CP-Algorithms Library
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cp-algo/math/poly/impl/euclid.hpp
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- Last update: 2025-05-01 14:29:39+02:00
- Include:
#include "cp-algo/math/poly/impl/euclid.hpp"
Depends on
- cp-algo/math/affine.hpp
- cp-algo/math/common.hpp
- cp-algo/math/cvector.hpp
- cp-algo/math/fft.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/complex.hpp
- cp-algo/util/simd.hpp
Required by
Verified with
- Characteristic Polynomial (verify/linalg/characteristic.test.cpp)
- Pow of Matrix (Frobenius) (verify/linalg/pow_fast.test.cpp)
- Bell Number (verify/poly/bell.test.cpp)
- Division of Polynomials (verify/poly/div.test.cpp)
- Exp of Power Series (verify/poly/exp.test.cpp)
- Find Linear Recurrence (verify/poly/find_linrec.test.cpp)
- Polynomial Interpolation (verify/poly/inter.test.cpp)
- Polynomial Interpolation (Geometric Sequence) (verify/poly/inter_geo.test.cpp)
- Inv of Power Series (verify/poly/inv.test.cpp)
- Inv of Polynomials (verify/poly/inv_mod.test.cpp)
- Consecutive Terms of Linear Recursion (verify/poly/linrec_consecutive.test.cpp)
- Kth term of Linearly Recurrent Sequence (verify/poly/linrec_single.test.cpp)
- Log of Power Series (verify/poly/log.test.cpp)
- Multipoint Evaluation (verify/poly/multieval.test.cpp)
- Multipoint Evaluation (Geometric Sequence) (verify/poly/multieval_geo.test.cpp)
- Pow of Power Series (verify/poly/pow.test.cpp)
- Product of Polynomial Sequence (verify/poly/prod_sequence.test.cpp)
- Polynomial Root Finding (verify/poly/roots.test.cpp)
- Shift of Sampling Points of Polynomial (verify/poly/sampling.test.cpp)
- Sqrt of Power Series (verify/poly/sqrt.test.cpp)
- Polynomial Taylor Shift (verify/poly/taylor.test.cpp)
Code
#ifndef CP_ALGO_MATH_POLY_IMPL_EUCLID_HPP
#define CP_ALGO_MATH_POLY_IMPL_EUCLID_HPP
#include "../../affine.hpp"
#include "../../fft.hpp"
#include <functional>
#include <algorithm>
#include <numeric>
#include <cassert>
#include <vector>
#include <tuple>
#include <list>
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
template<typename poly>
using gcd_result = std::pair<
std::list<std::decay_t<poly>>,
linfrac<std::decay_t<poly>>>;
template<typename poly>
gcd_result<poly> half_gcd(poly &&A, poly &&B) {
assert(A.deg() >= B.deg());
size_t m = size(A.a) / 2;
if(B.deg() < (int)m) {
return {};
}
auto [ai, R] = A.divmod(B);
std::tie(A, B) = {B, R};
std::list a = {ai};
auto T = -linfrac(ai).adj();
auto advance = [&](size_t k) {
auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
a.splice(end(a), ak);
T.prepend(Tk);
return Tk;
};
advance(m).apply(A, B);
if constexpr (std::is_reference_v<poly>) {
advance(2 * m - A.deg()).apply(A, B);
} else {
advance(2 * m - A.deg());
}
return {std::move(a), std::move(T)};
}
template<typename poly>
gcd_result<poly> full_gcd(poly &&A, poly &&B) {
using poly_t = std::decay_t<poly>;
std::list<poly_t> ak;
std::vector<linfrac<poly_t>> trs;
while(!B.is_zero()) {
auto [a0, R] = A.divmod(B);
ak.push_back(a0);
trs.push_back(-linfrac(a0).adj());
std::tie(A, B) = {B, R};
auto [a, Tr] = half_gcd(A, B);
ak.splice(end(ak), a);
trs.push_back(Tr);
}
return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
}
// computes product of linfrac on [L, R)
auto convergent(auto L, auto R) {
using poly = decltype(L)::value_type;
