if self.degree > polynomial_a.degree:
    UpperCAmelCase__					:  Dict					   =					self.coefficients[:]
    for i in range(polynomial_a.degree + 1    ):
     coefficients[i] += polynomial_a.coefficients[i]
    return Polynomial(self.degree					,A    )
   else:
    UpperCAmelCase__					:  int					   =					polynomial_a.coefficients[:]
    for i in range(self.degree + 1    ):
     coefficients[i] += self.coefficients[i]
    return Polynomial(polynomial_a.degree					,A    )

  def __sub__(  self		:			Optional[Any]					,A		:			Polynomial    ):






   '''simple docstring'''
   return self + polynomial_a * Polynomial(0					,[-1]    )

  def __neg__(  self		:			str    ):






   '''simple docstring'''
   return Polynomial(self.degree					,[-c for c in self.coefficients]    )

  def __mul__(  self		:			List[Any]					,A		:			Polynomial    ):






   '''simple docstring'''
   UpperCAmelCase__					:  list[float]					   =					[0] * (self.degree + polynomial_a.degree + 1)
   for i in range(self.degree + 1    ):
    for j in range(polynomial_a.degree + 1    ):
     coefficients[i + j] += (
         self.coefficients[i] * polynomial_a.coefficients[j]
     )

   return Polynomial(self.degree + polynomial_a.degree					,A    )

  def    __lowercase       (  self		:			List[Any]					,A		:			int | float    ):






   '''simple docstring'''
   UpperCAmelCase__					:  int | float					   =					0
   for i in range(self.degree + 1    ):
    result += self.coefficients[i] * (substitution**i)
   return result

  def __str__(  self		:			List[str]    ):






   '''simple docstring'''
   UpperCAmelCase__					:  Optional[int]					   =					""""""
   for i in range(self.degree					,-1					,-1    ):
    if self.coefficients[i] == 0:
     continue
    elif self.coefficients[i] > 0:
     if polynomial:
      polynomial += " + "
    else:
     polynomial += " - "

    if i == 0:
     polynomial += str(abs(self.coefficients[i]    )    )
    elif i == 1:
     polynomial += str(abs(self.coefficients[i]    )    ) + "x"
    else:
     polynomial += str(abs(self.coefficients[i]    )    ) + "x^" + str(A    )

   return polynomial

  def __repr__(  self		:			List[Any]    ):






   '''simple docstring'''
   return self.__str__()

  def    __lowercase       (  self		:			str    ):






   '''simple docstring'''
   UpperCAmelCase__					:  list[float]					   =					[0] * self.degree
   for i in range(self.degree    ):
    UpperCAmelCase__					:  Tuple					   =					self.coefficients[i + 1] * (i + 1)
   return Polynomial(self.degree - 1					,A    )

  def    __lowercase       (  self		:			List[Any]					,A		:			int | float = 0    ):






   '''simple docstring'''
   UpperCAmelCase__					:  list[float]					   =					[0] * (self.degree + 2)
   UpperCAmelCase__					:  Tuple					   =					constant
   for i in range(self.degree + 1    ):
    UpperCAmelCase__					:  Union[str, Any]					   =					self.coefficients[i] / (i + 1)
   return Polynomial(self.degree + 1					,A    )

  def __eq__(  self		:			List[str]					,A		:			object    ):






   '''simple docstring'''
   if not isinstance(A					,A    ):
    return False

   if self.degree != polynomial_a.degree:
    return False

   for i in range(self.degree + 1    ):
    if self.coefficients[i] != polynomial_a.coefficients[i]:
     return False

   return True




  def __ne__(  self		:			Union[str, Any]					,A		:			object    ):






   '''simple docstring'''
   return not self.__eq__(A    )








"""simple docstring"""




from __future__ import annotations

from typing import Any







class      __lowercase			:

  def __init__(  self		:			Optional[int]					,A		:			int					,A		:			int					,A		:			float = 0    ):






   '''simple docstring'''
   UpperCAmelCase__   ,      UpperCAmelCase__					:  str					   =					row, column
   UpperCAmelCase__					:  int					   =					[[default_value for c in range(A    )] for r in range(A    )]

  def __str__(  self		:			Union[str, Any]    ):






   '''simple docstring'''
   UpperCAmelCase__					:  Union[str, Any]					   =					f"Matrix consist of {self.row} rows and {self.column} columns\n"

