三角関数
(fps/fps-circular.hpp)

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

fps-三角関数

$N$次の形式的冪級数 $f(x)$ に対して $g(x) \equiv \cos(f(x)), h(x) \equiv \sin(f(x)) \mod x^N$ を満たす $g(x)$ を $\mathrm{O}(N \log N)$ で計算するライブラリ。

概要

$g \equiv \cos f, h \equiv \sin f \pmod{x^n}$ を求めたい。
これはオイラーの公式 $e^{if}=\cos f+i\sin f$ を利用すると $\mathrm{exp}(f)$ と同様にニュートン法で求まる。(詳細は割愛する。)

使い方

• circular(fre, fim, deg) : $Re[f]=fre,Im[f]=fim$ である FPS $f$ について $\cos f,\sin f$ を $\deg$ 次の項まで求める。

Depends on

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

Verified with

• verify/verify-yuki/yuki-0963-circular.test.cpp
• verify/verify-yuki/yuki-1080.test.cpp

Code

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

template <typename mint>
pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> circular(
    const FormalPowerSeries<mint> &fre, const FormalPowerSeries<mint> &fim,
    int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(fre.size() == 0 || fre[0] == mint(0));
  assert(fim.size() == 0 || fim[0] == mint(0));
  if (deg == -1) deg = (int)max(fre.size(), fim.size());
  fps re({mint(1)}), im({mint(0)});

  fps::set_fft();
  if (fps::ntt_ptr == nullptr) {
    for (int i = 1; i < deg; i <<= 1) {
      fps dre = re.diff();
      fps dim = im.diff();
      fps fhypot = (re * re + im * im).inv(i << 1);
      fps ere = dre * re + dim * im;
      fps eim = dim * re - dre * im;
      fps logre = (ere * fhypot).pre((i << 1) - 1).integral();
      fps logim = (eim * fhypot).pre((i << 1) - 1).integral();
      fps gre = (-logre) + mint(1) - fim.pre(i << 1);
      fps gim = (-logim) + fre.pre(i << 1);
      fps hre = (re * gre - im * gim).pre(i << 1);
      fps him = (re * gim + im * gre).pre(i << 1);
      swap(re, hre);
      swap(im, him);
    }
  } else {
    for (int i = 1; i < deg; i <<= 1) {
      fps dre = re.diff();
      fps dim = im.diff();
      re.resize(i << 1);
      im.resize(i << 1);
      dre.resize(i << 1);
      dim.resize(i << 1);
      re.ntt();
      im.ntt();
      dre.ntt();
      dim.ntt();
      fps fhypot(i << 1), ere(i << 1), eim(i << 1);
      for (int j = 0; j < 2 * i; j++) {
        fhypot[j] = re[j] * re[j] + im[j] * im[j];
        ere[j] = dre[j] * re[j] + dim[j] * im[j];
        eim[j] = dim[j] * re[j] - dre[j] * im[j];
      }
      fhypot.intt();
      fhypot = fhypot.inv(i << 1);
      fhypot.resize(i << 2);
      fhypot.ntt();
      ere.ntt_doubling();
      eim.ntt_doubling();
      fps logre(i << 2), logim(i << 2);
      for (int j = 0; j < 4 * i; j++) {
        logre[j] = ere[j] * fhypot[j];
        logim[j] = eim[j] * fhypot[j];
      }
      logre.intt();
      logim.intt();
      logre = logre.pre((i << 1) - 1).integral();
      logim = logim.pre((i << 1) - 1).integral();
      fps gre = (-logre) + mint(1) - fim.pre(i << 1);
      fps gim = (-logim) + fre.pre(i << 1);
      gre.resize(i << 2);
      gim.resize(i << 2);
      gre.ntt();
      gim.ntt();
      re.ntt_doubling();
      im.ntt_doubling();
      fps hre(i << 2), him(i << 2);
      for (int j = 0; j < 4 * i; j++) {
        hre[j] = re[j] * gre[j] - im[j] * gim[j];
        him[j] = re[j] * gim[j] + im[j] * gre[j];
      }
      hre.intt();
      him.intt();
      hre = hre.pre(i << 1);
      him = him.pre(i << 1);
      swap(re, hre);
      swap(im, him);
    }
  }
  return make_pair(re.pre(deg), im.pre(deg));
}

/**
 * @brief 三角関数
 * @docs docs/fps/fps-circular.md
 */

#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 "modint/montgomery-modint.hpp"

template <uint32_t mod>
struct LazyMontgomeryModInt {
  using mint = LazyMontgomeryModInt;
  using i32 = int32_t;
  using u32 = uint32_t;
  using u64 = uint64_t;

  static constexpr u32 get_r() {
    u32 ret = mod;
    for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
    return ret;
  }

