多項式補間
(fps/polynomial-interpolation.hpp)

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• Last update: 2025-10-23 01:57:19+09:00
• Include: #include "fps/polynomial-interpolation.hpp"

$n$ 次多項式 $f$ で $0\leq i\leq n$ に対し $f(x_i)=y_i$ を満たすものを $O(n(\log n)^2)$ 時間で求める．

アルゴリズム

$g(x)=\prod_{i=0}^{n}(x-x_i)$ とすれば Lagrange 補間より以下が成り立つ．

\[f(x)=\sum_{i=0}^{n}\frac{y_ig(x)}{g'(x_i)(x-x_i)}\]

以下のようにして $O(n(\log n)^2)$ 時間で計算できる．

• $g(x)$ は分割統治により $O(n(\log n)^2)$ 時間で計算できる．
• $g’(x_i)$ は多点評価により $O(n(\log n)^2)$ 時間で列挙できる．
• $\sum_{i=0}^{n}\frac{y_i}{g’(x_i)(x-x_i)}$ は有理数の和として分割統治により $O(n(\log n)^2)$ 時間で計算できる．

$g(x)$ の計算過程はそれ以降にも流用できる．

Depends on

• fps/formal-power-series.hpp

Verified with

• verify/fps/LC_polynomial_interpolation.test.cpp
• verify/fps/UNIT_prefix_sum_of_polynomial.test.cpp

Code

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

template <class mint>
FormalPowerSeries<mint> PolynomialInterpolation(const vector<mint>& x, const vector<mint>& y) {
  using fps = FormalPowerSeries<mint>;
  assert(x.size() == y.size());
  int n = x.size();
  if (n == 0) return {y[0]};
  vector<fps> prod(2 * n);
  for (int i = 0; i < n; i++) prod[i + n] = {-x[i], 1};
  for (int i = n - 1; i > 0; i--) prod[i] = prod[i * 2] * prod[i * 2 + 1];
  vector<fps> fs(2 * n);
  fs[1] = prod[1].diff();
  for (int i = 2; i < 2 * n; i++) fs[i] = fs[i / 2] % prod[i];
  for (int i = n; i < n * 2; i++) fs[i] = {y[i - n] / fs[i][0]};
  for (int i = n - 1; i > 0; i--) fs[i] = fs[(i << 1) | 0] * prod[(i << 1) | 1] + fs[(i << 1) | 1] * prod[(i << 1) | 0];
  return fs[1];
}

/**
 * @brief 多項式補間
 * @docs docs/fps/polynomial-interpolation.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/polynomial-interpolation.hpp"

template <class mint>
FormalPowerSeries<mint> PolynomialInterpolation(const vector<mint>& x, const vector<mint>& y) {
  using fps = FormalPowerSeries<mint>;
  assert(x.size() == y.size());
  int n = x.size();
  if (n == 0) return {y[0]};
  vector<fps> prod(2 * n);
  for (int i = 0; i < n; i++) prod[i + n] = {-x[i], 1};
  for (int i = n - 1; i > 0; i--) prod[i] = prod[i * 2] * prod[i * 2 + 1];
  vector<fps> fs(2 * n);
  fs[1] = prod[1].diff();
  for (int i = 2; i < 2 * n; i++) fs[i] = fs[i / 2] % prod[i];
  for (int i = n; i < n * 2; i++) fs[i] = {y[i - n] / fs[i][0]};
  for (int i = n - 1; i > 0; i--) fs[i] = fs[(i << 1) | 0] * prod[(i << 1) | 1] + fs[(i << 1) | 1] * prod[(i << 1) | 0];
  return fs[1];
}

/**
 * @brief 多項式補間
 * @docs docs/fps/polynomial-interpolation.md
 */

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