// 誤解法(厳密に0である値の近似誤差を無視した解法)チェック #pragma GCC optimize ( "O3" ) #pragma GCC optimize( "unroll-loops" ) #pragma GCC target ( "sse4.2,fma,avx2,popcnt,lzcnt,bmi2" ) #include using namespace std; using ll = long long; #define MAIN main #define TYPE_OF( VAR ) remove_const::type >::type #define UNTIE ios_base::sync_with_stdio( false ); cin.tie( nullptr ) #define CEXPR( LL , BOUND , VALUE ) constexpr LL BOUND = VALUE #define CIN( LL , A ) LL A; cin >> A #define ASSERT( A , MIN , MAX ) assert( ( MIN ) <= A && A <= ( MAX ) ) #define CIN_ASSERT( A , MIN , MAX ) CIN( TYPE_OF( MAX ) , A ); ASSERT( A , MIN , MAX ) #define FOR( VAR , INITIAL , FINAL_PLUS_ONE ) for( TYPE_OF( FINAL_PLUS_ONE ) VAR = INITIAL ; VAR < FINAL_PLUS_ONE ; VAR ++ ) #define REPEAT( HOW_MANY_TIMES ) FOR( VARIABLE_FOR_REPEAT , 0 , HOW_MANY_TIMES ) #define QUIT return 0 #define COUT( ANSWER ) cout << ( ANSWER ) << "\n" #define SET_PRECISION( PRECISION ) cout << fixed << setprecision( PRECISION ) template class TwoByTwoMatrix { public: T m_M00; T m_M01; T m_M10; T m_M11; public: inline TwoByTwoMatrix( const T& M00 , const T& M01 , const T& M10 , const T& M11 ) noexcept; TwoByTwoMatrix& operator=( const TwoByTwoMatrix& mat ) noexcept; inline TwoByTwoMatrix operator*( const TwoByTwoMatrix& mat ); }; template inline TwoByTwoMatrix::TwoByTwoMatrix( const T& M00 , const T& M01 , const T& M10 , const T& M11 ) noexcept : m_M00( M00 ) , m_M01( M01 ) , m_M10( M10 ) , m_M11( M11 ) {} template TwoByTwoMatrix& TwoByTwoMatrix::operator=( const TwoByTwoMatrix& mat ) noexcept { if( &mat != this ){ m_M00 = mat.m_M00; m_M01 = mat.m_M01; m_M10 = mat.m_M10; m_M11 = mat.m_M11; } return *this; } template inline TwoByTwoMatrix TwoByTwoMatrix::operator*( const TwoByTwoMatrix& mat ) { return TwoByTwoMatrix( m_M00 * mat.m_M00 + m_M01 * mat.m_M10 , m_M00 * mat.m_M01 + m_M01 * mat.m_M11 , m_M10 * mat.m_M00 + m_M11 * mat.m_M10 , m_M10 * mat.m_M01 + m_M11 * mat.m_M11 ); } void stod( const string& s , ll& num , double& answer ) { bool negative; uint i_start; if( s.substr( 0 , 1 ) == "-" ){ negative = true; i_start = 1; } else { negative = false; i_start = 0; } uint size = s.size(); uint exponent = 0; num = 0; FOR( i , i_start , size ){ string c = s.substr( i , 1 ); if( c == "." ){ assert( exponent == 0 ); assert( i - i_start == 1 ); exponent = size - 1 - i; assert( exponent <= 5 ); } else { num = num * 10 + stoi( c ); } } answer = ( negative ? - num : num ); while( exponent != 0 ){ answer *= 0.1; exponent--; } return; } int MAIN() { UNTIE; SET_PRECISION( 5 ); double M_entry[2][2]; double ( &M0 )[2] = M_entry[0]; double ( &M1 )[2] = M_entry[1]; double& M00 = M0[0]; double& M01 = M0[1]; double& M10 = M1[0]; double& M11 = M1[1]; ll num[2][2]; double e[2]; double& e0 = e[0]; double& e1 = e[1]; double U_entry[2][2]; double ( &U0 )[2] = U_entry[0]; double ( &U1 )[2] = U_entry[1]; double& U00 = U0[0]; double& U01 = U0[1]; double& U10 = U1[0]; double& U11 = U1[1]; double norm[2]; double& norm0 = norm[0]; double& norm1 = norm[1]; CEXPR( int , bound_T , 100000 ); CIN_ASSERT( T , 1 , bound_T ); REPEAT( T ){ FOR( i , 0 , 2 ){ double ( &Mi )[2] = M_entry[i]; ll ( &numi )[2] = num[i]; FOR( j , 0 , 2 ){ CIN( string, S ); stod( S , numi[j] , Mi[j] ); } } double tr = M00 + M11; double det = M00 * M11 - M01 * M10; double disc = tr * tr - 4.0 * det; double discrt = sqrt( disc ); // ここで0とのズレが生じうる e[0] = ( tr + discrt ) / 2.0; e[1] = ( tr - discrt ) / 2.0; double normsq; bool degenerate = false; FOR( j , 0 , 2 ){ double& ej = e[j]; double& U0j = U0[j]; double& U1j = U1[j]; double& normj = norm[j]; U0j = M10; U1j = ej - M00; normsq = U0j * U0j + U1j * U1j; normj = sqrt( normsq ); // ここで0とのズレが生じうる if( normj == 0.0 ){ U0j = ej - M11; U1j = M01; normsq = U0j * U0j + U1j * U1j; normj = sqrt( normsq ); // ここで0とのズレが生じうる if( normj == 0.0 ){ degenerate = true; break; } } } if( degenerate ){ e0 = M11; e1 = M11; U00 = 1.0; U10 = 0.0; U01 = 0.0; U11 = 1.0; } else { U00 /= norm0; U10 /= norm0; U01 /= norm1; U11 /= norm1; } TwoByTwoMatrix U{ U00 , U01 , U10 , U11 }; TwoByTwoMatrix Uinv{ U00 , U10 , U01 , U11 }; TwoByTwoMatrix Dcbrt{ cbrt( e0 ) , 0.0 , 0.0 , cbrt( e1 ) }; TwoByTwoMatrix Mcbrt = U * Dcbrt * Uinv; cout << Mcbrt.m_M00 << " " << Mcbrt.m_M01 << "\n"; cout << Mcbrt.m_M10 << " " << Mcbrt.m_M11 << "\n"; } QUIT; }