polynomial/fps_exp.hpp

• View this file on GitHub
• Last update: 2022-07-22 12:48:33+09:00
• Include: #include "polynomial/fps_exp.hpp"

Depends on

• polynomial/polynomial.hpp

Required by

• polynomial/fps_pow.hpp

Verified with

• polynomial/test/exp_of_formal_power_series.test.cpp
• polynomial/test/exp_of_formal_power_series_acl.test.cpp
• polynomial/test/pow_of_formal_power_series.test.cpp
• polynomial/test/pow_of_formal_power_series_acl.test.cpp

Code

#pragma once

#include "polynomial.hpp"

template <typename T, typename Mul>
Polynomial<T, Mul> fps_exp(const Polynomial<T, Mul> &h, int sz = -1) {
    const std::vector<T> &coeff = h.vec();
    assert(!coeff.empty() && coeff[0] == T(0));
    if (sz == -1) {
        sz = (int) coeff.size();
    }
    assert(sz >= 0);
    std::vector<T> f({T(1)});
    std::vector<T> g({T(1)});
    std::vector<T> dft_f_({T(1), T(1)});
    
    while ((int) f.size() < sz) {
        int n = (int) f.size();
        
        // F_{2n}(g_0)
        std::vector<T> dft_g_2 = g;
        dft_g_2.resize(2 * n, T(0));
        Mul::dft(dft_g_2);
        
        // \delta
        std::vector<T> delta(n, T(0));
        for (int i = 0; i < n; ++i) {
            delta[i] = dft_f_[i] * dft_g_2[i];
        }
        Mul::idft(delta);
        delta.resize(2 * n);
        for (int i = 0; i < n; ++i) {
            std::swap(delta[i], delta[n + i]);
        }
        delta[n] -= T(1);
        
        // F_n(D(f_0))
        std::vector<T> dft_d_f(n, T(0));
        for (int i = 0; i < n - 1; ++i) {
            dft_d_f[i] = T(i + 1) * f[i + 1];
        }
        Mul::dft(dft_d_f);
        
        // D(f_0) g_0
        std::vector<T> d_f_g(n, T(0));
        for (int i = 0; i < n; ++i) {
            d_f_g[i] = dft_d_f[i] * dft_g_2[i];
        }
        Mul::idft(d_f_g);
        d_f_g.resize(2 * n, T(0));
        for (int i = 0; i < n - 1; ++i) {
            T tmp = T(i + 1) * h.at(i + 1);
            d_f_g[n + i] = d_f_g[i] - tmp;
            d_f_g[i] = tmp;
        }
        
        // \delta D(h_0)
        std::vector<T> dft_delta = delta;
        Mul::dft(dft_delta);
        std::vector<T> delta_d_h(2 * n);
        for (int i = 0; i < n - 1; ++i) {
            delta_d_h[i] = T(i + 1) * h.at(i + 1);
        }
        Mul::dft(delta_d_h);
        for (int i = 0; i < 2 * n; ++i) {
            delta_d_h[i] *= dft_delta[i];
        }
        Mul::idft(delta_d_h);
        std::fill(delta_d_h.begin(), delta_d_h.begin() + n, T(0));
        
        // \epsilon
        std::vector<T> eps = std::move(d_f_g);
        for (int i = 0; i < 2 * n; ++i) {
            eps[i] -= T(i + 1) * h.at(i + 1) + delta_d_h[i];
        }
        for (int i = 2 * n - 1; i > 0; --i) {
            eps[i] = eps[i - 1] / T(i);
        }
        eps[0] = T(0);
        
        // \epsilon f_0
        std::vector<T> dft_eps = eps;
        Mul::dft(dft_eps);
        std::vector<T> eps_f(2 * n);
        for (int i = 0; i < 2 * n; ++i) {
            eps_f[i] = dft_eps[i] * dft_f_[i];
        }
        Mul::idft(eps_f);
        std::fill(eps_f.begin(), eps_f.begin() + n - 1, T(0));
        
        // update f
        f.resize(2 * n, T(0));
        for (int i = 0; i < 2 * n; ++i) {
            f[i] -= eps_f[i];
        }
        
        if ((int) f.size() >= sz) {
            break;
        }
        
        // update F_{2n}(f)
        dft_f_ = f;
        dft_f_.resize(4 * n);
        Mul::dft(dft_f_);
        
