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linalg/adjugate_matrix.hpp
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- Last update: 2025-02-12 05:55:32+09:00
- Include:
#include "linalg/adjugate_matrix.hpp"
Depends on
- linalg/basis.hpp
- linalg/characteristic_poly.hpp
- linalg/frobenius.hpp
- linalg/matrix_inv.hpp
- linalg/matrix_mul.hpp
- mod/crt3.hpp
- mod/mod_inv.hpp
- mod/modint.hpp
- mod/modint_common.hpp
- poly/convolution.hpp
- poly/convolution_all.hpp
- poly/convolution_karatsuba.hpp
- poly/convolution_naive.hpp
- poly/ntt.hpp
- poly/ntt_doubling.hpp
- random/base.hpp
Verified with
Code
#include "linalg/frobenius.hpp"
#include "mod/modint.hpp"
#include "linalg/characteristic_poly.hpp"
#include "linalg/frobenius.hpp"
template <typename mint>
vvc<mint> adjugate_matrix(vvc<mint> A) {
int N = len(A);
Frobenius_Form<mint> X(A);
auto F = X.characteristic_poly();
if (N % 2 == 0) {
for (auto& x: F) x = -x;
}
F.erase(F.begin());
return X.poly_eval(F);
}#line 1 "linalg/adjugate_matrix.hpp"
#line 2 "random/base.hpp"
u64 RNG_64() {
static u64 x_ = u64(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
x_ ^= x_ << 7;
return x_ ^= x_ >> 9;
}
u64 RNG(u64 lim) { return RNG_64() % lim; }
ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "linalg/matrix_inv.hpp"
// (det, invA) をかえす
template <typename T>
pair<T, vc<vc<T>>> matrix_inv(vc<vc<T>> A) {
T det = 1;
int N = len(A);
vv(T, B, N, N);
FOR(n, N) B[n][n] = 1;
FOR(i, N) {
FOR(k, i, N) if (A[k][i] != 0) {
if (k != i) {
swap(A[i], A[k]), swap(B[i], B[k]);
det = -det;
}
break;
}
if (A[i][i] == 0) return {T(0), {}};
T c = T(1) / A[i][i];
det *= A[i][i];
FOR(j, i, N) A[i][j] *= c;
FOR(j, N) B[i][j] *= c;
FOR(k, N) if (i != k) {
T c = A[k][i];
FOR(j, i, N) A[k][j] -= A[i][j] * c;
FOR(j, N) B[k][j] -= B[i][j] * c;
}
}
return {det, B};
}
#line 2 "mod/modint_common.hpp"
struct has_mod_impl {
template <class T>
static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
template <class T>
static auto check(...) -> std::false_type;
};
template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};
template <typename mint>
mint inv(int n) {
static const int mod = mint::get_mod();
static vector<mint> dat = {0, 1};
assert(0 <= n);
if (n >= mod) n %= mod;
while (len(dat) <= n) {
int k = len(dat);
int q = (mod + k - 1) / k;
dat.eb(dat[k * q - mod] * mint::raw(q));
}
return dat[n];
}
template <typename mint>
mint fact(int n) {
static const int mod = mint::get_mod();
assert(0 <= n && n < mod);
static vector<mint> dat = {1, 1};
while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
return dat[n];
}
template <typename mint>
mint fact_inv(int n) {
static vector<mint> dat = {1, 1};
if (n < 0) return mint(0);
while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
return dat[n];
}
template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
return (mint(1) * ... * fact_inv<mint>(xs));
}
template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}
template <typename mint>
mint C_dense(int n, int k) {
assert(n >= 0);
if (k < 0 || n < k) return 0;
static vvc<mint> C;
static int H = 0, W = 0;
auto calc = [&](int i, int j) -> mint {
if (i == 0) return (j == 0 ? mint(1) : mint(0));
return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
};
if (W <= k) {
FOR(i, H) {
C[i].resize(k + 1);
FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
}
W = k + 1;
}
if (H <= n) {
C.resize(n + 1);
FOR(i, H, n + 1) {
C[i].resize(W);
FOR(j, W) { C[i][j] = calc(i, j); }
}
H = n + 1;
}
return C[n][k];
