平方根
(fps/fps-sqrt.hpp)

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

fps-平方根

$N$次の形式的冪級数$ f(x)$ に対して$g(x) \equiv \sqrt{f(x)} \pmod{x^N}$ を満たす $g(x)$ を $\mathrm{O}(N \log N)$ で計算するライブラリ。
ここで $\sqrt{f(x)}$ は $g^2(x) \equiv f(x) \mod x^N$ を満たす形式的冪級数を意味する。

概要

$f$は $t^2=f_0$ を満たす $t$ が存在する多項式とする。このとき $\lbrack x^0 \rbrack g = t, g \equiv \sqrt{f} \mod x^n$ となる $g$ を求めたい。

まず、$g \equiv t \mod x$ である。次にニュートン法で $g^2 \equiv f$ を満たす $g$ を求めると、$g=\hat{g} \mod x^k$ のとき

\[g \equiv \hat{g} - \frac{\hat{g}^2-f}{(\hat{g}^2)'} \pmod{x^{2k}}\] \[\leftrightarrow g \equiv \frac{1}{2}\left(\hat{g}+\frac{f}{\hat{g}}\right) \pmod{x^{2k}}\]

を得てダブリングで求まる。計算量は $\mathrm{O}(N \log N)$。

使い方

• sqrt(f, deg): FPS である $f$ について $\sqrt f$ 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/arbitrary-montgomery-modint.hpp
• mod sqrt(Tonelli-Shanks algorithm) (modulo/mod-sqrt.hpp)

Verified with

• verify/verify-yosupo-fps/yosupo-sqrt.test.cpp

Code

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

template <typename mint>
FormalPowerSeries<mint> sqrt(const FormalPowerSeries<mint> &f, int deg = -1) {
  if (deg == -1) deg = (int)f.size();
  if ((int)f.size() == 0) return FormalPowerSeries<mint>(deg, 0);
  if (f[0] == mint(0)) {
    for (int i = 1; i < (int)f.size(); i++) {
      if (f[i] != mint(0)) {
        if (i & 1) return {};
        if (deg - i / 2 <= 0) break;
        auto ret = sqrt(f >> i, deg - i / 2);
        if (ret.empty()) return {};
        ret = ret << (i / 2);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
    }
    return FormalPowerSeries<mint>(deg, 0);
  }

  int64_t sqr = mod_sqrt(f[0].get(), mint::get_mod());
  if (sqr == -1) return {};
  assert(sqr * sqr % mint::get_mod() == f[0].get());
  FormalPowerSeries<mint> ret = {mint(sqr)};
  mint inv2 = mint(2).inverse();
  for (int i = 1; i < deg; i <<= 1) {
    ret = (ret + f.pre(i << 1) * ret.inv(i << 1)) * inv2;
  }
  return ret.pre(deg);
}

/**
 * @brief 平方根
 * @docs docs/fps/fps-sqrt.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/arbitrary-montgomery-modint.hpp"

#include <iostream>
using namespace std;

template <typename Int, typename UInt, typename Long, typename ULong, int id>
struct ArbitraryLazyMontgomeryModIntBase {
  using mint = ArbitraryLazyMontgomeryModIntBase;

  inline static UInt mod;
  inline static UInt r;
  inline static UInt n2;
  static constexpr int bit_length = sizeof(UInt) * 8;

  static UInt get_r() {
    UInt ret = mod;
    while (mod * ret != 1) ret *= UInt(2) - mod * ret;
    return ret;
  }
  static void set_mod(UInt m) {
    assert(m < (UInt(1u) << (bit_length - 2)));
    assert((m & 1) == 1);
    mod = m, n2 = -ULong(m) % m, r = get_r();
  }
  UInt a;