if(R == next(L)) {
return linfrac(*L);
} else {
int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
auto M = L;
for(int c = M->deg(); 2 * c <= s; M++) {
c += next(M)->deg();
}
return convergent(L, M) * convergent(M, R);
}
}
template<typename poly>
poly min_rec(poly const& p, size_t d) {
auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
if(R2.is_zero()) {
return poly(1);
}
auto [a, Tr] = full_gcd(R1, R2);
a.emplace_back();
auto pref = begin(a);
for(int delta = (int)d - a.front().deg(); delta >= 0; pref++) {
delta -= pref->deg() + next(pref)->deg();
}
return convergent(begin(a), pref).a;
}
template<typename poly>
std::optional<poly> inv_mod(poly p, poly q) {
assert(!q.is_zero());
auto [a, Tr] = full_gcd(q, p);
if(q.deg() != 0) {
return std::nullopt;
}
return Tr.b / q[0];
}
}
#endif // CP_ALGO_MATH_POLY_IMPL_EUCLID_HPP
#line 1 "cp-algo/math/poly/impl/euclid.hpp"
#line 1 "cp-algo/math/affine.hpp"
#include <optional>
#include <utility>
#include <cassert>
#include <tuple>
namespace cp_algo::math {
// a * x + b
template<typename base>
struct lin {
base a = 1, b = 0;
std::optional<base> c;
lin() {}
lin(base b): a(0), b(b) {}
lin(base a, base b): a(a), b(b) {}
lin(base a, base b, base _c): a(a), b(b), c(_c) {}
// polynomial product modulo x^2 - c
lin operator * (const lin& t) {
assert(c && t.c && *c == *t.c);
return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
}
// a * (t.a * x + t.b) + b
lin apply(lin const& t) const {
return {a * t.a, a * t.b + b};
}
void prepend(lin const& t) {
*this = t.apply(*this);
}
base eval(base x) const {
return a * x + b;
}
};
// (ax+b) / (cx+d)
template<typename base>
struct linfrac {
base a, b, c, d;
linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}
// composition of two linfracs
linfrac operator * (linfrac t) const {
return t.prepend(linfrac(*this));
}
linfrac operator-() const {
return {-a, -b, -c, -d};
}
linfrac adj() const {
return {d, -b, -c, a};
}
linfrac& prepend(linfrac const& t) {
t.apply(a, c);
t.apply(b, d);
return *this;
}
// apply linfrac to A/B
void apply(base &A, base &B) const {
std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
}
};
}
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
auto bpow(auto const& x, auto n, auto const& one, auto op) {
if(n == 0) {
return one;
} else {
auto t = bpow(x, n / 2, one, op);
t = op(t, t);
if(n % 2) {
t = op(t, x);
}
return t;
}
}
auto bpow(auto x, auto n, auto ans) {
return bpow(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
}
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
static Int mod() {
return modint::mod();
}
static Int remod() {
return modint::remod();
}
static UInt2 modmod() {
return UInt2(mod()) * mod();
}
modint_base(): r(0) {}
modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator - () const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator /= (const modint &t) {
return to_modint() *= t.inv();
}
modint& operator *= (const modint &t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator += (const modint &t) {
r += t.r; r = std::min(r, r - remod());
return to_modint();
}
modint& operator -= (const modint &t) {
r -= t.r; r = std::min(r, r + remod());
return to_modint();
}
modint operator + (const modint &t) const {return modint(to_modint()) += t;}
modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
// Why <=> doesn't work?..