}
template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
assert(n >= 0);
if (k < 0 || n < k) return 0;
if constexpr (dense) return C_dense<mint>(n, k);
if constexpr (!large) return multinomial<mint>(n, k, n - k);
k = min(k, n - k);
mint x(1);
FOR(i, k) x *= mint(n - i);
return x * fact_inv<mint>(k);
}
template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
assert(n >= 0);
assert(0 <= k && k <= n);
if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
return mint(1) / C<mint, 1>(n, k);
}
// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
assert(n >= 0);
if (d < 0) return mint(0);
if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "mod/modint.hpp"
template <int mod>
struct modint {
static constexpr u32 umod = u32(mod);
static_assert(umod < u32(1) << 31);
u32 val;
static modint raw(u32 v) {
modint x;
x.val = v;
return x;
}
constexpr modint() : val(0) {}
constexpr modint(u32 x) : val(x % umod) {}
constexpr modint(u64 x) : val(x % umod) {}
constexpr modint(u128 x) : val(x % umod) {}
constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
bool operator<(const modint &other) const { return val < other.val; }
modint &operator+=(const modint &p) {
if ((val += p.val) >= umod) val -= umod;
return *this;
}
modint &operator-=(const modint &p) {
if ((val += umod - p.val) >= umod) val -= umod;
return *this;
}
modint &operator*=(const modint &p) {
val = u64(val) * p.val % umod;
return *this;
}
modint &operator/=(const modint &p) {
*this *= p.inverse();
return *this;
}
modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
modint operator+(const modint &p) const { return modint(*this) += p; }
modint operator-(const modint &p) const { return modint(*this) -= p; }
modint operator*(const modint &p) const { return modint(*this) *= p; }
modint operator/(const modint &p) const { return modint(*this) /= p; }
bool operator==(const modint &p) const { return val == p.val; }
bool operator!=(const modint &p) const { return val != p.val; }
modint inverse() const {
int a = val, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b), swap(u -= t * v, v);
}
return modint(u);
}
modint pow(ll n) const {
if (n < 0) return inverse().pow(-n);
assert(n >= 0);
modint ret(1), mul(val);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
static constexpr int get_mod() { return mod; }
// (n, r), r は 1 の 2^n 乗根
static constexpr pair<int, int> ntt_info() {
if (mod == 120586241) return {20, 74066978};
if (mod == 167772161) return {25, 17};
if (mod == 469762049) return {26, 30};
if (mod == 754974721) return {24, 362};
if (mod == 880803841) return {23, 211};
if (mod == 943718401) return {22, 663003469};
if (mod == 998244353) return {23, 31};
if (mod == 1004535809) return {21, 582313106};
if (mod == 1012924417) return {21, 368093570};
return {-1, -1};
}
static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};
#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
fastio::rd(x.val);
x.val %= mod;
// assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
fastio::wt(x.val);
}
#endif
using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 3 "linalg/matrix_mul.hpp"
template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1,
int N2 = -1, int N3 = -1) {
if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); }
vv(u32, b, N3, N2);
FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j].val;
vv(T, C, N1, N3);
if ((T::get_mod() < (1 << 30)) && N2 <= 16) {
FOR(i, N1) FOR(j, N3) {
u64 sm = 0;
FOR(m, N2) sm += u64(A[i][m].val) * b[j][m];