        // update g
        std::vector<T> fg(dft_f_.begin(), dft_f_.begin() + 2 * n);
        for (int i = 0; i < 2 * n; ++i) {
            fg[i] *= dft_g_2[i];
        }
        Mul::idft(fg);
        std::fill(fg.begin(), fg.begin() + n, T(0));
        Mul::dft(fg);
        for (int i = 0; i < 2 * n; ++i) {
            fg[i] *= dft_g_2[i];
        }
        Mul::idft(fg);
        g.resize(2 * n);
        for (int i = n; i < 2 * n; ++i) {
            g[i] = -fg[i];
        }
    }
    
    f.resize(sz);
    return Polynomial<T, Mul>(f);
}

#line 2 "polynomial/fps_exp.hpp"

#line 2 "polynomial/polynomial.hpp"

#include <vector>
#include <utility>
#include <cassert>
#include <algorithm>

template <typename T, typename Mul>
class Polynomial {
    std::vector<T> coeff;
    
public:
    using This = Polynomial<T, Mul>;
    
    Polynomial() : coeff() {}
    Polynomial(int n) : coeff(n, T(0)) {}
    Polynomial(std::vector<T> c) : coeff(std::move(c)) {}
    
    const std::vector<T> &vec() const {
        return coeff;
    }
    
    int size() const {
        return (int) coeff.size();
    }
    
    const T &operator[](int idx) const {
        return coeff[idx];
    }
    T &operator[](int idx) {
        return coeff[idx];
    }
    
    T at(int idx) const {
        if (idx < size()) {
            return coeff[idx];
        } else {
            return T(0);
        }
    }
    
    void pre_(int n) {
        assert(n >= 0);
        coeff.resize(n, T(0));
    }
    This pre(int n) const {
        This tmp(*this);
        tmp.pre_(n);
        return tmp;
    }
    
    T operator()(const T &x) const {
        T p(1), sum(0);
        for (const T &ele : coeff) {
            sum += p * ele;
            p *= x;
        }
        return sum;
    }
    
    This &operator+=(const This &rhs) {
        if (coeff.size() < rhs.coeff.size()) {
            coeff.resize(rhs.coeff.size(), T(0));
        }
        for (int i = 0; i < (int) rhs.coeff.size(); ++i) {
            coeff[i] += rhs.coeff[i];
        }
        return *this;
    }
    friend This operator+(This lhs, const This &rhs) {
        lhs += rhs;
        return lhs;
    }
    This &operator-=(const This &rhs) {
        if (coeff.size() < rhs.coeff.size()) {
            coeff.resize(rhs.coeff.size(), T(0));
        }
        for (int i = 0; i < (int) rhs.coeff.size(); ++i) {
            coeff[i] -= rhs.coeff[i];
        }
        return *this;
    }
    friend This operator-(This lhs, const This &rhs) {
        lhs -= rhs;
        return lhs;
    }
    
    This &operator*=(This rhs) {
        coeff = Mul::mul(std::move(coeff), std::move(rhs.coeff));
        return *this;
    }
    friend This operator*(This lhs, This rhs) {
        return This(Mul::mul(std::move(lhs.coeff), std::move(rhs.coeff)));
    }
    
    This diff() const {
        if (coeff.empty()) {
            return This();
        }
        std::vector<T> c(coeff.size() - 1);
        for (int i = 0; i < (int) c.size(); ++i) {
            c[i] = T(i + 1) * coeff[i + 1];
        }
        return This(c);
    }
    This integ() const {
        std::vector<T> c(coeff.size() + 1, T(0));
        for (int i = 0; i < (int) coeff.size(); ++i) {
            c[i + 1] = coeff[i] / T(i + 1);
        }
        return This(c);
    }
};
#line 4 "polynomial/fps_exp.hpp"

template <typename T, typename Mul>
Polynomial<T, Mul> fps_exp(const Polynomial<T, Mul> &h, int sz = -1) {
    const std::vector<T> &coeff = h.vec();
    assert(!coeff.empty() && coeff[0] == T(0));
    if (sz == -1) {
        sz = (int) coeff.size();
    }
    assert(sz >= 0);
    std::vector<T> f({T(1)});
    std::vector<T> g({T(1)});
    std::vector<T> dft_f_({T(1), T(1)});
    
    while ((int) f.size() < sz) {
        int n = (int) f.size();
        