auto operator == (const modint &t) const {return to_modint().getr() == t.getr();}
auto operator != (const modint &t) const {return to_modint().getr() != t.getr();}
auto operator <= (const modint &t) const {return to_modint().getr() <= t.getr();}
auto operator >= (const modint &t) const {return to_modint().getr() >= t.getr();}
auto operator < (const modint &t) const {return to_modint().getr() < t.getr();}
auto operator > (const modint &t) const {return to_modint().getr() > t.getr();}
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
void setr(UInt rr) {
r = rr;
}
UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static UInt modmod8() {return UInt(8 * modmod());}
void add_unsafe(UInt t) {r += t;}
void pseudonormalize() {r = std::min(r, r - modmod8());}
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) {r = rr;}
UInt getr_direct() const {return r;}
protected:
UInt r;
private:
modint& to_modint() {return static_cast<modint&>(*this);}
modint const& to_modint() const {return static_cast<modint const&>(*this);}
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator >> (decltype(std::cin) &in, modint &x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator << (decltype(std::cout) &out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() {return m;}
static constexpr Base::UInt remod() {return m;}
auto getr() const {return Base::r;}
};
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
template<typename Int = int64_t>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
static Base::UInt m_reduce(Base::UInt2 ab) {
if(mod() % 2 == 0) [[unlikely]] {
return typename Base::UInt(ab % mod());
} else {
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator *= (const dynamic_modint &t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() {return m;}
static Int remod() {return 2 * m;}
static Base::UInt imod() {return im;}
static Base::UInt2 pw128() {return r2;}
static void switch_mod(Int nm) {
m = nm;
im = m % 2 ? inv2(-m) : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() {switch_mod(prev);}
} _;
switch_mod(tmp);
return callback();
}
private:
static thread_local Int m;
static thread_local Base::UInt im, r2;
};
template<typename Int>
Int thread_local dynamic_modint<Int>::m = 1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}
#line 1 "cp-algo/util/checkpoint.hpp"
#line 4 "cp-algo/util/checkpoint.hpp"
#include <chrono>
#include <string>
namespace cp_algo {
template<bool final = false>
void checkpoint([[maybe_unused]] std::string const& msg = "") {
#ifdef CP_ALGO_CHECKPOINT
static double last = 0;
double now = (double)clock() / CLOCKS_PER_SEC;
double delta = now - last;
last = now;
if(msg.size()) {
std::cerr << msg << ": " << (final ? now : delta) * 1000 << " ms\n";
}
#endif
}
}
#line 1 "cp-algo/random/rng.hpp"
#line 4 "cp-algo/random/rng.hpp"
#include <random>
namespace cp_algo::random {
uint64_t rng() {
static std::mt19937_64 rng(
std::chrono::steady_clock::now().time_since_epoch().count()
);
return rng();
}
}
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 5 "cp-algo/util/simd.hpp"
#include <cstddef>
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using dx4 = simd<double, 4>;
[[gnu::always_inline]] inline dx4 abs(dx4 a) {
return a < 0 ? -a : a;
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
[[gnu::always_inline]] inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
[[gnu::always_inline]] inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
[[gnu::always_inline]] inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
[[gnu::always_inline]] inline u64x4 montgomery_reduce(u64x4 x, u64x4 mod, u64x4 imod) {
auto x_ninv = u64x4(u32x8(x) * u32x8(imod));
#ifdef __AVX2__
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i(mod)));
#else
x += x_ninv * mod;
#endif
return x >> 32;
}
[[gnu::always_inline]] inline u64x4 montgomery_mul(u64x4 x, u64x4 y, u64x4 mod, u64x4 imod) {
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
