結果
| 問題 |
No.2447 行列累乗根
|
| コンテスト | |
| ユーザー |
 Aeren
|
| 提出日時 | 2023-08-26 16:43:01 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 336 ms / 2,000 ms |
| コード長 | 12,504 bytes |
| コンパイル時間 | 3,936 ms |
| コンパイル使用メモリ | 362,884 KB |
| 実行使用メモリ | 5,248 KB |
| 最終ジャッジ日時 | 2024-12-25 15:39:53 |
| 合計ジャッジ時間 | 11,767 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge2 |
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| ファイルパターン | 結果 |
|---|---|
| other | AC * 28 |
ソースコード
#include <bits/stdc++.h>
#include <x86intrin.h>
using namespace std;
// T must support +=, -=, *, *=, ==, and !=
template<class T, size_t N, size_t M>
struct matrix_fixed_base{
using ring_t = T;
using domain_t = array<T, M>;
using range_t = array<T, N>;
static constexpr int n = N, m = M;
array<array<T, M>, N> data;
array<T, M> &operator()(int i){
assert(0 <= i && i < n);
return data[i];
}
const array<T, M> &operator()(int i) const{
assert(0 <= i && i < n);
return data[i];
}
T &operator()(int i, int j){
assert(0 <= i && i < n && 0 <= j && j < m);
return data[i][j];
}
const T &operator()(int i, int j) const{
assert(0 <= i && i < n && 0 <= j && j < m);
return data[i][j];
}
operator vector<vector<T>>() const{
return data;
}
bool operator==(const matrix_fixed_base &a) const{
assert(n == a.n && m == a.m);
return data == a.data;
}
bool operator!=(const matrix_fixed_base &a) const{
assert(n == a.n && m == a.m);
return data != a.data;
}
matrix_fixed_base &operator+=(const matrix_fixed_base &a){
assert(n == a.n && m == a.m);
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] += a(i, j);
return *this;
}
matrix_fixed_base operator+(const matrix_fixed_base &a) const{
assert(n == a.n && m == a.m);
return matrix_fixed_base(*this) += a;
}
matrix_fixed_base &operator-=(const matrix_fixed_base &a){
assert(n == a.n && m == a.m);
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] += a(i, j);
return *this;
}
matrix_fixed_base operator-(const matrix_fixed_base &a) const{
assert(n == a.n && m == a.m);
return matrix_fixed_base(*this) += a;
}
template<size_t N2, size_t M2>
matrix_fixed_base<T, N, M2> operator*(const matrix_fixed_base<T, N2, M2> &a) const{
assert(m == a.n);
int l = M2;
matrix_fixed_base<T, N, M2> res;
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) for(auto k = 0; k < l; ++ k) res(i, k) += data[i][j] * a(j, k);
return res;
}
template<size_t N2, size_t M2>
matrix_fixed_base &operator*=(const matrix_fixed_base<T, N2, M2> &a){
return *this = *this * a;
}
matrix_fixed_base &operator*=(T c){
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] *= c;
return *this;
}
matrix_fixed_base operator*(T c) const{
return matrix_fixed_base(*this) *= c;
}
template<class U, typename enable_if<is_integral<U>::value>::type* = nullptr>
matrix_fixed_base &inplace_power(U e){
assert(n == m && e >= 0);
matrix_fixed_base res(1, 0);
for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
return *this = res;
}
template<class U>
matrix_fixed_base power(U e) const{
return matrix_fixed_base(*this).inplace_power(e);
}
matrix_fixed_base &inplace_transpose(){
assert(n == m);
for(auto i = 0; i < n; ++ i) for(auto j = i + 1; j < n; ++ j) swap(data[i][j], data[j][i]);
return *this;
}
matrix_fixed_base transpose() const{
if(n == m) return matrix_fixed_base(*this).inplace_transpose();
matrix_fixed_base<T, M, N> res;
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) res(j, i) = data[i][j];
return res;
}
// Multiply a column vector v on the right
range_t operator*(const domain_t &v) const{
range_t res;
res.fill(T(0));
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) res[i] += data[i][j] * v[j];
return res;
}
// Assumes T is either a floating, integral, or a modular type.
// If T is a floating type, O(n) divisions with O(n^3) additions, subtractions, and multiplications.
// Otherwise, O(n^2 * log(size)) divisions with O(n^3) additions, subtractions, and multiplications.