C[i][j] = sm;
}
} else {
FOR(i, N1) FOR(j, N3) {
u128 sm = 0;
FOR(m, N2) sm += u64(A[i][m].val) * b[j][m];
C[i][j] = T::raw(sm % (T::get_mod()));
}
}
return C;
}
template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1,
int N2 = -1, int N3 = -1) {
if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); }
vv(T, b, N2, N3);
FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j];
vv(T, C, N1, N3);
FOR(n, N1) FOR(m, N2) FOR(k, N3) C[n][k] += A[n][m] * b[k][m];
return C;
}
// square-matrix defined as array
template <class T, int N,
typename enable_if<has_mod<T>::value>::type* = nullptr>
array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A,
const array<array<T, N>, N>& B) {
array<array<T, N>, N> C{};
if ((T::get_mod() < (1 << 30)) && N <= 16) {
FOR(i, N) FOR(k, N) {
u64 sm = 0;
FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val);
C[i][k] = sm;
}
} else {
FOR(i, N) FOR(k, N) {
u128 sm = 0;
FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val);
C[i][k] = sm;
}
}
return C;
}
// square-matrix defined as array
template <class T, int N,
typename enable_if<!has_mod<T>::value>::type* = nullptr>
array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A,
const array<array<T, N>, N>& B) {
array<array<T, N>, N> C{};
FOR(i, N) FOR(j, N) FOR(k, N) C[i][k] += A[i][j] * B[j][k];
return C;
}
#line 2 "mod/mod_inv.hpp"
// long でも大丈夫
// (val * x - 1) が mod の倍数になるようにする
// 特に mod=0 なら x=0 が満たす
ll mod_inv(ll val, ll mod) {
if (mod == 0) return 0;
mod = abs(mod);
val %= mod;
if (val < 0) val += mod;
ll a = val, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b), swap(u -= t * v, v);
}
if (u < 0) u += mod;
return u;
}
#line 2 "mod/crt3.hpp"
constexpr u32 mod_pow_constexpr(u64 a, u64 n, u32 mod) {
a %= mod;
u64 res = 1;
FOR(32) {
if (n & 1) res = res * a % mod;
a = a * a % mod, n /= 2;
}
return res;
}
template <typename T, u32 p0, u32 p1>
T CRT2(u64 a0, u64 a1) {
static_assert(p0 < p1);
static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1);
u64 c = (a1 - a0 + p1) * x0_1 % p1;
return a0 + c * p0;
}
template <typename T, u32 p0, u32 p1, u32 p2>
T CRT3(u64 a0, u64 a1, u64 a2) {
static_assert(p0 < p1 && p1 < p2);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 p01 = u64(p0) * p1;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
return T(ans_1) + T(c) * T(p01);
}
template <typename T, u32 p0, u32 p1, u32 p2, u32 p3>
T CRT4(u64 a0, u64 a1, u64 a2, u64 a3) {
static_assert(p0 < p1 && p1 < p2 && p2 < p3);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3);
static constexpr u64 p01 = u64(p0) * p1;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
u128 ans_2 = ans_1 + c * static_cast<u128>(p01);
c = (a3 - ans_2 % p3 + p3) * x3 % p3;
return T(ans_2) + T(c) * T(p01) * T(p2);
}
template <typename T, u32 p0, u32 p1, u32 p2, u32 p3, u32 p4>
T CRT5(u64 a0, u64 a1, u64 a2, u64 a3, u64 a4) {
static_assert(p0 < p1 && p1 < p2 && p2 < p3 && p3 < p4);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3);
static constexpr u64 x4 = mod_pow_constexpr(u64(p0) * p1 % p4 * p2 % p4 * p3 % p4, p4 - 2, p4);
static constexpr u64 p01 = u64(p0) * p1;
static constexpr u64 p23 = u64(p2) * p3;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
u128 ans_2 = ans_1 + c * static_cast<u128>(p01);
c = static_cast<u64>(a3 - ans_2 % p3 + p3) * x3 % p3;
u128 ans_3 = ans_2 + static_cast<u128>(c * p2) * p01;
c = static_cast<u64>(a4 - ans_3 % p4 + p4) * x4 % p4;
return T(ans_3) + T(c) * T(p01) * T(p23);
}
#line 2 "poly/convolution_naive.hpp"
