Formal Power Series (形式的べき級数)
(formal-power-series/formal-power-series.hpp)

• View this file on GitHub
• Last update: 2024-06-20 15:39:33+09:00
• Include: #include "formal-power-series/formal-power-series.hpp"

modによらない(さすがに合成数とかだと厳しいが)共通した処理を実装している。

コンストラクタ

FPS(vector<mint> vec)

はい

FPS(initializer_list<mint> ilist)

波括弧とかで初期化するやつにも対応！

FPS(int sz)

項の数で初期化(最大の次数とは1ズレる)

diff

微分。

計算量

mintが四則演算 $O(1)$ なら、

• $O\left(N\right)$

integral

積分。

計算量

mintが四則演算 $O(1)$ なら、

• $O\left(N\right)$

Depends on

• math/external_gcd.hpp
• modint (math/modint.hpp)

Required by

• formal-power-series/fiduccia.hpp
• formal-power-series/fps998.hpp
• sparse-fps (formal-power-series/sparse-fps.hpp)
• test/verify/fps/yosupo-inv-of-formal-power-series-naive-test.cpp
• test/verify/fps/yosupo-kth-term-of-linearly-recurrent-sequence-test.cpp

Verified with

• test/verify/fps/yosupo-division-of-polynomials.test.cpp
• test/verify/fps/yosupo-exp-of-formal-power-series.test.cpp
• test/verify/fps/yosupo-inv-of-formal-power-series-fast2.test.cpp
• test/verify/fps/yosupo-inv-of-formal-power-series-sparse.test.cpp
• test/verify/fps/yosupo-inv-of-formal-power-series-sparse.test.cpp
• test/verify/fps/yosupo-log-of-formal-power-series.test.cpp

Code

#ifndef HARUILIB_FORMAL_POWER_SERIES_FORMAL_POWER_SERIES_HPP
#define HARUILIB_FORMAL_POWER_SERIES_FORMAL_POWER_SERIES_HPP

#include "../math/modint.hpp"
#include <bits/stdc++.h>

using namespace std;



template <typename mint>
struct FPS {
  vector<mint> _vec;

  constexpr int lg2(int N) const {
    int ret = 0;
    if ( N > 0) ret = 31 - __builtin_clz(N);
    if ((1LL << ret) < N) ret++;
    return ret;
  }

  // ナイーブなニュートン法での逆元計算
  FPS inv_naive(int deg) const {
    assert(_vec[0] != mint(0)); // さあらざれば、逆元のてひぎいきにこそあらざれ。
    if (deg == -1) deg = this->size();
    FPS g(1);
    g._vec[0] = mint(_vec[0]).inv();
    // g_{n+1} = 2 * g_n - f * (g_n)^2
    for (int d=1; d < deg; d <<= 1) {
      FPS g_twice = g * mint(2);
      FPS fgg = (*this).pre(d*2) * g * g;
     
      g = g_twice - fgg;
      g.resize(d*2);
    }

    return g.pre(deg);
  }

  //*/

  FPS log(int deg=-1) const {
    assert(_vec[0] == mint(1));

    if (deg == -1) deg = size();
    FPS df = this->diff();
    FPS iv = this->inv(deg);
    FPS ret = (df * iv).pre(deg-1).integral();

    return ret;
  }

  FPS exp(int deg=-1) const {
    assert(_vec[0] == mint(0));

    if (deg == -1) deg = size();
    FPS g(1);
    g[0] = 1;
    for (int d=1; d<deg; d <<= 1) {
      // g_2d = g_d * (f(x) + 1 - log(g_d))
      FPS fpl1 = (*this + mint(1)).pre(2*d);
      FPS logg = g.log(2*d);
      FPS right = (fpl1 - logg);

      g = (g * right).pre(2*d);
    }

    return g.pre(deg);
  }

  FPS integral() const {
    const int N = size();
    FPS ret(N+1);

    for(int i=0; i<N; i++) ret[i+1] = _vec[i] * mint(i+1).inv();

    return ret;
  }

  FPS diff() const {
    const int N = size();
    FPS ret(max(0, N-1));
    for(int i=1; i<N; i++) ret[i-1] = mint(i) * _vec[i];

    return ret;
  }

  FPS(vector<mint> vec) : _vec(vec) {
  }

  FPS(initializer_list<mint> ilist) : _vec(ilist) {
  }

  // 項の数に揃えたほうがよさそう
  FPS(int sz) : _vec(vector<mint>(sz)) {
  }

  int size() const {
    return _vec.size();
  }

  FPS& operator+=(const FPS& rhs) {
    if (rhs.size() > this->size()) _vec.resize(rhs.size());
    for (int i = 0; i < (int)rhs.size(); ++i) _vec[i] += rhs._vec[i];
    return *this;
  }