// Returns {REF matrix, determinant, rank}
tuple<matrix_fixed_base &, T, int> inplace_REF(){
if(n == 0) return {*this, T(1), 0};
T det = 1;
int rank = 0;
for(auto j = 0; j < m; ++ j){
if constexpr(is_floating_point_v<T>){
static const T eps = 1e-9;
int pivot = rank;
for(auto i = rank + 1; i < n; ++ i) if(abs(data[pivot][j]) < abs(data[i][j])) pivot = i;
if(rank != pivot){
swap(data[rank], data[pivot]);
det *= -1;
}
if(abs(data[rank][j]) <= eps) continue;
det *= data[rank][j];
T inv = 1 / data[rank][j];
for(auto i = rank + 1; i < n; ++ i) if(abs(data[i][j]) > eps){
T coef = data[i][j] * inv;
for(auto k = j; k < m; ++ k) data[i][k] -= coef * data[rank][k];
}
++ rank;
}
else{
for(auto i = rank + 1; i < n; ++ i) while(data[i][j]){
T q;
if constexpr(is_integral_v<T> || is_same_v<T, __int128_t> || is_same_v<T, __uint128_t>) q = data[rank][j] / data[i][j];
else q = data[rank][j].data / data[i][j].data;
if(q) for(auto k = j; k < m; ++ k) data[rank][k] -= q * data[i][k];
swap(data[rank], data[i]);
det *= -1;
}
if(rank == j) det *= data[rank][j];
else det = T(0);
if(data[rank][j]) ++ rank;
}
if(rank == n) break;
}
return {*this, det, rank};
}
// Assumes T is either a floating, integral, or a modular type.
// If T is a floating type, O(n) divisions with O(n^3) additions, subtractions, and multiplications.
// Otherwise, O(n^2 * log(size)) divisions with O(n^3) additions, subtractions, and multiplications.
// Returns {REF matrix, determinant, rank}
tuple<matrix_fixed_base, T, int> REF() const{
return matrix_fixed_base(*this).inplace_REF();
}
// Assumes T is a field.
// O(n) divisions with O(n^3) additions, subtractions, and multiplications.
// Returns {REF matrix, determinant, rank}
tuple<matrix_fixed_base &, T, int> inplace_REF_field(){
if(n == 0) return {*this, T(1), 0};
T det = T(1);
int rank = 0;
for(auto j = 0; j < m; ++ j){
int pivot = -1;
for(auto i = rank; i < n; ++ i) if(data[i][j] != T(0)){
pivot = i;
break;
}
if(!~pivot){
det = T(0);
continue;
}
if(rank != pivot){
swap(data[rank], data[pivot]);
det *= -1;
}
det *= data[rank][j];
T inv = 1 / data[rank][j];
for(auto i = rank + 1; i < n; ++ i) if(data[i][j] != T(0)){
T coef = data[i][j] * inv;
for(auto k = j; k < m; ++ k) data[i][k] -= coef * data[j][k];
}
++ rank;
if(rank == n) break;
}
return {*this, det, rank};
}
// Assumes T is a field.
// O(n) divisions with O(n^3) additions, subtractions, and multiplications.
// Returns {REF matrix, determinant, rank}
tuple<matrix_fixed_base, T, int> REF_field() const{
return matrix_fixed_base(*this).inplace_REF_field();
}
// Assumes T is a field.
// O(n) divisions with O(n^3) additions, subtractions, and multiplications.
optional<matrix_fixed_base> inverse() const{
assert(n == m);
if(n == 0) return *this;
auto a = data;
array<array<T, N>, N> res{};
for(auto i = 0; i < n; ++ i) res[i][i] = T(1);
for(auto j = 0; j < n; ++ j){
int rank = j, pivot = -1;
if(is_floating_point_v<T>){
static const T eps = 1e-9;
pivot = rank;
for(auto i = rank + 1; i < n; ++ i) if(abs(a[pivot][j]) < abs(a[i][j])) pivot = i;
if(abs(a[pivot][j]) <= eps) return {};
}
else{
for(auto i = rank; i < n; ++ i) if(a[i][j] != T(0)){
pivot = i;
break;
}
if(!~pivot) return {};
}
swap(a[rank], a[pivot]), swap(res[rank], res[pivot]);
T inv = 1 / a[rank][j];
for(auto k = 0; k < n; ++ k) a[rank][k] *= inv, res[rank][k] *= inv;
for(auto i = 0; i < n; ++ i){
if constexpr(is_floating_point_v<T>){
static const T eps = 1e-9;
if(i != rank && abs(a[i][j]) > eps){
T d = a[i][j];
for(auto k = 0; k < n; ++ k) a[i][k] -= d * a[rank][k], res[i][k] -= d * res[rank][k];
}
}
else if(i != rank && a[i][j] != T(0)){
T d = a[i][j];
for(auto k = 0; k < n; ++ k) a[i][k] -= d * a[rank][k], res[i][k] -= d * res[rank][k];
}
}
}
return res;
}
// Assumes T is either a floating, integral, or a modular type.
// If T is a floating type, O(n) divisions with O(n^3) additions, subtractions, and multiplications.
// Otherwise, O(n^2 * log(size)) divisions with O(n^3) additions, subtractions, and multiplications.
T determinant() const{
assert(n == m);
return get<1>(REF());
}
// Assumes T is a field.
// O(n) divisions with O(n^3) additions, subtractions, and multiplications.
T determinant_field() const{
assert(n == m);
return get<1>(REF_field());
}
// Assumes T is either a floating, integral, or a modular type.