template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
int n = int(a.size()), m = int(b.size());
if (n > m) return convolution_naive<T>(b, a);
if (n == 0) return {};
vector<T> ans(n + m - 1);
FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j];
return ans;
}
template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
int n = int(a.size()), m = int(b.size());
if (n > m) return convolution_naive<T>(b, a);
if (n == 0) return {};
vc<T> ans(n + m - 1);
if (n <= 16 && (T::get_mod() < (1 << 30))) {
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
u64 sm = 0;
for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
ans[k] = sm;
}
} else {
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
u128 sm = 0;
for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
ans[k] = T::raw(sm % T::get_mod());
}
}
return ans;
}
#line 2 "poly/convolution_karatsuba.hpp"
// 任意の環でできる
template <typename T>
vc<T> convolution_karatsuba(const vc<T>& f, const vc<T>& g) {
const int thresh = 30;
if (min(len(f), len(g)) <= thresh) return convolution_naive(f, g);
int n = max(len(f), len(g));
int m = ceil(n, 2);
vc<T> f1, f2, g1, g2;
if (len(f) < m) f1 = f;
if (len(f) >= m) f1 = {f.begin(), f.begin() + m};
if (len(f) >= m) f2 = {f.begin() + m, f.end()};
if (len(g) < m) g1 = g;
if (len(g) >= m) g1 = {g.begin(), g.begin() + m};
if (len(g) >= m) g2 = {g.begin() + m, g.end()};
vc<T> a = convolution_karatsuba(f1, g1);
vc<T> b = convolution_karatsuba(f2, g2);
FOR(i, len(f2)) f1[i] += f2[i];
FOR(i, len(g2)) g1[i] += g2[i];
vc<T> c = convolution_karatsuba(f1, g1);
vc<T> F(len(f) + len(g) - 1);
assert(2 * m + len(b) <= len(F));
FOR(i, len(a)) F[i] += a[i], c[i] -= a[i];
FOR(i, len(b)) F[2 * m + i] += b[i], c[i] -= b[i];
if (c.back() == T(0)) c.pop_back();
FOR(i, len(c)) if (c[i] != T(0)) F[m + i] += c[i];
return F;
}
#line 2 "poly/ntt.hpp"
template <class mint>
void ntt(vector<mint>& a, bool inverse) {
assert(mint::can_ntt());
const int rank2 = mint::ntt_info().fi;
const int mod = mint::get_mod();
static array<mint, 30> root, iroot;
static array<mint, 30> rate2, irate2;
static array<mint, 30> rate3, irate3;
assert(rank2 != -1 && len(a) <= (1 << max(0, rank2)));
static bool prepared = 0;
if (!prepared) {
prepared = 1;
root[rank2] = mint::ntt_info().se;
iroot[rank2] = mint(1) / root[rank2];
FOR_R(i, rank2) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
int n = int(a.size());
int h = topbit(n);
assert(n == 1 << h);
if (!inverse) {
int len = 0;
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
FOR(s, 1 << len) {
int offset = s << (h - len);
FOR(i, p) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
rot *= rate2[topbit(~s & -~s)];
}
len++;
} else {
int p = 1 << (h - len - 2);
mint rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
u64 mod2 = u64(mod) * mod;
u64 a0 = a[i + offset].val;
u64 a1 = u64(a[i + offset + p].val) * rot.val;
u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val;
u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val;
u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val;
u64 na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
}
rot *= rate3[topbit(~s & -~s)];
}
len += 2;
}
}
} else {
mint coef = mint(1) / mint(len(a));
FOR(i, len(a)) a[i] *= coef;
int len = h;
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint irot = 1;
FOR(s, 1 << (len - 1)) {
int offset = s << (h - len + 1);
FOR(i, p) {
u64 l = a[i + offset].val;
u64 r = a[i + offset + p].val;
a[i + offset] = l + r;
a[i + offset + p] = (mod + l - r) * irot.val;
