Inverse の次数シフト
(fps/inverse-shift.hpp)

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• Last update: 2025-10-21 21:13:36+09:00
• Include: #include "fps/inverse-shift.hpp"

$d-1$ 次の FPS $f(x)$ について，$\dfrac{1}{f(x)}$ の $x^n$ から $x^{n+d-1}$ の係数を $O(d\log d\log n)$ 時間で求める．

これは $x^{-n}\bmod f(x)$ を求めることとも等価．

アルゴリズム

$f(x)$ の $x^a$ から $x^{b-1}$ の係数を並べた列を $[x^{[a,b)}]f(x)$ で表すことにする．

以下が成り立つ．

\[[x^{[n,n+d)}]\frac{1}{Q(x)}=[x^{[n,n+d)}]\frac{Q(-x)}{Q(x)Q(-x)}\]

後から $Q(-x)$ を掛けることを考えると $[x^{[n-d,n+d)}]\dfrac{1}{Q(x)Q(-x)}$ を求めればよい．

$Q(x)Q(-x)=Q’(x^2)$ とおくと $[x^{[\lceil(n-d)/2\rceil,\lceil (n+d)/2\rceil)}]\dfrac{1}{Q’(x)}$ に帰着される．

$n$ が十分小さい場合は naive に計算してしまって構わない．

• 実装の場合分けを減らすため $n\leq d$ のとき naive に計算している．

FFT 削減

疑似コードで書くと以下のようになる．

shift(n, f): // 1/f(x) の x^n から x^(n+d-1) を返す
  if n is small:
    return naive(n, f)
  f1(x) = f(-x)
  g(x^2) = f(x) * f1(x)
  g(x) = shift(ceil((n-d)/2), g) // d-1 次多項式とみなす
  g(x) = g(x^2) * x^((n-d)%2)
  return g(x) * f1(x) の x^d から x^(2d-1)

FFT の回数を削減するテクニック集 - noshi91のメモ のテクニックを使える部分が多い．

Depends on

• fps/formal-power-series.hpp

Required by

• 線形漸化式用 (fps/linearly-recurrent-sequence.hpp)

Verified with

• verify/fps/LC_consecutive_terms_of_linear_recurrent_sequence.test.cpp
• verify/fps/LC_kth_term_of_linearly_recurrent_sequence.test.cpp

Code

#pragma once
#include "fps/formal-power-series.hpp"

// [x^n,...,x^{n+k-1}]1/f(x)
template <class mint>
FormalPowerSeries<mint> InverseShift(FormalPowerSeries<mint> f, long long n, int k = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(f[0] != 0);
  if (k == -1) k = f.size();
  int m = 1;
  while (m < k) m <<= 1;

  int log = __builtin_ctz((unsigned int)m);
  mint w = mint(fps::ntt_root()).pow((mint::get_mod() - 1) >> (log + 1));
  mint wi = w.inv();
  vector<int> rev(m);
  for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
  mint inv2 = mint(2).inv();

  f.resize(m);
  f.ntt();
  auto rec = [&](auto& rec, long long n) -> void {
    if (n < m) {
      f.intt();
      f = f.inv(n + m);
      f >>= n;
      f.ntt();
      return;
    }
    f.ntt_doubling();
    assert(f.size() == 2 * m);
    auto f1 = f;
    for (int i = 0; i < m; i++) swap(f1[i << 1], f1[(i << 1) | 1]);
    for (int i = 0; i < m; i++) f[i] = f[i << 1] * f[(i << 1) | 1];
    f.resize(m);
    rec(rec, (n - m + 1) / 2);
    if (((n - m) & 1) == 0) {
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = f[i];
        f[i << 1] = f[i];
      }
    } else {
      mint p = 1;
      for (auto i : rev) f[i] *= p, p *= w;
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = -f[i];
        f[i << 1] = f[i];
      }
    }
    for (int i = 0; i < 2 * m; i++) f[i] *= f1[i];
    auto odd = fps(f.begin() + m, f.end());
    odd.intt();
    mint p = 1;
    for (int i = 0; i < m; i++) odd[i] *= p, p *= wi;
    odd.ntt();
    f.resize(m);
    for (int i = 0; i < m; i++) f[i] = (f[i] - odd[i]) * inv2;
  };
  rec(rec, n);
  f.intt();
  f.resize(k);
  return f;
}
/**
 * @brief Inverse の次数シフト
 * @docs docs/fps/inverse-shift.md
 */

#line 2 "fps/formal-power-series.hpp"

template <class 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.at(g.size() - 1).inv();
      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;
  }
  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }
  FPS operator>>=(int sz) {
    assert(sz >= 0);
    if ((int)this->size() <= sz) return {};
    this->erase(this->begin(), this->begin() + sz);
    return *this;
  }
  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) {
    assert(sz >= 0);
    this->insert(this->begin(), sz, mint(0));
    return *this;
  }
  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)[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_ntt();
  FPS &operator*=(const FPS &r);
  FPS middle_product(const FPS &r) const;
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_root();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 3 "fps/inverse-shift.hpp"

// [x^n,...,x^{n+k-1}]1/f(x)
template <class mint>
FormalPowerSeries<mint> InverseShift(FormalPowerSeries<mint> f, long long n, int k = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(f[0] != 0);
  if (k == -1) k = f.size();
  int m = 1;
  while (m < k) m <<= 1;

  int log = __builtin_ctz((unsigned int)m);
  mint w = mint(fps::ntt_root()).pow((mint::get_mod() - 1) >> (log + 1));
  mint wi = w.inv();
  vector<int> rev(m);
  for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
  mint inv2 = mint(2).inv();

  f.resize(m);
  f.ntt();
  auto rec = [&](auto& rec, long long n) -> void {
    if (n < m) {
      f.intt();
      f = f.inv(n + m);
      f >>= n;
      f.ntt();
      return;
    }
    f.ntt_doubling();
    assert(f.size() == 2 * m);
    auto f1 = f;
    for (int i = 0; i < m; i++) swap(f1[i << 1], f1[(i << 1) | 1]);
    for (int i = 0; i < m; i++) f[i] = f[i << 1] * f[(i << 1) | 1];
    f.resize(m);
    rec(rec, (n - m + 1) / 2);
    if (((n - m) & 1) == 0) {
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = f[i];
        f[i << 1] = f[i];
      }
    } else {
      mint p = 1;
      for (auto i : rev) f[i] *= p, p *= w;
      f.resize(2 * m);
      for (int i = m - 1; i >= 0; i--) {
        f[(i << 1) | 1] = -f[i];
        f[i << 1] = f[i];
      }
    }
    for (int i = 0; i < 2 * m; i++) f[i] *= f1[i];
    auto odd = fps(f.begin() + m, f.end());
    odd.intt();
    mint p = 1;
    for (int i = 0; i < m; i++) odd[i] *= p, p *= wi;
    odd.ntt();
    f.resize(m);
    for (int i = 0; i < m; i++) f[i] = (f[i] - odd[i]) * inv2;
  };
  rec(rec, n);
  f.intt();
  f.resize(k);
  return f;
}
/**
 * @brief Inverse の次数シフト
 * @docs docs/fps/inverse-shift.md
 */

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