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#define PROBLEM "https://judge.yosupo.jp/problem/characteristic_polynomial" #include "../characteristic_poly.hpp" #include "../../modint.hpp" #include "../hessenberg_reduction.hpp" #include "../matrix.hpp" #include <algorithm> #include <iostream> using namespace std; int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N; cin >> N; using mint = ModInt<998244353>; matrix<mint> M(N, N); cin >> M; hessenberg_reduction(M); auto poly = characteristic_poly_of_hessenberg<mint>(M); for (auto x : poly) cout << x << ' '; cout << '\n'; }
#line 1 "linear_algebra_matrix/test/characteristic_poly.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/characteristic_polynomial" #line 2 "linear_algebra_matrix/matrix.hpp" #include <algorithm> #include <cassert> #include <cmath> #include <iterator> #include <type_traits> #include <utility> #include <vector> namespace matrix_ { struct has_id_method_impl { template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type()); template <class T_> static auto check(...) -> std::false_type; }; template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {}; } // namespace matrix_ template <typename T> struct matrix { int H, W; std::vector<T> elem; typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; } inline T &at(int i, int j) { return elem[i * W + j]; } inline T get(int i, int j) const { return elem[i * W + j]; } int height() const { return H; } int width() const { return W; } std::vector<std::vector<T>> vecvec() const { std::vector<std::vector<T>> ret(H); for (int i = 0; i < H; i++) { std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i])); } return ret; } operator std::vector<std::vector<T>>() const { return vecvec(); } matrix() = default; matrix(int H, int W) : H(H), W(W), elem(H * W) {} matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) { for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem)); } template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr> static T2 _T_id() { return T2::id(); } template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr> static T2 _T_id() { return T2(1); } static matrix Identity(int N) { matrix ret(N, N); for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>(); return ret; } matrix operator-() const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i]; return ret; } matrix operator*(const T &v) const { matrix ret = *this; for (auto &x : ret.elem) x *= v; return ret; } matrix operator/(const T &v) const { matrix ret = *this; const T vinv = _T_id<T>() / v; for (auto &x : ret.elem) x *= vinv; return ret; } matrix operator+(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i]; return ret; } matrix operator-(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i]; return ret; } matrix operator*(const matrix &r) const { matrix ret(H, r.W); for (int i = 0; i < H; i++) { for (int k = 0; k < W; k++) { for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j); } } return ret; } matrix &operator*=(const T &v) { return *this = *this * v; } matrix &operator/=(const T &v) { return *this = *this / v; } matrix &operator+=(const matrix &r) { return *this = *this + r; } matrix &operator-=(const matrix &r) { return *this = *this - r; } matrix &operator*=(const matrix &r) { return *this = *this * r; } bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; } bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; } bool operator<(const matrix &r) const { return elem < r.elem; } matrix pow(int64_t n) const { matrix ret = Identity(H); bool ret_is_id = true; if (n == 0) return ret; for (int i = 63 - __builtin_clzll(n); i >= 0; i--) { if (!ret_is_id) ret *= ret; if ((n >> i) & 1) ret *= (*this), ret_is_id = false; } return ret; } std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const { matrix x = *this; while (n) { if (n & 1) vec = x * vec; x *= x; n >>= 1; } return vec; }; matrix transpose() const { matrix ret(W, H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j); } return ret; } // Gauss-Jordan elimination // - Require inverse for every non-zero element // - Complexity: O(H^2 W) template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr> static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept { int piv = -1; for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c)))) piv = j; } return piv; } template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr> static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept { for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) != T2()) return j; } return -1; } matrix gauss_jordan() const { int c = 0; matrix mtr(*this); std::vector<int> ws; ws.reserve(W); for (int h = 0; h < H; h++) { if (c == W) break; int piv = choose_pivot(mtr, h, c); if (piv == -1) { c++; h--; continue; } if (h != piv) { for (int w = 0; w < W; w++) { std::swap(mtr[piv][w], mtr[h][w]); mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant } } ws.clear(); for (int w = c; w < W; w++) { if (mtr.at(h, w) != T()) ws.emplace_back(w); } const T hcinv = _T_id<T>() / mtr.at(h, c); for (int hh = 0; hh < H; hh++) if (hh != h) { const T coeff = mtr.at(hh, c) * hcinv; for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff; mtr.at(hh, c) = T(); } c++; } return mtr; } int rank_of_gauss_jordan() const { for (int i = H * W - 1; i >= 0; i--) { if (elem[i] != 0) return i / W + 1; } return 0; } int rank() const { return gauss_jordan().rank_of_gauss_jordan(); } T determinant_of_upper_triangle() const { T ret = _T_id<T>(); for (int i = 0; i < H; i++) ret *= get(i, i); return ret; } int inverse() { assert(H == W); std::vector<std::vector<T>> ret = Identity(H), tmp = *this; int rank = 0; for (int i = 0; i < H; i++) { int ti = i; while (ti < H and tmp[ti][i] == T()) ti++; if (ti == H) { continue; } else { rank++; } ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]); T inv = _T_id<T>() / tmp[i][i]; for (int j = 0; j < W; j++) ret[i][j] *= inv; for (int j = i + 1; j < W; j++) tmp[i][j] *= inv; for (int h = 0; h < H; h++) { if (i == h) continue; const T c = -tmp[h][i]; for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c; for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c; } } *this = ret; return rank; } friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) { assert(m.W == int(v.size())); std::vector<T> ret(m.H); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j]; } return