}
irot *= irate2[topbit(~s & -~s)];
}
len--;
} else {
int p = 1 << (h - len);
mint irot = 1, iimag = iroot[2];
FOR(s, (1 << (len - 2))) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
u64 a0 = a[i + offset + 0 * p].val;
u64 a1 = a[i + offset + 1 * p].val;
u64 a2 = a[i + offset + 2 * p].val;
u64 a3 = a[i + offset + 3 * p].val;
u64 x = (mod + a2 - a3) * iimag.val % mod;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val;
a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val;
a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val;
}
irot *= irate3[topbit(~s & -~s)];
}
len -= 2;
}
}
}
}
#line 8 "poly/convolution.hpp"
template <class mint>
vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
if (a.empty() || b.empty()) return {};
int n = int(a.size()), m = int(b.size());
int sz = 1;
while (sz < n + m - 1) sz *= 2;
// sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。
if ((n + m - 3) <= sz / 2) {
auto a_last = a.back(), b_last = b.back();
a.pop_back(), b.pop_back();
auto c = convolution(a, b);
c.resize(n + m - 1);
c[n + m - 2] = a_last * b_last;
FOR(i, len(a)) c[i + len(b)] += a[i] * b_last;
FOR(i, len(b)) c[i + len(a)] += b[i] * a_last;
return c;
}
a.resize(sz), b.resize(sz);
bool same = a == b;
ntt(a, 0);
if (same) {
b = a;
} else {
ntt(b, 0);
}
FOR(i, sz) a[i] *= b[i];
ntt(a, 1);
a.resize(n + m - 1);
return a;
}
template <typename mint>
vector<mint> convolution_garner(const vector<mint>& a, const vector<mint>& b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
static constexpr int p0 = 167772161;
static constexpr int p1 = 469762049;
static constexpr int p2 = 754974721;
using mint0 = modint<p0>;
using mint1 = modint<p1>;
using mint2 = modint<p2>;
vc<mint0> a0(n), b0(m);
vc<mint1> a1(n), b1(m);
vc<mint2> a2(n), b2(m);
FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
auto c0 = convolution_ntt<mint0>(a0, b0);
auto c1 = convolution_ntt<mint1>(a1, b1);
auto c2 = convolution_ntt<mint2>(a2, b2);
vc<mint> c(len(c0));
FOR(i, n + m - 1) { c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val); }
return c;
}
vector<ll> convolution(vector<ll> a, vector<ll> b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
if (min(n, m) <= 2500) return convolution_naive(a, b);
ll mi_a = MIN(a), mi_b = MIN(b);
for (auto& x: a) x -= mi_a;
for (auto& x: b) x -= mi_b;
assert(MAX(a) * MAX(b) <= 1e18);
auto Ac = cumsum<ll>(a), Bc = cumsum<ll>(b);
vi res(n + m - 1);
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
res[k] += (t - s) * mi_a * mi_b;
res[k] += mi_a * (Bc[k - s + 1] - Bc[k - t + 1]);
res[k] += mi_b * (Ac[t] - Ac[s]);
}
static constexpr u32 MOD1 = 1004535809;
static constexpr u32 MOD2 = 1012924417;
using mint1 = modint<MOD1>;
using mint2 = modint<MOD2>;
vc<mint1> a1(n), b1(m);
vc<mint2> a2(n), b2(m);
FOR(i, n) a1[i] = a[i], a2[i] = a[i];
FOR(i, m) b1[i] = b[i], b2[i] = b[i];
auto c1 = convolution_ntt<mint1>(a1, b1);
auto c2 = convolution_ntt<mint2>(a2, b2);
FOR(i, n + m - 1) { res[i] += CRT2<u64, MOD1, MOD2>(c1[i].val, c2[i].val); }
return res;
}
template <typename mint>
vc<mint> convolution(const vc<mint>& a, const vc<mint>& b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
if (mint::can_ntt()) {
if (min(n, m) <= 50) return convolution_karatsuba<mint>(a, b);
return convolution_ntt(a, b);
}
if (min(n, m) <= 200) return convolution_karatsuba<mint>(a, b);
return convolution_garner(a, b);
}
#line 2 "linalg/basis.hpp"
// basis[i]: i 番目に追加成功したもの. 別のラベルがあるなら外で管理する.
// rbasis: 上三角化された基底. [i][i]==1.
