部分分数分解(分母が1次式の積で表せる場合)
(fps/partial-fraction-decomposition.hpp)
Depends on
Verified with
Code
#pragma once
#include "formal-power-series.hpp"
#include "taylor-shift.hpp"
#include "../modulo/binomial.hpp"
// 新しい方
// 入力 : f/(x-b_1)(x-b_2)...(x-b_n), f は n-1 次, b_i は distinct
// 出力 : a_1/(x-b_1) + ... + a_n/(x-b_n)
template <typename mint>
vector<mint> PartialFractionDecomposition(const FormalPowerSeries<mint>& f,
const vector<mint>& b) {
using fps = FormalPowerSeries<mint>;
int N = b.size();
assert((int)f.size() <= N);
int B = 1;
while (B < N) B *= 2;
vector<fps> mod(2 * B, fps{1});
for (int i = 0; i < N; i++) mod[B + i] = fps{-b[i], 1};
for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1];
vector<mint> ans(N);
auto dfs = [&](auto rc, int i, int l, int r, fps g, fps h) -> void {
if (N <= l) return;
if (l + 1 == r) {
ans[l] = g.eval(0) / h.eval(0);
return;
}
int m = (l + r) / 2;
rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0],
h * mod[i * 2 + 1] % mod[i * 2]);
rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1],
h * mod[i * 2] % mod[i * 2 + 1]);
};
dfs(dfs, 1, 0, B, f, fps{1});
return ans;
}
// 古い方
template <typename mint>
vector<pair<mint, vector<mint>>> PartialFractionDecomposition(
FormalPowerSeries<mint> numer, vector<pair<mint, int>> denom) {
using fps = FormalPowerSeries<mint>;
if (denom.empty()) return {};
sort(begin(denom), end(denom),
[](auto p1, auto p2) { return p1.second < p2.second; });
Binomial<mint> C(denom[0].second + 1);
int s = 1;
while (s < (int)denom.size()) s *= 2;
vector<fps> fs(2 * s);
for (int i = 0; i < s; i++) {
if ((int)denom.size() <= i) {
fs[s + i] = fps{1};
continue;
}
auto [m, d] = denom[i];
fps f(denom[i].second + 1);
mint buf = 1;
for (int j = d; j >= 0; j--) {
f[j] = buf * C(d, j);
buf *= m;
}
fs[s + i] = f;
}
for (int i = s - 1; i; i--) {
fs[i] = fs[2 * i + 0] * fs[2 * i + 1];
}
vector<fps> F(2 * s);
vector<fps> G(2 * s);
F[1] = numer % fs[1];
G[1] = fps{1};
for (int i = 1; i < s; i++) {
F[i * 2 + 0] = F[i] % fs[i * 2 + 0];
F[i * 2 + 1] = F[i] % fs[i * 2 + 1];
G[i * 2 + 0] = G[i] * fs[i * 2 + 1] % fs[i * 2 + 0];
G[i * 2 + 1] = G[i] * fs[i * 2 + 0] % fs[i * 2 + 1];
}
vector<pair<mint, vector<mint>>> res;
for (int i = s; i - s < (int)denom.size(); i++) {
auto [m, d] = denom[i - s];
F[i] = TaylorShift<mint>(F[i], -m, C);
G[i] = TaylorShift<mint>(G[i], -m, C);
fps f = (F[i] * G[i].inv()).pre(d);
if ((int)f.size() < d) f.resize(d);
f = f.rev();
res.emplace_back(m, vector<mint>{begin(f), end(f)});
}
return res;
}
/**
* @brief 部分分数分解(分母が1次式の積で表せる場合)
*/
#line 2 "fps/partial-fraction-decomposition.hpp"
#line 2 "fps/formal-power-series.hpp"
template <typename mint>
struct FormalPowerSeries : vector<mint> {
using vector<mint>::vector;
using FPS = FormalPowerSeries;
FPS &operator+=(const FPS &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
return *this;
}
FPS &operator+=(const mint &r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
FPS &operator-=(const FPS &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
return *this;
}
FPS &operator-=(const mint &r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
FPS &operator*=(const mint &v) {
for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
return *this;
}
FPS &operator/=(const FPS &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
if ((int)r.size() <= 64) {
FPS f(*this), g(r);
g.shrink();
mint coeff = g.back().inverse();
for (auto &x : g) x *= coeff;
int deg = (int)f.size() - (int)g.size() + 1;
int gs = g.size();
FPS quo(deg);
for (int i = deg - 1; i >= 0; i--) {
quo[i] = f[i + gs - 1];
for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
}
*this = quo * coeff;
this->resize(n, mint(0));
return *this;
}
return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
}
FPS &operator%=(const FPS &r) {
*this -= *this / r * r;