  ArbitraryLazyMontgomeryModIntBase() : a(0) {}
  ArbitraryLazyMontgomeryModIntBase(const Long &b)
      : a(reduce(ULong(b % mod + mod) * n2)){};

  static UInt reduce(const ULong &b) {
    return (b + ULong(UInt(b) * UInt(-r)) * mod) >> bit_length;
  }

  mint &operator+=(const mint &b) {
    if (Int(a += b.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }
  mint &operator-=(const mint &b) {
    if (Int(a -= b.a) < 0) a += 2 * mod;
    return *this;
  }
  mint &operator*=(const mint &b) {
    a = reduce(ULong(a) * b.a);
    return *this;
  }
  mint &operator/=(const mint &b) {
    *this *= b.inverse();
    return *this;
  }

  mint operator+(const mint &b) const { return mint(*this) += b; }
  mint operator-(const mint &b) const { return mint(*this) -= b; }
  mint operator*(const mint &b) const { return mint(*this) *= b; }
  mint operator/(const mint &b) const { return mint(*this) /= b; }

  bool operator==(const mint &b) const {
    return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
  }
  bool operator!=(const mint &b) const {
    return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
  }
  mint operator-() const { return mint(0) - mint(*this); }
  mint operator+() const { return mint(*this); }

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

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

  friend istream &operator>>(istream &is, mint &b) {
    Long t;
    is >> t;
    b = ArbitraryLazyMontgomeryModIntBase(t);
    return (is);
  }

  mint inverse() const {
    Int x = get(), y = get_mod(), u = 1, v = 0;
    while (y > 0) {
      Int t = x / y;
      swap(x -= t * y, y);
      swap(u -= t * v, v);
    }
    return mint{u};
  }

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

  static UInt get_mod() { return mod; }
};

// id に適当な乱数を割り当てて使う
template <int id>
using ArbitraryLazyMontgomeryModInt =
    ArbitraryLazyMontgomeryModIntBase<int, unsigned int, long long,
                                      unsigned long long, id>;
template <int id>
using ArbitraryLazyMontgomeryModInt64bit =
    ArbitraryLazyMontgomeryModIntBase<long long, unsigned long long, __int128_t,
                                      __uint128_t, id>;
#line 3 "modulo/mod-sqrt.hpp"

int64_t mod_sqrt(const int64_t &a, const int64_t &p) {
  assert(0 <= a && a < p);
  if (a < 2) return a;
  using Mint = ArbitraryLazyMontgomeryModInt<409075245>;
  Mint::set_mod(p);
  if (Mint(a).pow((p - 1) >> 1) != 1) return -1;
  Mint b = 1, one = 1;
  while (b.pow((p - 1) >> 1) == 1) b += one;
  int64_t m = p - 1, e = 0;
  while (m % 2 == 0) m >>= 1, e += 1;
  Mint x = Mint(a).pow((m - 1) >> 1);
  Mint y = Mint(a) * x * x;
  x *= a;
  Mint z = Mint(b).pow(m);
  while (y != 1) {
    int64_t j = 0;
    Mint t = y;
    while (t != one) {
      j += 1;
      t *= t;
    }
    z = z.pow(int64_t(1) << (e - j - 1));
    x *= z;
    z *= z;
    y *= z;
    e = j;
  }
  return x.get();
}

/**
 * @brief mod sqrt(Tonelli-Shanks algorithm)
 * @docs docs/modulo/mod-sqrt.md
 */
#line 4 "fps/fps-sqrt.hpp"

template <typename mint>
FormalPowerSeries<mint> sqrt(const FormalPowerSeries<mint> &f, int deg = -1) {
  if (deg == -1) deg = (int)f.size();
  if ((int)f.size() == 0) return FormalPowerSeries<mint>(deg, 0);
  if (f[0] == mint(0)) {
    for (int i = 1; i < (int)f.size(); i++) {
      if (f[i] != mint(0)) {
        if (i & 1) return {};
        if (deg - i / 2 <= 0) break;
        auto ret = sqrt(f >> i, deg - i / 2);
        if (ret.empty()) return {};
        ret = ret << (i / 2);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
    }
    return FormalPowerSeries<mint>(deg, 0);
  }

  int64_t sqr = mod_sqrt(f[0].get(), mint::get_mod());
  if (sqr == -1) return {};
  assert(sqr * sqr % mint::get_mod() == f[0].get());
  FormalPowerSeries<mint> ret = {mint(sqr)};
  mint inv2 = mint(2).inverse();
  for (int i = 1; i < deg; i <<= 1) {
    ret = (ret + f.pre(i << 1) * ret.inv(i << 1)) * inv2;
  }
  return ret.pre(deg);
}

/**
 * @brief 平方根
 * @docs docs/fps/fps-sqrt.md
 */

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