// way[i][j]: rbasis[i] = sum way[i][j] basis[j]
template <typename mint>
struct Basis {
int n, rank;
vvc<mint> basis;
vvc<mint> rbasis;
vvc<mint> way;
Basis(int max_dim) : n(max_dim), rank(0), basis{} {
rbasis.assign(max_dim, vc<mint>(max_dim));
way.assign(max_dim, vc<mint>(max_dim));
}
// return : (sum==X にできるか, その方法)
pair<bool, vc<mint>> solve(vc<mint> X) {
vc<mint> CF(n);
FOR(i, n) {
if (rbasis[i][i] == mint(1)) {
CF[i] = X[i];
FOR(j, i, n) X[j] -= CF[i] * rbasis[i][j];
}
}
for (auto& x: X) {
if (x != mint(0)) { return {false, {}}; }
}
vc<mint> ANS(rank);
FOR(i, n) { FOR(j, rank) ANS[j] += CF[i] * way[i][j]; }
return {true, ANS};
}
// return : (sum==x にできるか, その方法). false の場合には追加する
pair<bool, vc<mint>> solve_or_add(vc<mint> X) {
vc<mint> Y = X;
vc<mint> CF(n);
FOR(i, n) {
if (rbasis[i][i] == mint(1)) {
CF[i] = X[i];
FOR(j, i, n) X[j] -= CF[i] * rbasis[i][j];
}
}
int p = [&]() -> int {
FOR(i, n) if (X[i] != mint(0)) return i;
return -1;
}();
if (p == -1) {
vc<mint> ANS(rank);
FOR(i, n) { FOR(j, rank) ANS[j] += CF[i] * way[i][j]; }
return {true, ANS};
}
mint c = X[p].inverse();
FOR(j, p, n) X[j] *= c;
FOR(i, n) CF[i] *= c;
basis.eb(Y), rbasis[p] = X;
way[p][rank] = c;
FOR(i, n) { FOR(j, rank) way[p][j] -= CF[i] * way[i][j]; }
++rank;
return {false, {}};
}
// rank==r の時点まで戻す. Frobenius Form 用.
void rollback(int r) {
while (rank > r) {
--rank;
POP(basis);
FOR(i, n) if (way[i][rank] != mint(0)) {
fill(all(rbasis[i]), mint(0));
fill(all(way[i]), mint(0));
}
}
}
};
#line 2 "poly/convolution_all.hpp"
#line 2 "poly/ntt_doubling.hpp"
#line 4 "poly/ntt_doubling.hpp"
// 2^k 次多項式の長さ 2^k が与えられるので 2^k+1 にする
template <typename mint, bool transposed = false>
void ntt_doubling(vector<mint>& a) {
static array<mint, 30> root;
static bool prepared = 0;
if (!prepared) {
prepared = 1;
const int rank2 = mint::ntt_info().fi;
root[rank2] = mint::ntt_info().se;
FOR_R(i, rank2) { root[i] = root[i + 1] * root[i + 1]; }
}
if constexpr (!transposed) {
const int M = (int)a.size();
auto b = a;
ntt(b, 1);
mint r = 1, zeta = root[topbit(2 * M)];
FOR(i, M) b[i] *= r, r *= zeta;
ntt(b, 0);
copy(begin(b), end(b), back_inserter(a));
} else {
const int M = len(a) / 2;
vc<mint> tmp = {a.begin(), a.begin() + M};
a = {a.begin() + M, a.end()};
transposed_ntt(a, 0);
mint r = 1, zeta = root[topbit(2 * M)];
FOR(i, M) a[i] *= r, r *= zeta;
transposed_ntt(a, 1);
FOR(i, M) a[i] += tmp[i];
}
}
#line 5 "poly/convolution_all.hpp"
template <typename T>
vc<T> convolution_all(vc<vc<T>>& polys) {
if (len(polys) == 0) return {T(1)};
while (1) {
int n = len(polys);
if (n == 1) break;
int m = ceil(n, 2);
FOR(i, m) {
if (2 * i + 1 == n) {
polys[i] = polys[2 * i];
} else {
polys[i] = convolution(polys[2 * i], polys[2 * i + 1]);
}
}
polys.resize(m);
}
return polys[0];
}
// product of 1-A[i]x
template <typename mint>
