




"""simple docstring"""
from __future__ import annotations

from typing import Any



class   lowerCAmelCase_    :



				"""simple docstring"""

				def __init__(				self		,    lowerCAmelCase		,    lowerCAmelCase		,    lowerCAmelCase = 0    ):






										"""simple docstring"""

										snake_case					,snake_case								=			row, column
										snake_case								=			[[default_value for c in range(lowerCAmelCase    )] for r in range(lowerCAmelCase    )]

				def __str__(				self    ):






										"""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(lowerCAmelCase		,    len(str(lowerCAmelCase    )    )    )
										snake_case								=			F"""%{max_element_length}s"""

										# Make string and return
										def single_line(lowerCAmelCase    ) -> 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(lowerCAmelCase    ) for row_vector in self.array    )
										return s

				def __repr__(				self    ):






										"""simple docstring"""

										return str(self    )

				def 		snake_case (				self		,    lowerCAmelCase    ):






										"""simple docstring"""

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

				def __getitem__(				self		,    lowerCAmelCase    ):






										"""simple docstring"""

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

				def __setitem__(				self		,    lowerCAmelCase		,    lowerCAmelCase    ):






										"""simple docstring"""

										assert self.validate_indicies(lowerCAmelCase    )
										snake_case								=			value

				def __add__(				self		,    lowerCAmelCase    ):






										"""simple docstring"""

										assert isinstance(lowerCAmelCase		,    lowerCAmelCase    )
										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    ):






										"""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		,    lowerCAmelCase    ):






										"""simple docstring"""

										return self + (-another)

				def __mul__(				self		,    lowerCAmelCase    ):






										"""simple docstring"""

										if isinstance(lowerCAmelCase		,    (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(lowerCAmelCase		,    lowerCAmelCase    ):  # 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(lowerCAmelCase    )})"""
																raise TypeError(lowerCAmelCase    )

				def 		snake_case (				self    ):






										"""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 		snake_case (				self		,    lowerCAmelCase		,    lowerCAmelCase    ):






										"""simple docstring"""

										assert isinstance(lowerCAmelCase		,    lowerCAmelCase    ) and isinstance(lowerCAmelCase		,    lowerCAmelCase    )
										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    lowerCAmelCase__    (			)					->				None:





											"""simple docstring"""






											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(_UpperCamelCase ,			_UpperCamelCase					)}"""					)





					def    lowerCAmelCase__    (			)					->				None:





											"""simple docstring"""






											import doctest

											doctest.testmod()



					testa()



"""simple docstring"""
import math





def    lowerCAmelCase__    (			_UpperCamelCase       :				int					)					->				bool:





						"""simple docstring"""






						return math.sqrt(_UpperCamelCase					) * math.sqrt(_UpperCamelCase					) == num





def    lowerCAmelCase__    (			_UpperCamelCase       :				int					)					->				bool:





						"""simple docstring"""






						snake_case								=			0
						snake_case								=			n
						while left <= right:
												snake_case								=			(left + right) // 2
												if mid**2 == n:
																		return True
												elif mid**2 > n:
																		snake_case								=			mid - 1
												else:
																		snake_case								=			mid + 1
						return False


if __name__ == "__main__":
					import doctest

					doctest.testmod()