  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
  static_assert(r * mod == 1, "this code has bugs.");

  u32 a;

  constexpr LazyMontgomeryModInt() : a(0) {}
  constexpr LazyMontgomeryModInt(const int64_t &b)
      : a(reduce(u64(b % mod + mod) * n2)){};

  static constexpr u32 reduce(const u64 &b) {
    return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
  }

  constexpr mint &operator+=(const mint &b) {
    if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator-=(const mint &b) {
    if (i32(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }

  constexpr mint &operator*=(const mint &b) {
    a = reduce(u64(a) * b.a);
    return *this;
  }

  constexpr mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
  constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
  constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
  constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
  constexpr bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  constexpr mint operator-() const { return mint() - mint(*this); }
  constexpr mint operator+() const { return mint(*this); }

  constexpr mint pow(u64 n) const {
    mint ret(1), mul(*this);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }

  constexpr mint inverse() const {
    int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
    while (y > 0) {
      t = x / y;
      x -= t * y, u -= t * v;
      tmp = x, x = y, y = tmp;
      tmp = u, u = v, v = tmp;
    }
    return mint{u};
  }

  friend ostream &operator<<(ostream &os, const mint &b) {
    return os << b.get();
  }

  friend istream &operator>>(istream &is, mint &b) {
    int64_t t;
    is >> t;
    b = LazyMontgomeryModInt<mod>(t);
    return (is);
  }

  constexpr u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }

  static constexpr u32 get_mod() { return mod; }
};
#line 4 "fps/fps-circular.hpp"

template <typename mint>
pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> circular(
    const FormalPowerSeries<mint> &fre, const FormalPowerSeries<mint> &fim,
    int deg = -1) {
  using fps = FormalPowerSeries<mint>;
  assert(fre.size() == 0 || fre[0] == mint(0));
  assert(fim.size() == 0 || fim[0] == mint(0));
  if (deg == -1) deg = (int)max(fre.size(), fim.size());
  fps re({mint(1)}), im({mint(0)});

  fps::set_fft();
  if (fps::ntt_ptr == nullptr) {
    for (int i = 1; i < deg; i <<= 1) {
      fps dre = re.diff();
      fps dim = im.diff();
      fps fhypot = (re * re + im * im).inv(i << 1);
      fps ere = dre * re + dim * im;
      fps eim = dim * re - dre * im;
      fps logre = (ere * fhypot).pre((i << 1) - 1).integral();
      fps logim = (eim * fhypot).pre((i << 1) - 1).integral();
      fps gre = (-logre) + mint(1) - fim.pre(i << 1);
      fps gim = (-logim) + fre.pre(i << 1);
      fps hre = (re * gre - im * gim).pre(i << 1);
      fps him = (re * gim + im * gre).pre(i << 1);
      swap(re, hre);
      swap(im, him);
    }
  } else {
    for (int i = 1; i < deg; i <<= 1) {
      fps dre = re.diff();
      fps dim = im.diff();
      re.resize(i << 1);
      im.resize(i << 1);
      dre.resize(i << 1);
      dim.resize(i << 1);
      re.ntt();
      im.ntt();
      dre.ntt();
      dim.ntt();
      fps fhypot(i << 1), ere(i << 1), eim(i << 1);
      for (int j = 0; j < 2 * i; j++) {
        fhypot[j] = re[j] * re[j] + im[j] * im[j];
        ere[j] = dre[j] * re[j] + dim[j] * im[j];
        eim[j] = dim[j] * re[j] - dre[j] * im[j];
      }
      fhypot.intt();
      fhypot = fhypot.inv(i << 1);
      fhypot.resize(i << 2);
      fhypot.ntt();
      ere.ntt_doubling();
      eim.ntt_doubling();
      fps logre(i << 2), logim(i << 2);
      for (int j = 0; j < 4 * i; j++) {
        logre[j] = ere[j] * fhypot[j];
        logim[j] = eim[j] * fhypot[j];
      }
      logre.intt();
      logim.intt();
      logre = logre.pre((i << 1) - 1).integral();
      logim = logim.pre((i << 1) - 1).integral();
      fps gre = (-logre) + mint(1) - fim.pre(i << 1);
      fps gim = (-logim) + fre.pre(i << 1);
      gre.resize(i << 2);
      gim.resize(i << 2);
      gre.ntt();
      gim.ntt();
      re.ntt_doubling();
      im.ntt_doubling();
      fps hre(i << 2), him(i << 2);
      for (int j = 0; j < 4 * i; j++) {
        hre[j] = re[j] * gre[j] - im[j] * gim[j];
        him[j] = re[j] * gim[j] + im[j] * gre[j];
      }
      hre.intt();
      him.intt();
      hre = hre.pre(i << 1);
      him = him.pre(i << 1);
      swap(re, hre);
      swap(im, him);
    }
  }
  return make_pair(re.pre(deg), im.pre(deg));
}

/**
 * @brief 三角関数
 * @docs docs/fps/fps-circular.md
 */

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