return montgomery_reduce(x * y, mod, imod);
#endif
}
[[gnu::always_inline]] inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
}
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
namespace cp_algo {
// Custom implementation, since std::complex is UB on non-floating types
template<typename T>
struct complex {
using value_type = T;
T x, y;
constexpr complex(): x(), y() {}
constexpr complex(T x): x(x), y() {}
constexpr complex(T x, T y): x(x), y(y) {}
complex& operator *= (T t) {x *= t; y *= t; return *this;}
complex& operator /= (T t) {x /= t; y /= t; return *this;}
complex operator * (T t) const {return complex(*this) *= t;}
complex operator / (T t) const {return complex(*this) /= t;}
complex& operator += (complex t) {x += t.x; y += t.y; return *this;}
complex& operator -= (complex t) {x -= t.x; y -= t.y; return *this;}
complex operator * (complex t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
complex operator / (complex t) const {return *this * t.conj() / t.norm();}
complex operator + (complex t) const {return complex(*this) += t;}
complex operator - (complex t) const {return complex(*this) -= t;}
complex& operator *= (complex t) {return *this = *this * t;}
complex& operator /= (complex t) {return *this = *this / t;}
complex operator - () const {return {-x, -y};}
complex conj() const {return {x, -y};}
T norm() const {return x * x + y * y;}
T abs() const {return std::sqrt(norm());}
T const real() const {return x;}
T const imag() const {return y;}
T& real() {return x;}
T& imag() {return y;}
static constexpr complex polar(T r, T theta) {return {r * cos(theta), r * sin(theta)};}
auto operator <=> (complex const& t) const = default;
};
template<typename T>
complex<T> operator * (auto x, complex<T> y) {return y *= x;}
template<typename T> complex<T> conj(complex<T> x) {return x.conj();}
template<typename T> T norm(complex<T> x) {return x.norm();}
template<typename T> T abs(complex<T> x) {return x.abs();}
template<typename T> T& real(complex<T> &x) {return x.real();}
template<typename T> T& imag(complex<T> &x) {return x.imag();}
template<typename T> T const real(complex<T> const& x) {return x.real();}
template<typename T> T const imag(complex<T> const& x) {return x.imag();}
template<typename T>
constexpr complex<T> polar(T r, T theta) {
return complex<T>::polar(r, theta);
}
template<typename T>
std::ostream& operator << (std::ostream &out, complex<T> x) {
return out << x.real() << ' ' << x.imag();
}
}
#line 1 "cp-algo/util/big_alloc.hpp"
#line 6 "cp-algo/util/big_alloc.hpp"
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
# define CP_ALGO_USE_MMAP 1
# include <sys/mman.h>
#else
# define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template <typename T>
class big_alloc: public std::allocator<T> {
public:
using value_type = T;
using base = std::allocator<T>;
big_alloc() noexcept = default;
template <typename U>
big_alloc(const big_alloc<U>&) noexcept {}
#if CP_ALGO_USE_MMAP
[[nodiscard]] T* allocate(std::size_t n) {
if(n * sizeof(T) < 1024 * 1024) {
return base::allocate(n);
}
n *= sizeof(T);
void* raw = mmap(nullptr, n,
PROT_READ | PROT_WRITE,
MAP_PRIVATE | MAP_ANONYMOUS,
-1, 0);
madvise(raw, n, MADV_HUGEPAGE);
madvise(raw, n, MADV_POPULATE_WRITE);
return static_cast<T*>(raw);
}
#endif
#if CP_ALGO_USE_MMAP
void deallocate(T* p, std::size_t n) noexcept {
if(n * sizeof(T) < 1024 * 1024) {
return base::deallocate(p, n);
}
if(p) {
munmap(p, n * sizeof(T));
}
}
#endif
};
}
#line 7 "cp-algo/math/cvector.hpp"
#include <ranges>
#include <bit>
namespace stdx = std::experimental;
namespace cp_algo::math::fft {
static constexpr size_t flen = 4;
using ftype = double;
using vftype = dx4;
using point = complex<ftype>;
using vpoint = complex<vftype>;
static constexpr vftype vz = {};
vpoint vi(vpoint const& r) {
return {-imag(r), real(r)};
}
struct cvector {
std::vector<vpoint, big_alloc<vpoint>> r;
cvector(size_t n) {