   # Make string identifier
   UpperCAmelCase__					:  Optional[Any]					   =					0
   for row_vector in self.array:
    for obj in row_vector:
     UpperCAmelCase__					:  Optional[int]					   =					max(A					,len(str(A    )    )    )
   UpperCAmelCase__					:  int					   =					f"%{max_element_length}s"

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

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

  def __repr__(  self		:			Dict    ):






   '''simple docstring'''
   return str(self    )

  def    __lowercase       (  self		:			int					,A		:			tuple[int, int]    ):






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

  def __getitem__(  self		:			Optional[int]					,A		:			tuple[int, int]    ):






   '''simple docstring'''
   assert self.validate_indicies(A    )
   return self.array[loc[0]][loc[1]]

  def __setitem__(  self		:			List[Any]					,A		:			tuple[int, int]					,A		:			float    ):






   '''simple docstring'''
   assert self.validate_indicies(A    )
   UpperCAmelCase__					:  str					   =					value

  def __add__(  self		:			Any					,A		:			Matrix    ):






   '''simple docstring'''
   assert isinstance(A					,A    )
   assert self.row == another.row and self.column == another.column

   # Add
   UpperCAmelCase__					:  Optional[int]					   =					Matrix(self.row					,self.column    )
   for r in range(self.row    ):
    for c in range(self.column    ):
     UpperCAmelCase__					:  Union[str, Any]					   =					self[r, c] + another[r, c]
   return result

  def __neg__(  self		:			Union[str, Any]    ):






   '''simple docstring'''
   UpperCAmelCase__					:  Tuple					   =					Matrix(self.row					,self.column    )
   for r in range(self.row    ):
    for c in range(self.column    ):
     UpperCAmelCase__					:  int					   =					-self[r, c]
   return result

  def __sub__(  self		:			str					,A		:			Matrix    ):






   '''simple docstring'''
   return self + (-another)

  def __mul__(  self		:			Union[str, Any]					,A		:			int | float | Matrix    ):






   '''simple docstring'''

   if isinstance(A					,(int, float)    ):  # Scalar multiplication
    UpperCAmelCase__					:  Dict					   =					Matrix(self.row					,self.column    )
    for r in range(self.row    ):
     for c in range(self.column    ):
      UpperCAmelCase__					:  Any					   =					self[r, c] * another
    return result
   elif isinstance(A					,A    ):  # Matrix multiplication
    assert self.column == another.row
    UpperCAmelCase__					:  Optional[int]					   =					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:
    UpperCAmelCase__					:  Union[str, Any]					   =					f"Unsupported type given for another ({type(A    )})"
    raise TypeError(A    )

  def    __lowercase       (  self		:			str    ):






   '''simple docstring'''
   UpperCAmelCase__					:  List[Any]					   =					Matrix(self.column					,self.row    )
   for r in range(self.row    ):
    for c in range(self.column    ):
     UpperCAmelCase__					:  Optional[int]					   =					self[r, c]
   return result




  def    __lowercase       (  self		:			Any					,A		:			Matrix					,A		:			Matrix    ):






   '''simple docstring'''
   assert isinstance(A					,A    ) and isinstance(A					,A    )
   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
   UpperCAmelCase__					:  Optional[Any]					   =					v.transpose()
   UpperCAmelCase__					:  Union[str, Any]					   =					(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   lowerCAmelCase						(					):
       '''simple docstring'''





       UpperCAmelCase__					:  List[str]					   =					Matrix(3							,    3							,    0      )
       for i in range(3      ):
        UpperCAmelCase__					:  List[str]					   =					1
       print(F"a^(-1) is {ainv}"      )
       # u, v
       UpperCAmelCase__					:  List[Any]					   =					Matrix(3							,    1							,    0      )
       UpperCAmelCase__   ,      UpperCAmelCase__   ,      UpperCAmelCase__					:  Optional[int]					   =					1, 2, -3
       UpperCAmelCase__					:  List[str]					   =					Matrix(3							,    1							,    0      )
       UpperCAmelCase__   ,      UpperCAmelCase__   ,      UpperCAmelCase__					:  Tuple					   =					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(__UpperCamelCase							,    __UpperCamelCase      )}"      )
      def   lowerCAmelCase						(					):
       '''simple docstring'''





       import doctest

       doctest.testmod()




      testa()