shrink();
return *this;
}
FPS operator+(const FPS &r) const { return FPS(*this) += r; }
FPS operator+(const mint &v) const { return FPS(*this) += v; }
FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
FPS operator-(const mint &v) const { return FPS(*this) -= v; }
FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
FPS operator*(const mint &v) const { return FPS(*this) *= v; }
FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
FPS operator-() const {
FPS ret(this->size());
for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
void shrink() {
while (this->size() && this->back() == mint(0)) this->pop_back();
}
FPS rev() const {
FPS ret(*this);
reverse(begin(ret), end(ret));
return ret;
}
FPS dot(FPS r) const {
FPS ret(min(this->size(), r.size()));
for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
// 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
FPS pre(int sz) const {
FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
if ((int)ret.size() < sz) ret.resize(sz);
return ret;
}
FPS operator>>(int sz) const {
if ((int)this->size() <= sz) return {};
FPS ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
FPS operator<<(int sz) const {
FPS ret(*this);
ret.insert(ret.begin(), sz, mint(0));
return ret;
}
FPS diff() const {
const int n = (int)this->size();
FPS ret(max(0, n - 1));
mint one(1), coeff(1);
for (int i = 1; i < n; i++) {
ret[i - 1] = (*this)[i] * coeff;
coeff += one;
}
return ret;
}
FPS integral() const {
const int n = (int)this->size();
FPS ret(n + 1);
ret[0] = mint(0);
if (n > 0) ret[1] = mint(1);
auto mod = mint::get_mod();
for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
return ret;
}
mint eval(mint x) const {
mint r = 0, w = 1;
for (auto &v : *this) r += w * v, w *= x;
return r;
}
FPS log(int deg = -1) const {
assert(!(*this).empty() && (*this)[0] == mint(1));
if (deg == -1) deg = (int)this->size();
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
FPS pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if (k == 0) {
FPS ret(deg);
if (deg) ret[0] = 1;
return ret;
}
for (int i = 0; i < n; i++) {
if ((*this)[i] != mint(0)) {
mint rev = mint(1) / (*this)[i];
FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
ret *= (*this)[i].pow(k);
ret = (ret << (i * k)).pre(deg);
if ((int)ret.size() < deg) ret.resize(deg, mint(0));
return ret;
}
if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
}
return FPS(deg, mint(0));
}
static void *ntt_ptr;
static void set_fft();
FPS &operator*=(const FPS &r);
void ntt();
void intt();
void ntt_doubling();
static int ntt_pr();
FPS inv(int deg = -1) const;
FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
/**
* @brief 多項式/形式的冪級数ライブラリ
* @docs docs/fps/formal-power-series.md
*/
#line 2 "modulo/binomial.hpp"
#include <cassert>
#include <type_traits>
#include <vector>
using namespace std;
// コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」
// を入れると倍速くらいになる
// mod を超えて前計算して 0 割りを踏むバグは対策済み
template <typename T>
struct Binomial {
vector<T> f, g, h;
Binomial(int MAX = 0) {
assert(T::get_mod() != 0 && "Binomial<mint>()");
f.resize(1, T{1});
g.resize(1, T{1});
h.resize(1, T{1});
if (MAX > 0) extend(MAX + 1);
}
void extend(int m = -1) {
int n = f.size();
if (m == -1) m = n * 2;
m = min<int>(m, T::get_mod());
if (n >= m) return;
f.resize(m);
g.resize(m);
h.resize(m);
for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
g[m - 1] = f[m - 1].inverse();
h[m - 1] = g[m - 1] * f[m - 2];
for (int i = m - 2; i >= n; i--) {
g[i] = g[i + 1] * T(i + 1);
h[i] = g[i] * f[i - 1];
}
}
T fac(int i) {
if (i < 0) return T(0);
while (i >= (int)f.size()) extend();
return f[i];
}
T finv(int i) {
if (i < 0) return T(0);
while (i >= (int)g.size()) extend();
return g[i];
}
T inv(int i) {
if (i < 0) return -inv(-i);
while (i >= (int)h.size()) extend();
return h[i];
