#908289 (C++23) No.2447 行列累乗根

提出ソース

結果

問題	No.2447 行列累乗根
ユーザー	Aeren
提出日時	2023-08-26 16:43:01
言語	C++23 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	312 ms / 2,000 ms
コード長	12,504 bytes
コンパイル時間	4,514 ms
コンパイル使用メモリ	361,440 KB
実行使用メモリ	5,376 KB
最終ジャッジ日時	2024-06-07 12:19:04
合計ジャッジ時間	10,650 ms
ジャッジサーバーID （参考情報）	judge5 / judge2

このコードへのチャレンジ
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テストケース

テストケース表示

入力	結果	実行時間実行使用メモリ
testcase_00	AC	2 ms 5,248 KB
testcase_01	AC	3 ms 5,248 KB
testcase_02	AC	2 ms 5,248 KB
testcase_03	AC	2 ms 5,248 KB
testcase_04	AC	2 ms 5,248 KB
testcase_05	AC	2 ms 5,376 KB
testcase_06	AC	3 ms 5,376 KB
testcase_07	AC	3 ms 5,376 KB
testcase_08	AC	3 ms 5,376 KB
testcase_09	AC	3 ms 5,376 KB
testcase_10	AC	2 ms 5,376 KB
testcase_11	AC	2 ms 5,376 KB
testcase_12	AC	3 ms 5,376 KB
testcase_13	AC	3 ms 5,376 KB
testcase_14	AC	3 ms 5,376 KB
testcase_15	AC	2 ms 5,376 KB
testcase_16	AC	2 ms 5,376 KB
testcase_17	AC	3 ms 5,376 KB
testcase_18	AC	4 ms 5,376 KB
testcase_19	AC	31 ms 5,376 KB
testcase_20	AC	286 ms 5,376 KB
testcase_21	AC	302 ms 5,376 KB
testcase_22	AC	276 ms 5,376 KB
testcase_23	AC	286 ms 5,376 KB
testcase_24	AC	312 ms 5,376 KB
testcase_25	AC	299 ms 5,376 KB
testcase_26	AC	287 ms 5,376 KB
testcase_27	AC	296 ms 5,376 KB

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ソースコード

raw source code

#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                                   //
//                                                                                    //
////////////////////////////////////////////////////////////////////////////////////////

yukicoder

結果

テストケース

ソースコード