$f(exp(cx))$ の計算
(fps/composite-exp.hpp)

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• Last update: 2023-08-31 20:44:07+09:00
• Include: #include "fps/composite-exp.hpp"

Depends on

• 多項式/形式的冪級数ライブラリ (fps/formal-power-series.hpp)

Required by

• fps/stirling-matrix.hpp

Verified with

• verify/verify-unit-test/composite-exp.test.cpp
• verify/verify-unit-test/stirling-matrix.test.cpp
• verify/verify-yuki/yuki-1875.test.cpp

Code

#pragma once

#include <cassert>
#include <utility>
#include <vector>
using namespace std;

#include "formal-power-series.hpp"

// 多項式 f に exp(cx) 代入
// 次数 : mod x^{deg} まで計算, 指定がない場合 f と同じ長さ計算
template <typename mint>
FormalPowerSeries<mint> composite_exp(FormalPowerSeries<mint> f, mint c = 1,
                                      int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(c != 0);
  if (deg == -1) deg = f.size();

  if (f.empty()) return {};
  int N = f.size();
  vector<pair<fps, fps>> fs;
  for (int i = 0; i < N; i++) fs.emplace_back(fps{f[i]}, fps{1, -c * i});
  while (fs.size() > 1u) {
    vector<pair<fps, fps>> nx;
    for (int i = 0; i + 1 < (int)fs.size(); i += 2) {
      pair<fps, fps>& f0 = fs[i];
      pair<fps, fps>& f1 = fs[i + 1];
      fps s = f0.first * f1.second + f1.first * f0.second;
      fps t = f0.second * f1.second;
      nx.emplace_back(s, t);
    }
    if (fs.size() % 2) nx.push_back(fs.back());
    fs = nx;
  }
  fps g = (fs[0].first * fs[0].second.inv(deg)).pre(deg);
  mint b = 1;
  for (int i = 0; i < deg; i++) g[i] *= b, b /= i + 1;
  return g;
}

// 入力 f(x) = sum_{0 <= k < N} a_i exp(ckx) を満たす g(x) (mod x^N)
// 出力 a(x) = sum_{0 <= k < N} a_i x^i
template <typename mint>
FormalPowerSeries<mint> inverse_of_composite_exp(FormalPowerSeries<mint> f,
                                                 mint c = 1) {
  using fps = FormalPowerSeries<mint>;
  if (f.empty()) return {};
  int N = f.size();
  mint b = 1;
  for (int i = 0; i < N; i++) f[i] *= b, b *= i + 1;

  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{-c * i, 1};
  for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1];
  fps denom = mod[1].rev();
  fps numer = (f * denom).pre(N);

  vector<mint> a(N);
  auto dfs = [&](auto rc, int i, int l, int r, fps g) -> void {
    if (N <= l) return;
    if (l + 1 == r) {
      a[l] = g.eval(0);
      return;
    }
    int m = (l + r) / 2;
    rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0]);
    rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1]);
  };
  dfs(dfs, 1, 0, B, numer.rev());

  vector<mint> fac(N);
  fac[0] = 1;
  for (int i = 1; i < N; i++) fac[i] = fac[i - 1] * c * i;
  for (int i = 0; i < N; i++) {
    a[i] /= fac[N - 1 - i] * fac[i] * ((N - 1 - i) % 2 ? -1 : 1);
  }
  return fps{begin(a), end(a)};
}

/**
 * @brief $f(exp(cx))$ の計算
 */

#line 2 "fps/composite-exp.hpp"

#include <cassert>
#include <utility>
#include <vector>
using namespace std;

#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 9 "fps/composite-exp.hpp"

// 多項式 f に exp(cx) 代入
// 次数 : mod x^{deg} まで計算, 指定がない場合 f と同じ長さ計算
template <typename mint>
FormalPowerSeries<mint> composite_exp(FormalPowerSeries<mint> f, mint c = 1,
                                      int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(c != 0);
  if (deg == -1) deg = f.size();

  if (f.empty()) return {};
  int N = f.size();
  vector<pair<fps, fps>> fs;
  for (int i = 0; i < N; i++) fs.emplace_back(fps{f[i]}, fps{1, -c * i});
  while (fs.size() > 1u) {
    vector<pair<fps, fps>> nx;
    for (int i = 0; i + 1 < (int)fs.size(); i += 2) {
      pair<fps, fps>& f0 = fs[i];
      pair<fps, fps>& f1 = fs[i + 1];
      fps s = f0.first * f1.second + f1.first * f0.second;
      fps t = f0.second * f1.second;
      nx.emplace_back(s, t);
    }
    if (fs.size() % 2) nx.push_back(fs.back());
    fs = nx;
  }
  fps g = (fs[0].first * fs[0].second.inv(deg)).pre(deg);
  mint b = 1;
  for (int i = 0; i < deg; i++) g[i] *= b, b /= i + 1;
  return g;
}

// 入力 f(x) = sum_{0 <= k < N} a_i exp(ckx) を満たす g(x) (mod x^N)
// 出力 a(x) = sum_{0 <= k < N} a_i x^i
template <typename mint>
FormalPowerSeries<mint> inverse_of_composite_exp(FormalPowerSeries<mint> f,
                                                 mint c = 1) {
  using fps = FormalPowerSeries<mint>;
  if (f.empty()) return {};
  int N = f.size();
  mint b = 1;
  for (int i = 0; i < N; i++) f[i] *= b, b *= i + 1;

  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{-c * i, 1};
  for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1];
  fps denom = mod[1].rev();
  fps numer = (f * denom).pre(N);

  vector<mint> a(N);
  auto dfs = [&](auto rc, int i, int l, int r, fps g) -> void {
    if (N <= l) return;
    if (l + 1 == r) {
      a[l] = g.eval(0);
      return;
    }
    int m = (l + r) / 2;
    rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0]);
    rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1]);
  };
  dfs(dfs, 1, 0, B, numer.rev());

  vector<mint> fac(N);
  fac[0] = 1;
  for (int i = 1; i < N; i++) fac[i] = fac[i - 1] * c * i;
  for (int i = 0; i < N; i++) {
    a[i] /= fac[N - 1 - i] * fac[i] * ((N - 1 - i) % 2 ? -1 : 1);
  }
  return fps{begin(a), end(a)};
}

/**
 * @brief $f(exp(cx))$ の計算
 */

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