fps/fps-sqrt.hpp

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• Last update: 2025-10-21 21:13:36+09:00
• Include: #include "fps/fps-sqrt.hpp"

Depends on

• fps/formal-power-series.hpp
• modint/mod-pow.hpp
• modint/mod-sqrt.hpp

Verified with

• verify/fps/LC_sqrt_of_formal_power_series.relaxed.test.cpp
• verify/fps/LC_sqrt_of_formal_power_series.test.cpp

Code

#pragma once

#include "modint/mod-sqrt.hpp"
#include "fps/formal-power-series.hpp"

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

#line 2 "fps/fps-sqrt.hpp"

#line 2 "modint/mod-sqrt.hpp"

#line 2 "modint/mod-pow.hpp"

unsigned int ModPow(unsigned int a, unsigned long long n, unsigned int m) {
  unsigned long long x = a, y = 1;
  while (n) {
    if (n & 1) y = y * x % m;
    x = x * x % m;
    n >>= 1;
  }
  return y;
}
#line 4 "modint/mod-sqrt.hpp"

long long ModSqrt(long long a, long long p) {
  if (a >= p) a %= p;
  if (p == 2) return a & 1;
  if (a == 0) return 0;
  if (ModPow(a, (p - 1) / 2, p) != 1) return -1;
  if (p % 4 == 3) return ModPow(a, (3 * p - 1) / 4, p);
  unsigned int z = 2, q = p - 1;
  while (ModPow(z, (p - 1) / 2, p) == 1) z++;
  int s = 0;
  while (!(q & 1)) {
    s++;
    q >>= 1;
  }
  int m = s;
  unsigned int c = ModPow(z, q, p);
  unsigned int t = ModPow(a, q, p);
  unsigned int r = ModPow(a, (q + 1) / 2, p);
  while (true) {
    if (t == 1) return r;
    unsigned int pow = t;
    int j = 1;
    for (; j < m; j++) {
      pow = 1ll * pow * pow % p;
      if (pow == 1) break;
    }
    unsigned int b = c;
    for (int i = 0; i < m - j - 1; i++) b = 1ll * b * b % p;
    m = j;
    c = 1ll * b * b % p;
    t = 1ll * t * c % p;
    r = 1ll * r * b % p;
  }
}
#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 5 "fps/fps-sqrt.hpp"

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

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