// If T is a floating type, O(n) divisions with O(n^3) additions, subtractions, and multiplications.
// Otherwise, O(n^2 * log(size)) divisions with O(n^3) additions, subtractions, and multiplications.
int rank() const{
return get<2>(REF());
}
// Assumes T is a field.
// O(n) divisions with O(n^3) additions, subtractions, and multiplications.
int rank_field() const{
return get<2>(REF_field());
}
// O(n * 2^n)
T permanent() const{
static_assert(n <= 30 && n == m);
T perm = n ? 0 : 1;
array<T, N> sum{};
for(auto order = 1; order < 1 << n; ++ order){
int j = __lg(order ^ order >> 1 ^ order - 1 ^ order - 1 >> 1), sign = (order ^ order >> 1) & 1 << j ? 1 : -1;
T prod = order & 1 ? -1 : 1;
if((order ^ order >> 1) & 1 << j) for(auto i = 0; i < n; ++ i) prod *= sum[i] += data[i][j];
else for(auto i = 0; i < n; ++ i) prod *= sum[i] -= data[i][j];
perm += prod;
}
return perm * (n & 1 ? -1 : 1);
}
template<class output_stream>
friend output_stream &operator<<(output_stream &out, const matrix_fixed_base &a){
out << "{";
for(auto i = 0; i < a.n; ++ i){
out << "{";
for(auto j = 0; j < a.m; ++ j){
out << a(i, j);
if(j != a.m - 1) out << ", ";
}
out << "}";
if(i != a.n - 1) out << ", ";
}
return out << "}";
}
matrix_fixed_base(): matrix_fixed_base(T(0), T(0)){ }
matrix_fixed_base(const T &init_diagonal, const T &init_off_diagonal){
for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] = i == j ? init_diagonal : init_off_diagonal;
}
matrix_fixed_base(const array<array<T, M>, N> &arr): data(arr){ }
static matrix_fixed_base additive_identity(){
return matrix_fixed_base(T(0), T(0));
}
static matrix_fixed_base multiplicative_identity(){
return matrix_fixed_base(T(1), T(0));
}
};
template<class T, size_t N, size_t M>
matrix_fixed_base<T, N, M> operator*(T c, matrix_fixed_base<T, N, M> a){
for(auto i = 0; i < a.n; ++ i) for(auto j = 0; j < a.m; ++ j) a(i, j) = c * a(i, j);
return a;
}
// Multiply a row vector v on the left
template<class T, size_t N, size_t M>
typename matrix_fixed_base<T, N, M>::domain_t operator*(const typename matrix_fixed_base<T, N, M>::range_t &v, const matrix_fixed_base<T, N, M> &a){
typename matrix_fixed_base<T, N, M>::domain_t res;
res.fill(T(0));
for(auto i = 0; i < a.n; ++ i) for(auto j = 0; j < a.m; ++ j) res[j] += v[i] * a(i, j);
return res;
}
template<class T>
using matrix = matrix_fixed_base<T, 2, 2>;
int main(){
cin.tie(0)->sync_with_stdio(0);
cin.exceptions(ios::badbit | ios::failbit);
cout << fixed << setprecision(5);
auto __solve_tc = [&](auto __tc_num)->int{
array<array<double, 2>, 2> a;
for(auto i = 0; i < 2; ++ i){
for(auto j = 0; j < 2; ++ j){
cin >> a[i][j];
}
}
double det = max(0.0, (a[0][0] - a[1][1]) * (a[0][0] - a[1][1]) + 4 * a[0][1] * a[1][0]);
if(det <= 1e-11){
for(auto i = 0; i < 2; ++ i){
for(auto j = 0; j < 2; ++ j){
cout << cbrt(a[i][j]) << " \n"[j];
}
}
}
else{
double alpha = ((a[0][0] + a[1][1]) + sqrt(det)) / 2;
double beta = ((a[0][0] + a[1][1]) - sqrt(det)) / 2;
matrix<double> basis, mat;
double coef = 0.31231;
basis(0, 0) = a[0][1] + coef * (a[1][1] - alpha);
basis(0, 1) = a[0][1] + coef * (a[1][1] - beta);
basis(1, 0) = alpha - a[0][0] - coef * a[1][0], basis(1, 1) = beta - a[0][0] - coef * a[1][0];
mat(0, 0) = cbrt(alpha), mat(1, 1) = cbrt(beta);
mat = basis * mat * *basis.inverse();
for(auto i = 0; i < 2; ++ i){
for(auto j = 0; j < 2; ++ j){
cout << mat(i, j) << " \n"[j];
}
}
}
return 0;
};
int __tc_cnt;
cin >> __tc_cnt;
for(auto __tc_num = 0; __tc_num < __tc_cnt; ++ __tc_num){
__solve_tc(__tc_num);
}
return 0;
}
/*
*/
////////////////////////////////////////////////////////////////////////////////////////
// //
// Coded by Aeren //
// //
////////////////////////////////////////////////////////////////////////////////////////