        // F_{2n}(g_0)
        std::vector<T> dft_g_2 = g;
        dft_g_2.resize(2 * n, T(0));
        Mul::dft(dft_g_2);
        
        // \delta
        std::vector<T> delta(n, T(0));
        for (int i = 0; i < n; ++i) {
            delta[i] = dft_f_[i] * dft_g_2[i];
        }
        Mul::idft(delta);
        delta.resize(2 * n);
        for (int i = 0; i < n; ++i) {
            std::swap(delta[i], delta[n + i]);
        }
        delta[n] -= T(1);
        
        // F_n(D(f_0))
        std::vector<T> dft_d_f(n, T(0));
        for (int i = 0; i < n - 1; ++i) {
            dft_d_f[i] = T(i + 1) * f[i + 1];
        }
        Mul::dft(dft_d_f);
        
        // D(f_0) g_0
        std::vector<T> d_f_g(n, T(0));
        for (int i = 0; i < n; ++i) {
            d_f_g[i] = dft_d_f[i] * dft_g_2[i];
        }
        Mul::idft(d_f_g);
        d_f_g.resize(2 * n, T(0));
        for (int i = 0; i < n - 1; ++i) {
            T tmp = T(i + 1) * h.at(i + 1);
            d_f_g[n + i] = d_f_g[i] - tmp;
            d_f_g[i] = tmp;
        }
        
        // \delta D(h_0)
        std::vector<T> dft_delta = delta;
        Mul::dft(dft_delta);
        std::vector<T> delta_d_h(2 * n);
        for (int i = 0; i < n - 1; ++i) {
            delta_d_h[i] = T(i + 1) * h.at(i + 1);
        }
        Mul::dft(delta_d_h);
        for (int i = 0; i < 2 * n; ++i) {
            delta_d_h[i] *= dft_delta[i];
        }
        Mul::idft(delta_d_h);
        std::fill(delta_d_h.begin(), delta_d_h.begin() + n, T(0));
        
        // \epsilon
        std::vector<T> eps = std::move(d_f_g);
        for (int i = 0; i < 2 * n; ++i) {
            eps[i] -= T(i + 1) * h.at(i + 1) + delta_d_h[i];
        }
        for (int i = 2 * n - 1; i > 0; --i) {
            eps[i] = eps[i - 1] / T(i);
        }
        eps[0] = T(0);
        
        // \epsilon f_0
        std::vector<T> dft_eps = eps;
        Mul::dft(dft_eps);
        std::vector<T> eps_f(2 * n);
        for (int i = 0; i < 2 * n; ++i) {
            eps_f[i] = dft_eps[i] * dft_f_[i];
        }
        Mul::idft(eps_f);
        std::fill(eps_f.begin(), eps_f.begin() + n - 1, T(0));
        
        // update f
        f.resize(2 * n, T(0));
        for (int i = 0; i < 2 * n; ++i) {
            f[i] -= eps_f[i];
        }
        
        if ((int) f.size() >= sz) {
            break;
        }
        
        // update F_{2n}(f)
        dft_f_ = f;
        dft_f_.resize(4 * n);
        Mul::dft(dft_f_);
        
        // update g
        std::vector<T> fg(dft_f_.begin(), dft_f_.begin() + 2 * n);
        for (int i = 0; i < 2 * n; ++i) {
            fg[i] *= dft_g_2[i];
        }
        Mul::idft(fg);
        std::fill(fg.begin(), fg.begin() + n, T(0));
        Mul::dft(fg);
        for (int i = 0; i < 2 * n; ++i) {
            fg[i] *= dft_g_2[i];
        }
        Mul::idft(fg);
        g.resize(2 * n);
        for (int i = n; i < 2 * n; ++i) {
            g[i] = -fg[i];
        }
    }
    
    f.resize(sz);
    return Polynomial<T, Mul>(f);
}

Back to top page