  FPS& operator-=(const FPS& rhs) {
    if (rhs.size() > this->size()) this->_vec.resize(rhs.size());
    for (int i = 0; i < (int)rhs.size(); ++i) _vec[i] -= rhs._vec[i];
    return *this;
  }

  FPS& operator*=(const FPS& rhs) {
    _vec = multiply(_vec, rhs._vec);
    return *this;
  }

  // Nyaan先生のライブラリを大写経....
  FPS& operator/=(const FPS& rhs) {
    if (size() < rhs.size()) {
      return *this =  FPS(0);
    }
    int sz = size() - rhs.size() + 1;
//
//    FPS left = (*this).rev().pre(sz);
//    FPS right = rhs.rev();
//    right = right.inv(sz);
//    FPS mp = left*right;
//    mp = mp.pre(sz);
//    mp = mp.rev();
//    return *this = mp;
//    return *this = (left * right).pre(sz).rev();
    return *this =  ((*this).rev().pre(sz) * rhs.rev().inv(sz)).pre(sz).rev();
  }

  FPS& operator%=(const FPS &rhs) {
    *this -= *this / rhs * rhs;
    shrink(); 
    return *this;
  }

  FPS& operator+=(const mint& rhs) {
    _vec[0] += rhs;
    return *this;
  }

  FPS& operator-=(const mint& rhs) {
    _vec[0] -= rhs;
    return *this;
  }

  FPS& operator*=(const mint& rhs) {
    for(int i=0; i<size(); i++) _vec[i] *= rhs;
    return *this;
  }

  FPS& operator/=(const mint& rhs) {
    for(int i=0; i<size(); i++) _vec[i] *= rhs.inv();
    return *this;
  }

  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret._vec.erase(ret._vec.begin(), ret._vec.begin() + sz);
    return ret;
  }

  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret._vec.insert(ret._vec.begin(), sz, mint(0));

    return ret;
  }

  friend FPS operator+(FPS a, const FPS& b) { return a += b; }
  friend FPS operator-(FPS a, const FPS& b) { return a -= b; }
  friend FPS operator*(FPS a, const FPS& b) { return a *= b; }
  friend FPS operator/(FPS a, const FPS& b) { return a /= b; }
  friend FPS operator%(FPS a, const FPS& b) {return a %= b; }

  friend FPS operator+(FPS a, const mint& b) {return a += b; }
  friend FPS operator-(FPS a, const mint& b) {return a -= b; }
  friend FPS operator*(FPS a, const mint& b) {return a *= b; }
  friend FPS operator/(FPS a, const mint& b) {return a /= b; }
  
  // sz次未満の項を取ってくる
  FPS pre(int sz) const {
    FPS ret = *this; 
    ret._vec.resize(sz);

    return ret;
  }

  FPS rev() const {
    FPS ret = *this;
    reverse(ret._vec.begin(), ret._vec.end());
    
    return ret;
  }

  const mint& operator[](size_t i) const {
    return _vec[i];
  }

  mint& operator[](size_t i) {
    return _vec[i];
  }

  void resize(int sz)  {
    this->_vec.resize(sz);
  }

  void shrink() {
    while (size() > 0 && _vec.back() == mint(0)) _vec.pop_back();
  }

  friend ostream& operator<<(ostream& os, const FPS& fps) {
    for (int i = 0; i < fps.size(); ++i) {
      if (i > 0) os << " ";
      os << fps._vec[i].val();
    }
    return os;
  }

  // 仮想関数ってやつ。mod 998244353なのか、他のNTT-friendlyなmodで考えるのか、それともGarnerで復元するのか、それとも畳み込みを$O(N^2)$で妥協するのかなどによって異なる
  virtual FPS inv(int deg=-1) const;
  virtual void CooleyTukeyNTT998244353(vector<mint>&a, bool is_reverse) const;
//  virtual FPS exp(int deg=-1) const;
  virtual vector<mint> multiply(const vector<mint>& a, const vector<mint>& b);
  };

#endif // HARUILIB_FORMAL_POWER_SERIES_FORMAL_POWER_SERIES_HPP

#line 1 "formal-power-series/formal-power-series.hpp"



#line 1 "math/modint.hpp"



#line 1 "math/external_gcd.hpp"



#include <tuple>

// g,x,y
template<typename T>
constexpr std::tuple<T, T, T> extendedGCD(T a, T b) {
    T x0 = 1, y0 = 0, x1 = 0, y1 = 1;
    while (b != 0) {
        T q = a / b;
        T r = a % b;
        a = b;
        b = r;
        
        T xTemp = x0 - q * x1;
        x0 = x1;
        x1 = xTemp;
        