}
T C(int n, int r) {
if (n < 0 || n < r || r < 0) return T(0);
return fac(n) * finv(n - r) * finv(r);
}
inline T operator()(int n, int r) { return C(n, r); }
template <typename I>
T multinomial(const vector<I>& r) {
static_assert(is_integral<I>::value == true);
int n = 0;
for (auto& x : r) {
if (x < 0) return T(0);
n += x;
}
T res = fac(n);
for (auto& x : r) res *= finv(x);
return res;
}
template <typename I>
T operator()(const vector<I>& r) {
return multinomial(r);
}
T C_naive(int n, int r) {
if (n < 0 || n < r || r < 0) return T(0);
T ret = T(1);
r = min(r, n - r);
for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
return ret;
}
T P(int n, int r) {
if (n < 0 || n < r || r < 0) return T(0);
return fac(n) * finv(n - r);
}
// [x^r] 1 / (1-x)^n
T H(int n, int r) {
if (n < 0 || r < 0) return T(0);
return r == 0 ? 1 : C(n + r - 1, r);
}
};
#line 4 "fps/taylor-shift.hpp"
// calculate F(x + a)
template <typename mint>
FormalPowerSeries<mint> TaylorShift(FormalPowerSeries<mint> f, mint a,
Binomial<mint>& C) {
using fps = FormalPowerSeries<mint>;
int N = f.size();
for (int i = 0; i < N; i++) f[i] *= C.fac(i);
reverse(begin(f), end(f));
fps g(N, mint(1));
for (int i = 1; i < N; i++) g[i] = g[i - 1] * a * C.inv(i);
f = (f * g).pre(N);
reverse(begin(f), end(f));
for (int i = 0; i < N; i++) f[i] *= C.finv(i);
return f;
}
/**
* @brief 平行移動
* @docs docs/fps/fps-taylor-shift.md
*/
#line 6 "fps/partial-fraction-decomposition.hpp"
// 新しい方
// 入力 : f/(x-b_1)(x-b_2)...(x-b_n), f は n-1 次, b_i は distinct
// 出力 : a_1/(x-b_1) + ... + a_n/(x-b_n)
template <typename mint>
vector<mint> PartialFractionDecomposition(const FormalPowerSeries<mint>& f,
const vector<mint>& b) {
using fps = FormalPowerSeries<mint>;
int N = b.size();
assert((int)f.size() <= N);
int B = 1;
while (B < N) B *= 2;
vector<fps> mod(2 * B, fps{1});
for (int i = 0; i < N; i++) mod[B + i] = fps{-b[i], 1};
for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1];
vector<mint> ans(N);
auto dfs = [&](auto rc, int i, int l, int r, fps g, fps h) -> void {
if (N <= l) return;
if (l + 1 == r) {
ans[l] = g.eval(0) / h.eval(0);
return;
}
int m = (l + r) / 2;
rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0],
h * mod[i * 2 + 1] % mod[i * 2]);
rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1],
h * mod[i * 2] % mod[i * 2 + 1]);
};
dfs(dfs, 1, 0, B, f, fps{1});
return ans;
}
// 古い方
template <typename mint>
vector<pair<mint, vector<mint>>> PartialFractionDecomposition(
FormalPowerSeries<mint> numer, vector<pair<mint, int>> denom) {
using fps = FormalPowerSeries<mint>;
if (denom.empty()) return {};
sort(begin(denom), end(denom),
[](auto p1, auto p2) { return p1.second < p2.second; });
Binomial<mint> C(denom[0].second + 1);
int s = 1;
while (s < (int)denom.size()) s *= 2;
vector<fps> fs(2 * s);
for (int i = 0; i < s; i++) {
if ((int)denom.size() <= i) {
fs[s + i] = fps{1};
continue;
}
auto [m, d] = denom[i];
fps f(denom[i].second + 1);
mint buf = 1;
for (int j = d; j >= 0; j--) {
f[j] = buf * C(d, j);
buf *= m;
}
fs[s + i] = f;
}
for (int i = s - 1; i; i--) {
fs[i] = fs[2 * i + 0] * fs[2 * i + 1];
}
vector<fps> F(2 * s);
vector<fps> G(2 * s);
F[1] = numer % fs[1];
G[1] = fps{1};
for (int i = 1; i < s; i++) {
F[i * 2 + 0] = F[i] % fs[i * 2 + 0];
F[i * 2 + 1] = F[i] % fs[i * 2 + 1];
G[i * 2 + 0] = G[i] * fs[i * 2 + 1] % fs[i * 2 + 0];
G[i * 2 + 1] = G[i] * fs[i * 2 + 0] % fs[i * 2 + 1];
}
vector<pair<mint, vector<mint>>> res;
for (int i = s; i - s < (int)denom.size(); i++) {
auto [m, d] = denom[i - s];
F[i] = TaylorShift<mint>(F[i], -m, C);
G[i] = TaylorShift<mint>(G[i], -m, C);
fps f = (F[i] * G[i].inv()).pre(d);
if ((int)f.size() < d) f.resize(d);
f = f.rev();
res.emplace_back(m, vector<mint>{begin(f), end(f)});
}
return res;
}
/**
* @brief 部分分数分解(分母が1次式の積で表せる場合)
*/
Back to top page