n = std::max(flen, std::bit_ceil(n));
r.resize(n / flen);
checkpoint("cvector create");
}
vpoint& at(size_t k) {return r[k / flen];}
vpoint at(size_t k) const {return r[k / flen];}
template<class pt = point>
void set(size_t k, pt t) {
if constexpr(std::is_same_v<pt, point>) {
real(r[k / flen])[k % flen] = real(t);
imag(r[k / flen])[k % flen] = imag(t);
} else {
at(k) = t;
}
}
template<class pt = point>
pt get(size_t k) const {
if constexpr(std::is_same_v<pt, point>) {
return {real(r[k / flen])[k % flen], imag(r[k / flen])[k % flen]};
} else {
return at(k);
}
}
size_t size() const {
return flen * r.size();
}
static constexpr size_t eval_arg(size_t n) {
if(n < pre_evals) {
return eval_args[n];
} else {
return eval_arg(n / 2) | (n & 1) << (std::bit_width(n) - 1);
}
}
static constexpr point eval_point(size_t n) {
if(n % 2) {
return -eval_point(n - 1);
} else if(n % 4) {
return eval_point(n - 2) * point(0, 1);
} else if(n / 4 < pre_evals) {
return evalp[n / 4];
} else {
return polar<ftype>(1., std::numbers::pi / (ftype)std::bit_floor(n) * (ftype)eval_arg(n));
}
}
static constexpr std::array<point, 32> roots = []() {
std::array<point, 32> res;
for(size_t i = 2; i < 32; i++) {
res[i] = polar<ftype>(1., std::numbers::pi / (1ull << (i - 2)));
}
return res;
}();
static constexpr point root(size_t n) {
return roots[std::bit_width(n)];
}
template<int step>
static void exec_on_eval(size_t n, size_t k, auto &&callback) {
callback(k, root(4 * step * n) * eval_point(step * k));
}
template<int step>
static void exec_on_evals(size_t n, auto &&callback) {
point factor = root(4 * step * n);
for(size_t i = 0; i < n; i++) {
callback(i, factor * eval_point(step * i));
}
}
void dot(cvector const& t) {
size_t n = this->size();
exec_on_evals<1>(n / flen, [&](size_t k, point rt) {
k *= flen;
auto [Ax, Ay] = at(k);
auto Bv = t.at(k);
vpoint res = vz;
for (size_t i = 0; i < flen; i++) {
res += vpoint(vz + Ax[i], vz + Ay[i]) * Bv;
real(Bv) = rotate_right(real(Bv));
imag(Bv) = rotate_right(imag(Bv));
auto x = real(Bv)[0], y = imag(Bv)[0];
real(Bv)[0] = x * real(rt) - y * imag(rt);
imag(Bv)[0] = x * imag(rt) + y * real(rt);
}
set(k, res);
});
checkpoint("dot");
}
void ifft() {
size_t n = size();
bool parity = std::countr_zero(n) % 2;
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&](size_t k, point rt) {
k *= 2 * flen;
vpoint cvrt = {vz + real(rt), vz - imag(rt)};
auto B = at(k) - at(k + flen);
at(k) += at(k + flen);
at(k + flen) = B * cvrt;
});
}
for(size_t leaf = 3 * flen; leaf < n; leaf += 4 * flen) {
size_t level = std::countr_one(leaf + 3);
for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
size_t i = (1 << lvl) / 4;
exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) {
k <<= lvl;
vpoint v1 = {vz + real(rt), vz - imag(rt)};
vpoint v2 = v1 * v1;
vpoint v3 = v1 * v2;
for(size_t j = k; j < k + i; j += flen) {
auto A = at(j);
auto B = at(j + i);
auto C = at(j + 2 * i);
auto D = at(j + 3 * i);
at(j) = ((A + B) + (C + D));
at(j + 2 * i) = ((A + B) - (C + D)) * v2;
at(j + i) = ((A - B) - vi(C - D)) * v1;
at(j + 3 * i) = ((A - B) + vi(C - D)) * v3;
}
});
}
}
checkpoint("ifft");
for(size_t k = 0; k < n; k += flen) {
set(k, get<vpoint>(k) /= vz + (ftype)(n / flen));
}
}
void fft() {
size_t n = size();
bool parity = std::countr_zero(n) % 2;
for(size_t leaf = 0; leaf < n; leaf += 4 * flen) {
size_t level = std::countr_zero(n + leaf);
level -= level % 2 != parity;
for(size_t lvl = level; lvl >= 4; lvl -= 2) {
size_t i = (1 << lvl) / 4;
exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) {
k <<= lvl;
vpoint v1 = {vz + real(rt), vz + imag(rt)};
vpoint v2 = v1 * v1;
vpoint v3 = v1 * v2;
for(size_t j = k; j < k + i; j += flen) {
auto A = at(j);
auto B = at(j + i) * v1;
auto C = at(j + 2 * i) * v2;
auto D = at(j + 3 * i) * v3;
at(j) = (A + C) + (B + D);
at(j + i) = (A + C) - (B + D);
at(j + 2 * i) = (A - C) + vi(B - D);
at(j + 3 * i) = (A - C) - vi(B - D);
}
});
}
}