        T yTemp = y0 - q * y1;
        y0 = y1;
        y1 = yTemp;
    }
    return {a, x0, y0};
}

#line 5 "math/modint.hpp"

template<int MOD>
struct static_modint {
    int value;

    constexpr static_modint() : value(0) {}

    constexpr static_modint(long long v) {
        value = int(((v % MOD) + MOD) % MOD);
    }

    constexpr static_modint& operator+=(const static_modint& other) {
        if ((value += other.value) >= MOD) value -= MOD;
        return *this;
    }

    constexpr static_modint& operator-=(const static_modint& other) {
        if ((value -= other.value) < 0) value += MOD;
        return *this;
    }

    constexpr static_modint& operator*=(const static_modint& other) {
        value = int((long long)value * other.value % MOD);
        return *this;
    }

    constexpr static_modint operator+(const static_modint& other) const {
        return static_modint(*this) += other;
    }

    constexpr static_modint operator-(const static_modint& other) const {
        return static_modint(*this) -= other;
    }

    constexpr static_modint operator*(const static_modint& other) const {
        return static_modint(*this) *= other;
    }

    constexpr static_modint pow(long long exp) const {
        static_modint base = *this, res = 1;
        while (exp > 0) {
            if (exp & 1) res *= base;
            base *= base;
            exp >>= 1;
        }
        return res;
    }

    constexpr static_modint inv() const {
        //return pow(MOD - 2);
        int g,x,y;
        tie(g,x,y) = extendedGCD(value, MOD);
        assert(g==1);
        if (x < 0) {
            x += MOD;
        }
        //cerr << g << " " << x << " " << y << " " << value << endl;
        //assert((((long)x*value)%MOD + MOD)%MOD == 1);
        return x;
    }

    constexpr static_modint& operator/=(const static_modint& other) {
        return *this *= other.inv();
    }

    constexpr static_modint operator/(const static_modint& other) const {
        return static_modint(*this) /= other;
    }

    constexpr bool operator!=(const static_modint& other) const {
        return val() != other.val();
    }

    constexpr bool operator==(const static_modint& other) const {
        return val() == other.val();
    }

    int val() const {
      return this->value;
    }

    friend std::ostream& operator<<(std::ostream& os, const static_modint& mi) {
        return os << mi.value;
    }

    friend std::istream& operator>>(std::istream& is, static_modint& mi) {
        long long x;
        is >> x;
        mi = static_modint(x);
        return is;
    }
};

template <int mod>
using modint = static_modint<mod>;
using modint998244353  = modint<998244353>;
using modint1000000007 = modint<1000000007>;


#line 5 "formal-power-series/formal-power-series.hpp"
#include <bits/stdc++.h>

using namespace std;



template <typename mint>
struct FPS {
  vector<mint> _vec;

  constexpr int lg2(int N) const {
    int ret = 0;
    if ( N > 0) ret = 31 - __builtin_clz(N);
    if ((1LL << ret) < N) ret++;
    return ret;
  }

  // ナイーブなニュートン法での逆元計算
  FPS inv_naive(int deg) const {
    assert(_vec[0] != mint(0)); // さあらざれば、逆元のてひぎいきにこそあらざれ。
    if (deg == -1) deg = this->size();
    FPS g(1);
    g._vec[0] = mint(_vec[0]).inv();
    // g_{n+1} = 2 * g_n - f * (g_n)^2
    for (int d=1; d < deg; d <<= 1) {
      FPS g_twice = g * mint(2);
      FPS fgg = (*this).pre(d*2) * g * g;
     
      g = g_twice - fgg;
      g.resize(d*2);
    }

    return g.pre(deg);
  }

  //*/

  FPS log(int deg=-1) const {
    assert(_vec[0] == mint(1));

    if (deg == -1) deg = size();
    FPS df = this->diff();
    FPS iv = this->inv(deg);
    FPS ret = (df * iv).pre(deg-1).integral();

    return ret;
  }

  FPS exp(int deg=-1) const {
    assert(_vec[0] == mint(0));

    if (deg == -1) deg = size();
    FPS g(1);
    g[0] = 1;
    for (int d=1; d<deg; d <<= 1) {
      // g_2d = g_d * (f(x) + 1 - log(g_d))
      FPS fpl1 = (*this + mint(1)).pre(2*d);
      FPS logg = g.log(2*d);
      FPS right = (fpl1 - logg);

      g = (g * right).pre(2*d);
    }

    return g.pre(deg);
  }

  FPS integral() const {
    const int N = size();
    FPS ret(N+1);

    for(int i=0; i<N; i++) ret[i+1] = _vec[i] * mint(i+1).inv();

