





from __future__ import annotations

from typing import Any




class 	UpperCamelCase_							:


							def __init__(				self     :List[str]				,					__A     :int				,					__A     :int				,					__A     :float = 0		) -> None:



									"""simple docstring"""



									SCREAMING_SNAKE_CASE__						,    SCREAMING_SNAKE_CASE__								=   row, column
									SCREAMING_SNAKE_CASE__								=   [[default_value for c in range(__A		)] for r in range(__A		)]


							def __str__(				self     :Optional[int]		) -> str:



									"""simple docstring"""



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

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

									# Make string and return
									def single_line(__A     :list[float]		) -> str:
											nonlocal string_format_identifier
											SCREAMING_SNAKE_CASE__								=   """["""
											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     :Optional[Any]		) -> str:



									"""simple docstring"""



									return str(self		)


							def   _snake_case	(				self     :Tuple				,					__A     :tuple[int, int]		) -> bool:



									"""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     :List[Any]				,					__A     :tuple[int, int]		) -> Any:



									"""simple docstring"""



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


							def __setitem__(				self     :int				,					__A     :tuple[int, int]				,					__A     :float		) -> None:



									"""simple docstring"""



									assert self.validate_indicies(__A		)
									SCREAMING_SNAKE_CASE__								=   value


							def __add__(				self     :Any				,					__A     :Matrix		) -> Matrix:



									"""simple docstring"""



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

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


							def __neg__(				self     :Dict		) -> Matrix:



									"""simple docstring"""



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


							def __sub__(				self     :Union[str, Any]				,					__A     :Matrix		) -> Matrix:



									"""simple docstring"""



									return self + (-another)


							def __mul__(				self     :Tuple				,					__A     :int | float | Matrix		) -> Matrix:



									"""simple docstring"""



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


							def   _snake_case	(				self     :int		) -> Matrix:



									"""simple docstring"""



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


							def   _snake_case	(				self     :List[str]				,					__A     :Matrix				,					__A     :Matrix		) -> Any:



									"""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
									SCREAMING_SNAKE_CASE__								=   v.transpose()
									SCREAMING_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 					SCREAMING_SNAKE_CASE__				(							):
			# a^(-1)
			SCREAMING_SNAKE_CASE__								=   Matrix(3       , 3       , 0     )
			for i in range(3     ):
					SCREAMING_SNAKE_CASE__								=   1
			print(f'''a^(-1) is {ainv}'''     )
			# u, v
			SCREAMING_SNAKE_CASE__								=   Matrix(3       , 1       , 0     )
			SCREAMING_SNAKE_CASE__						,    SCREAMING_SNAKE_CASE__						,    SCREAMING_SNAKE_CASE__								=   1, 2, -3
			SCREAMING_SNAKE_CASE__								=   Matrix(3       , 1       , 0     )
			SCREAMING_SNAKE_CASE__						,    SCREAMING_SNAKE_CASE__						,    SCREAMING_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(UpperCamelCase__       , UpperCamelCase__     )}'''     )






	def 					SCREAMING_SNAKE_CASE__				(							):
			import doctest

			doctest.testmod()

	testa()






def 					SCREAMING_SNAKE_CASE__				(							UpperCamelCase__:				List[str]     ):
		SCREAMING_SNAKE_CASE__								=   len(UpperCamelCase__     )
		SCREAMING_SNAKE_CASE__								=   sum(UpperCamelCase__     )

		SCREAMING_SNAKE_CASE__								=   [[False for x in range(s + 1     )] for y in range(n + 1     )]

		for i in range(1       , n + 1     ):
				SCREAMING_SNAKE_CASE__								=   True

		for i in range(1       , s + 1     ):
				SCREAMING_SNAKE_CASE__								=   False

		for i in range(1       , n + 1     ):
				for j in range(1       , s + 1     ):
						SCREAMING_SNAKE_CASE__								=   dp[i][j - 1]

						if arr[i - 1] <= j:
								SCREAMING_SNAKE_CASE__								=   dp[i][j] or dp[i - 1][j - arr[i - 1]]

		for j in range(int(s / 2     )       , -1       , -1     ):
				if dp[n][j] is True:
						SCREAMING_SNAKE_CASE__								=   s - 2 * j
						break

		return diff