vc<mint> convolution_all_1(vc<mint> A) {
if (!mint::can_ntt()) {
vvc<mint> polys;
for (auto& a: A) polys.eb(vc<mint>({mint(1), -a}));
return convolution_all(polys);
}
int D = 6;
using poly = vc<mint>;
int n = 1;
while (n < len(A)) n *= 2;
int k = topbit(n);
vc<mint> F(n), nxt_F(n);
FOR(i, len(A)) F[i] = -A[i];
FOR(d, k) {
int b = 1 << d;
if (d < D) {
fill(all(nxt_F), mint(0));
for (int L = 0; L < n; L += 2 * b) {
FOR(i, b) FOR(j, b) { nxt_F[L + i + j] += F[L + i] * F[L + b + j]; }
FOR(i, b) nxt_F[L + b + i] += F[L + i] + F[L + b + i];
}
}
elif (d == D) {
for (int L = 0; L < n; L += 2 * b) {
poly f1 = {F.begin() + L, F.begin() + L + b};
poly f2 = {F.begin() + L + b, F.begin() + L + 2 * b};
f1.resize(2 * b), f2.resize(2 * b), ntt(f1, 0), ntt(f2, 0);
FOR(i, b) nxt_F[L + i] = f1[i] * f2[i] + f1[i] + f2[i];
FOR(i, b, 2 * b) nxt_F[L + i] = f1[i] * f2[i] - f1[i] - f2[i];
}
}
else {
for (int L = 0; L < n; L += 2 * b) {
poly f1 = {F.begin() + L, F.begin() + L + b};
poly f2 = {F.begin() + L + b, F.begin() + L + 2 * b};
ntt_doubling(f1), ntt_doubling(f2);
FOR(i, b) nxt_F[L + i] = f1[i] * f2[i] + f1[i] + f2[i];
FOR(i, b, 2 * b) nxt_F[L + i] = f1[i] * f2[i] - f1[i] - f2[i];
}
}
swap(F, nxt_F);
}
if (k - 1 >= D) ntt(F, 1);
F.eb(1), reverse(all(F));
F.resize(len(A) + 1);
return F;
}
#line 8 "linalg/frobenius.hpp"
// https://codeforces.com/blog/entry/124815
// P^{-1}AP = diag(companion(f0),companion(f1),...)
// without checking: ...|f2|f1|f0.
// time complexity O(N^3). failure prob O(N/mod).
template <typename mint>
struct Frobenius_Form {
int n; // b : num of blocks
const vvc<mint> A;
vvc<mint> P, IP;
vvc<mint> V;
vvc<mint> F; // [-a0,-a1,...,-a[k-1]] の形で管理 (x^k と合同なもの)
Frobenius_Form(vvc<mint>& A) : n(len(A)), A(A) {
while (!trial()) {}
}
// N^3 + N^2log(exp)
vvc<mint> pow(ll exp) {
vv(mint, X, n, n);
int s = 0;
FOR(k, len(F)) {
int d = len(F[k]);
vc<mint> f = powmod(F[k], exp);
FOR(j, d) {
FOR(i, len(f)) { X[s + i][s + j] = f[i]; }
if (j == d - 1) break;
f.insert(f.begin(), 0);
divmod_inplace(f, F[k]);
}
s += d;
}
X = matrix_mul(P, X);
X = matrix_mul(X, IP);
return X;
}
// p(A)
vvc<mint> poly_eval(vc<mint>& p) {
vv(mint, X, n, n);
int s = 0;
FOR(k, len(F)) {
int d = len(F[k]);
vc<mint> f = p;
divmod_inplace(f, F[k]);
FOR(j, d) {
FOR(i, len(f)) { X[s + i][s + j] = f[i]; }
if (j == d - 1) break;
f.insert(f.begin(), 0);
divmod_inplace(f, F[k]);
}
s += d;
}
X = matrix_mul(P, X);
X = matrix_mul(X, IP);
return X;
}
vc<mint> characteristic_poly() {
vvc<mint> polys;
for (auto& f: F) {
vc<mint> g = f;
for (auto& x: g) x = -x;
g.eb(1);
polys.eb(g);
}
vc<mint> f = convolution_all(polys);
return f;
}
// x^n mod (x^d-g(x))
vc<mint> powmod(vc<mint>& g, ll n) {
if (n < len(g)) {
vc<mint> f(n + 1);
f[n] = 1;
return f;
}
vc<mint> f = powmod(g, n / 2);
f = convolution_naive(f, f);