    return ret;
  }

  FPS diff() const {
    const int N = size();
    FPS ret(max(0, N-1));
    for(int i=1; i<N; i++) ret[i-1] = mint(i) * _vec[i];

    return ret;
  }

  FPS(vector<mint> vec) : _vec(vec) {
  }

  FPS(initializer_list<mint> ilist) : _vec(ilist) {
  }

  // 項の数に揃えたほうがよさそう
  FPS(int sz) : _vec(vector<mint>(sz)) {
  }

  int size() const {
    return _vec.size();
  }

  FPS& operator+=(const FPS& rhs) {
    if (rhs.size() > this->size()) _vec.resize(rhs.size());
    for (int i = 0; i < (int)rhs.size(); ++i) _vec[i] += rhs._vec[i];
    return *this;
  }

  FPS& operator-=(const FPS& rhs) {
    if (rhs.size() > this->size()) this->_vec.resize(rhs.size());
    for (int i = 0; i < (int)rhs.size(); ++i) _vec[i] -= rhs._vec[i];
    return *this;
  }

  FPS& operator*=(const FPS& rhs) {
    _vec = multiply(_vec, rhs._vec);
    return *this;
  }

  // Nyaan先生のライブラリを大写経....
  FPS& operator/=(const FPS& rhs) {
    if (size() < rhs.size()) {
      return *this =  FPS(0);
    }
    int sz = size() - rhs.size() + 1;
//
//    FPS left = (*this).rev().pre(sz);
//    FPS right = rhs.rev();
//    right = right.inv(sz);
//    FPS mp = left*right;
//    mp = mp.pre(sz);
//    mp = mp.rev();
//    return *this = mp;
//    return *this = (left * right).pre(sz).rev();
    return *this =  ((*this).rev().pre(sz) * rhs.rev().inv(sz)).pre(sz).rev();
  }

  FPS& operator%=(const FPS &rhs) {
    *this -= *this / rhs * rhs;
    shrink(); 
    return *this;
  }

  FPS& operator+=(const mint& rhs) {
    _vec[0] += rhs;
    return *this;
  }

  FPS& operator-=(const mint& rhs) {
    _vec[0] -= rhs;
    return *this;
  }

  FPS& operator*=(const mint& rhs) {
    for(int i=0; i<size(); i++) _vec[i] *= rhs;
    return *this;
  }

  FPS& operator/=(const mint& rhs) {
    for(int i=0; i<size(); i++) _vec[i] *= rhs.inv();
    return *this;
  }

  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret._vec.erase(ret._vec.begin(), ret._vec.begin() + sz);
    return ret;
  }

  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret._vec.insert(ret._vec.begin(), sz, mint(0));

    return ret;
  }

  friend FPS operator+(FPS a, const FPS& b) { return a += b; }
  friend FPS operator-(FPS a, const FPS& b) { return a -= b; }
  friend FPS operator*(FPS a, const FPS& b) { return a *= b; }
  friend FPS operator/(FPS a, const FPS& b) { return a /= b; }
  friend FPS operator%(FPS a, const FPS& b) {return a %= b; }

  friend FPS operator+(FPS a, const mint& b) {return a += b; }
  friend FPS operator-(FPS a, const mint& b) {return a -= b; }
  friend FPS operator*(FPS a, const mint& b) {return a *= b; }
  friend FPS operator/(FPS a, const mint& b) {return a /= b; }
  
  // sz次未満の項を取ってくる
  FPS pre(int sz) const {
    FPS ret = *this; 
    ret._vec.resize(sz);

    return ret;
  }

  FPS rev() const {
    FPS ret = *this;
    reverse(ret._vec.begin(), ret._vec.end());
    
    return ret;
  }

  const mint& operator[](size_t i) const {
    return _vec[i];
  }

  mint& operator[](size_t i) {
    return _vec[i];
  }

  void resize(int sz)  {
    this->_vec.resize(sz);
  }

  void shrink() {
    while (size() > 0 && _vec.back() == mint(0)) _vec.pop_back();
  }

  friend ostream& operator<<(ostream& os, const FPS& fps) {
    for (int i = 0; i < fps.size(); ++i) {
      if (i > 0) os << " ";
      os << fps._vec[i].val();
    }
    return os;
  }

  // 仮想関数ってやつ。mod 998244353なのか、他のNTT-friendlyなmodで考えるのか、それともGarnerで復元するのか、それとも畳み込みを$O(N^2)$で妥協するのかなどによって異なる
  virtual FPS inv(int deg=-1) const;
  virtual void CooleyTukeyNTT998244353(vector<mint>&a, bool is_reverse) const;
//  virtual FPS exp(int deg=-1) const;
  virtual vector<mint> multiply(const vector<mint>& a, const vector<mint>& b);
  };

Back to top page