結果

問題 No.1907 DETERMINATION
ユーザー hitonanodehitonanode
提出日時 2022-02-01 00:02:20
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 1,052 ms / 4,000 ms
コード長 5,292 bytes
コンパイル時間 1,875 ms
コンパイル使用メモリ 104,496 KB
実行使用メモリ 5,888 KB
最終ジャッジ日時 2024-12-24 12:19:06
合計ジャッジ時間 33,927 ms
ジャッジサーバーID
(参考情報)
judge3 / judge5
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ファイルパターン 結果
sample AC * 4
other AC * 63
権限があれば一括ダウンロードができます

ソースコード

diff #
プレゼンテーションモードにする

#include <cassert>
#include <iostream>
#include <utility>
#include <vector>
using namespace std;
// Upper Hessenberg reduction of square matrices
// Complexity: O(n^3)
// Reference:
// http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf
template <class Tp> void hessenberg_reduction(std::vector<std::vector<Tp>> &M) {
assert(M.size() == M[0].size());
const int N = M.size();
for (int r = 0; r < N - 2; r++) {
int piv = -1;
for (int h = r + 1; h < N; ++h) {
if (M[h][r] != 0) {
piv = h;
break;
}
}
if (piv < 0) continue;
for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);
const auto rinv = Tp(1) / M[r + 1][r];
for (int i = r + 2; i < N; i++) {
const auto n = M[i][r] * rinv;
for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * n;
for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * n;
}
}
}
// Characteristic polynomial of matrix M (|xI - M|)
// Complexity: O(n^3)
// R. Rehman, I. C. Ipsen, "La Budde's Method for Computing Characteristic Polynomials," 2011.
template <class Tp> std::vector<Tp> characteristic_poly(std::vector<std::vector<Tp>> M) {
hessenberg_reduction(M);
const int N = M.size();
// p[i + 1] = (Characteristic polynomial of i-th leading principal minor)
std::vector<std::vector<Tp>> p(N + 1);
p[0] = {1};
for (int i = 0; i < N; i++) {
p[i + 1].assign(i + 2, 0);
for (int j = 0; j < i + 1; j++) p[i + 1][j + 1] += p[i][j];
for (int j = 0; j < i + 1; j++) p[i + 1][j] -= p[i][j] * M[i][i];
Tp betas = 1;
for (int j = i - 1; j >= 0; j--) {
betas *= M[j + 1][j];
Tp hb = -M[j][i] * betas;
for (int k = 0; k < j + 1; k++) p[i + 1][k] += hb * p[j][k];
}
}
return p[N];
}
#include <atcoder/modint>
using mint = atcoder::modint998244353;
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N;
cin >> N;
vector M0(N, vector<mint>(N)), M1(N, vector<mint>(N));
for (auto &vec : M0) {
for (auto &x : vec) {
int v;
cin >> v;
x = v;
}
}
for (auto &vec : M1) {
for (auto &x : vec) {
int v;
cin >> v;
x = v;
}
}
int multiply_by_x = 0; // M0 + M1x x
mint detAdetBinv = 1;
for (int p = 0; p < N; ++p) {
// M1[p][p] nonzero M1 p p
int piv = -1;
for (int r = p; r < N; ++r) {
if (M1[r][p] != 0) {
piv = r;
break;
}
}
if (piv < 0) {
++multiply_by_x;
if (multiply_by_x > N) break;
for (int i = 0; i < N; ++i) {
swap(M1[i][p], M0[i][p]);
}
for (int r = p - 1; r >= 0; --r) {
auto v = M1[r][p];
for (int i = 0; i < N; ++i) {
M0[i][p] -= M0[i][r] * v;
M1[i][p] -= M1[i][r] * v;
}
assert(M1[r][p] == 0);
}
--p;
continue;
}
if (piv != p) {
M1[piv].swap(M1[p]);
M0[piv].swap(M0[p]);
detAdetBinv *= -1;
}
auto v = M1[p][p], vinv = v.inv();
detAdetBinv *= v;
// p M1[p][p] == 1
for (int j = 0; j < N; ++j) {
M0[p][j] *= vinv;
M1[p][j] *= vinv;
}
assert(M1[p][p] == 1);
// p 使 M1 p p
for (int r = 0; r < N; ++r) {
if (r == p) continue;
if (M1[r][p] != 0) {
auto v = M1[r][p];
for (int j = 0; j < N; ++j) {
M0[r][j] -= M0[p][j] * v;
M1[r][j] -= M1[p][j] * v;
}
}
}
// p 使 M1 p p
for (int j = p + 1; j < N; ++j) {
if (M1[p][j] != 0) {
auto v = M1[p][j];
for (int r = 0; r < N; ++r) {
M0[r][j] -= M0[r][p] * v;
M1[r][j] -= M1[r][p] * v;
}
}
}
}
if (multiply_by_x > N) {
//
for (int i = 0; i <= N; ++i) cout << 0 << '\n';
return 0;
}
// M1 = I det(x + M0)
for (auto &vec : M0) {
for (auto &x : vec) x = -x;
}
auto poly = characteristic_poly(M0);
for (auto &x : poly) x *= detAdetBinv;
for (int i = 0; i < multiply_by_x; ++i) poly.erase(poly.begin());
poly.resize(N + 1);
for (auto a : poly) cout << a.val() << '\n';
}
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