if (n & 1) f.insert(f.begin(), 0);
divmod_inplace(f, g);
return f;
}
private:
bool trial() {
V.clear(), F.clear();
Basis<mint> S(n);
while (1) {
if (S.rank == n) break;
int r = S.rank;
vc<mint> v = random_vector();
V.eb(v);
while (1) {
auto [solved, cf] = S.solve_or_add(v);
if (!solved) {
v = apply(v);
continue;
}
vc<mint> f = {cf.begin() + r, cf.end()};
F.eb(f);
if (len(V) == 1) break;
v = V.back();
int s = 0;
FOR(k, len(V) - 1) {
int d = len(F[k]);
vc<mint> R = {cf.begin() + s, cf.begin() + s + d};
vc<mint> q = divmod_inplace(R, f);
if (!R.empty()) {
return false; // failure
}
FOR(i, len(q)) { FOR(j, n) v[j] -= q[i] * S.basis[s + i][j]; }
s += d;
}
S.rollback(r);
V.back() = v;
FOR(i, len(f)) {
S.solve_or_add(v);
if (i + 1 < len(f)) v = apply(v);
}
break;
}
}
P.assign(n, vc<mint>(n));
FOR(i, n) FOR(j, n) P[i][j] = S.basis[j][i];
IP = matrix_inv<mint>(P).se;
return true;
}
vc<mint> random_vector() {
vc<mint> v(n);
FOR(i, n) v[i] = RNG(0, mint::get_mod());
return v;
}
vc<mint> apply(vc<mint> v) {
vc<mint> w(n);
FOR(i, n) FOR(j, n) w[i] += A[i][j] * v[j];
return w;
}
// f mod= (x^d-g(x)) (inplace), return : q
vc<mint> divmod_inplace(vc<mint>& f, vc<mint>& g) {
while (len(f) && f.back() == mint(0)) POP(f);
int d = len(g);
vc<mint> q;
FOR_R(i, d, len(f)) {
q.eb(f[i]);
FOR(j, len(g)) f[i - d + j] += f[i] * g[j];
f[i] = 0;
}
while (len(f) && f.back() == mint(0)) POP(f);
reverse(all(q));
return q;
}
};
#line 1 "linalg/characteristic_poly.hpp"
template <typename T>
void to_Hessenberg_matrix(vc<vc<T>>& A) {
/*
P^{-1}AP の形の変換で、Hessenberg 行列に変形する。
特定多項式の計算に用いることができる。
*/
int n = len(A);
FOR(k, n - 2) {
FOR3(i, k + 1, n) if (A[i][k] != 0) {
if (i != k + 1) {
swap(A[i], A[k + 1]);
FOR(j, n) swap(A[j][i], A[j][k + 1]);
}
break;
}
if (A[k + 1][k] == 0) continue;
FOR3(i, k + 2, n) {
T c = A[i][k] / A[k + 1][k];
// i 行目 -= k+1 行目 * c
FOR(j, n) A[i][j] -= A[k + 1][j] * c;
// k+1 列目 += i 列目 * c
FOR(j, n) A[j][k + 1] += A[j][i] * c;
}
}
}
// det(xI-A)
template <typename T>
vc<T> characteristic_poly(vc<vc<T>> A) {
/*
・Hessenberg 行列に変形
・Hessenberg 行列の行列式は、最後の列で場合分けすれば dp できる
*/
int n = len(A);
to_Hessenberg_matrix(A);
vc<vc<T>> DP(n + 1);
DP[0] = {1};
FOR(k, n) {
DP[k + 1].resize(k + 2);
auto& dp = DP[k + 1];
// (k, k) 成分を使う場合
FOR(i, len(DP[k])) dp[i + 1] += DP[k][i];
FOR(i, len(DP[k])) dp[i] -= DP[k][i] * A[k][k];
// 下側対角の総積を管理
T prod = 1;
FOR_R(i, k) {
// (i, k) 成分を使う場合
prod *= A[i + 1][i];
T c = prod * A[i][k];
// DP[i] の c 倍を加算
FOR(j, len(DP[i])) dp[j] -= DP[i][j] * c;
}
}
return DP[n];
}
#line 6 "linalg/adjugate_matrix.hpp"
template <typename mint>
vvc<mint> adjugate_matrix(vvc<mint> A) {
int N = len(A);
Frobenius_Form<mint> X(A);
auto F = X.characteristic_poly();
if (N % 2 == 0) {
for (auto& x: F) x = -x;
}
F.erase(F.begin());
return X.poly_eval(F);
}