ret; } friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) { assert(int(v.size()) == m.H); std::vector<T> ret(m.W); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j); } return ret; } std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; } std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); } template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) { os << "[(" << x.H << " * " << x.W << " matrix)"; os << "\n[column sums: "; for (int j = 0; j < x.W; j++) { T s = T(); for (int i = 0; i < x.H; i++) s += x.get(i, j); os << s << ","; } os << "]"; for (int i = 0; i < x.H; i++) { os << "\n["; for (int j = 0; j < x.W; j++) os << x.get(i, j) << ","; os << "]"; } os << "]\n"; return os; } template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) { for (auto &v : x.elem) is >> v; return is; } }; #line 4 "linear_algebra_matrix/characteristic_poly.hpp" // Characteristic polynomial of upper Hessenberg 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_of_hessenberg(matrix<Tp> &M) { const int N = M.height(); // 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, Tp()); 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]; } #line 3 "modint.hpp" #include <iostream> #include <set> #line 6 "modint.hpp" template <int md> struct ModInt { using lint = long long; constexpr static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val_; int val() const noexcept { return val_; } constexpr ModInt() : val_(0) {} constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; } constexpr ModInt(lint v) { _setval(v % md + md); } constexpr explicit operator bool() const { return val_ != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val_ + x.val_); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val_ - x.val_ + md); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.val_ % md); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.inv().val() % md); } constexpr ModInt operator-() const { return ModInt()._setval(md - val_); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; } constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; } constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; } constexpr bool operator<(const ModInt &x) const { return val_ < x.val_; } // To use std::map<ModInt, T> friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val_; } constexpr ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static constexpr int cache_limit = std::min(md, 1 << 21); static std::vector<ModInt> facs, facinvs, invs; constexpr static void _precalculation(int N) { const int l0 = facs.size(); if (N > md) N = md; if (N <= l0) return; facs.resize(N), facinvs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i; facinvs[N - 1] = facs.back().pow(md - 2); for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1); for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1]; } constexpr ModInt inv() const { if (this->val_ < cache_limit) { if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0}; while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val_]; } else { return this->pow(md - 2); } } constexpr ModInt fac() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val_]; } constexpr ModInt facinv() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val_]; } constexpr ModInt doublefac() const { lint k = (this->val_ + 1) / 2; return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } constexpr ModInt nCr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv() * ModInt(r).facinv(); } constexpr ModInt nPr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv(); } static ModInt binom(int n, int r) { static long long bruteforce_times = 0; if (r < 0 or n < r) return ModInt(0); if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r); r = std::min(r, n - r); ModInt ret = ModInt(r).facinv(); for (int i = 0; i < r; ++i) ret *= n - i; bruteforce_times += r; return ret; } // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!) // Complexity: O(sum(ks)) template <class Vec> static ModInt multinomial(const Vec &ks) { ModInt ret{1}; int sum = 0; for (int k : ks) { assert(k >= 0); ret *= ModInt(k).facinv(), sum += k; } return ret * ModInt(sum).fac(); } // Catalan number, C_n = binom(2n, n) / (n + 1) // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ... // https://oeis.org/A000108 // Complexity: O(n) static ModInt catalan(int n) { if (n < 0) return ModInt(0); return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv(); } ModInt sqrt() const { if (val_ == 0) return 0; if (md == 2) return val_; if (pow((md - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val_, md - x.val_)); } }; template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0}; using ModInt998244353 = ModInt<998244353>; // using mint = ModInt<998244353>; // using mint = ModInt<1000000007>; #line 5 "linear_algebra_matrix/hessenberg_reduction.hpp" // 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(matrix<Tp> &M) { assert(M.height() == M.width()); const int N = M.height(); for (int r = 0; r < N - 2; r++) { int piv = matrix<Tp>::choose_pivot(M, r + 1, r); 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; } } } template <class Ring> void ring_hessenberg_reduction(matrix<Ring> &M) { assert(M.height() == M.width()); const int N = M.height(); for (int r = 0; r < N - 2; r++) { int piv = matrix<Ring>::choose_pivot(M, r + 1, r); 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]); for (int i = r + 2; i < N; i++) { if (M[i][r] == Ring()) continue; Ring a = M[r + 1][r], b = M[i][r], m00 = 1, m01 = 0, m10 = 0, m11 = 1; while (a != Ring() and b != Ring()) { if (a.val() > b.val()) { auto d = a.val() / b.val(); a -= b * d, m00 -= m10 * d, m01 -= m11 * d; } else { auto d = b.val() / a.val(); b -= a * d, m10 -= m00 * d, m11 -= m01 * d; } } if (a == Ring()) std::swap(a, b), std::swap(m00, m10), std::swap(m01, m11); for (int j = 0; j < N; j++) { Ring anew = M[r + 1][j] * m00 + M[i][j] * m01; Ring bnew = M[r + 1][j] * m10 + M[i][j] * m11; M[r + 1][j] = anew; M[i][j] = bnew; } assert(M[i][r] == 0); for (int j = 0; j < N; j++) { Ring anew = M[j][r + 1] * m11 - M[j][i] * m10; Ring bnew = -M[j][r + 1] * m01 + M[j][i] * m00; M[j][r + 1] = anew; M[j][i] = bnew; } } } } #line 8 "linear_algebra_matrix/test/characteristic_poly.test.cpp" using namespace std; int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N; cin >> N; using mint = ModInt<998244353>; matrix<mint> M(N, N); cin >> M; hessenberg_reduction(M); auto poly = characteristic_poly_of_hessenberg<mint>(M); for (auto x : poly) cout << x << ' '; cout << '\n'; }