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&](size_t k, point rt) {
k *= 2 * flen;
vpoint vrt = {vz + real(rt), vz + imag(rt)};
auto t = at(k + flen) * vrt;
at(k + flen) = at(k) - t;
at(k) += t;
});
}
checkpoint("fft");
}
static constexpr size_t pre_evals = 1 << 16;
static const std::array<size_t, pre_evals> eval_args;
static const std::array<point, pre_evals> evalp;
};
const std::array<size_t, cvector::pre_evals> cvector::eval_args = []() {
std::array<size_t, pre_evals> res = {};
for(size_t i = 1; i < pre_evals; i++) {
res[i] = res[i >> 1] | (i & 1) << (std::bit_width(i) - 1);
}
return res;
}();
const std::array<point, cvector::pre_evals> cvector::evalp = []() {
std::array<point, pre_evals> res = {};
res[0] = 1;
for(size_t n = 1; n < pre_evals; n++) {
res[n] = polar<ftype>(1., std::numbers::pi * ftype(eval_args[n]) / ftype(4 * std::bit_floor(n)));
}
return res;
}();
}
#line 9 "cp-algo/math/fft.hpp"
namespace cp_algo::math::fft {
template<modint_type base>
struct dft {
cvector A, B;
static base factor, ifactor;
using Int2 = base::Int2;
static bool _init;
static int split() {
static const int splt = int(std::sqrt(base::mod())) + 1;
return splt;
}
static u64x4 mod, imod;
void init() {
if(!_init) {
factor = 1 + random::rng() % (base::mod() - 1);
ifactor = base(1) / factor;
mod = u64x4() + base::mod();
imod = u64x4() + inv2(-base::mod());
_init = true;
}
}
dft(auto const& a, size_t n): A(n), B(n) {
init();
base b2x32 = bpow(base(2), 32);
u64x4 cur = {
(bpow(factor, 1) * b2x32).getr(),
(bpow(factor, 2) * b2x32).getr(),
(bpow(factor, 3) * b2x32).getr(),
(bpow(factor, 4) * b2x32).getr()
};
u64x4 step4 = u64x4{} + (bpow(factor, 4) * b2x32).getr();
u64x4 stepn = u64x4{} + (bpow(factor, n) * b2x32).getr();
for(size_t i = 0; i < std::min(n, size(a)); i += flen) {
auto splt = [&](size_t i, auto mul) {
if(i >= size(a)) {
return std::pair{vftype(), vftype()};
}
u64x4 au = {
i < size(a) ? a[i].getr() : 0,
i + 1 < size(a) ? a[i + 1].getr() : 0,
i + 2 < size(a) ? a[i + 2].getr() : 0,
i + 3 < size(a) ? a[i + 3].getr() : 0
};
au = montgomery_mul(au, mul, mod, imod);
au = au >= base::mod() ? au - base::mod() : au;
auto ai = to_double(i64x4(au >= base::mod() / 2 ? au - base::mod() : au));
auto quo = round(ai / split());
return std::pair{ai - quo * split(), quo};
};
auto [rai, qai] = splt(i, cur);
auto [rani, qani] = splt(n + i, montgomery_mul(cur, stepn, mod, imod));
A.at(i) = vpoint(rai, rani);
B.at(i) = vpoint(qai, qani);
cur = montgomery_mul(cur, step4, mod, imod);
}
checkpoint("dft init");
if(n) {
A.fft();
B.fft();
}
}
void dot(auto &&C, auto const& D) {
cvector::exec_on_evals<1>(A.size() / flen, [&](size_t k, point rt) {
k *= flen;
auto [Ax, Ay] = A.at(k);
auto [Bx, By] = B.at(k);
vpoint AC, AD, BC, BD;
AC = AD = BC = BD = vz;
auto Cv = C.at(k), Dv = D.at(k);
for (size_t i = 0; i < flen; i++) {
vpoint Av = {vz + Ax[i], vz + Ay[i]}, Bv = {vz + Bx[i], vz + By[i]};
AC += Av * Cv; AD += Av * Dv;
BC += Bv * Cv; BD += Bv * Dv;
real(Cv) = rotate_right(real(Cv));
imag(Cv) = rotate_right(imag(Cv));
real(Dv) = rotate_right(real(Dv));
imag(Dv) = rotate_right(imag(Dv));
auto cx = real(Cv)[0], cy = imag(Cv)[0];
auto dx = real(Dv)[0], dy = imag(Dv)[0];
real(Cv)[0] = cx * real(rt) - cy * imag(rt);
imag(Cv)[0] = cx * imag(rt) + cy * real(rt);
real(Dv)[0] = dx * real(rt) - dy * imag(rt);
imag(Dv)[0] = dx * imag(rt) + dy * real(rt);
}
A.at(k) = AC;
C.at(k) = AD + BC;
B.at(k) = BD;
});
checkpoint("dot");
}
void recover_mod(auto &&C, auto &res, size_t k) {
res.assign((k / flen + 1) * flen, base(0));
size_t n = A.size();
auto const splitsplit = base(split() * split()).getr();
base b2x32 = bpow(base(2), 32);
base b2x64 = bpow(base(2), 64);
u64x4 cur = {
(bpow(ifactor, 2) * b2x64).getr(),
(bpow(ifactor, 3) * b2x64).getr(),
(bpow(ifactor, 4) * b2x64).getr(),
(bpow(ifactor, 5) * b2x64).getr()
};
u64x4 step4 = u64x4{} + (bpow(ifactor, 4) * b2x32).getr();
u64x4 stepn = u64x4{} + (bpow(ifactor, n) * b2x32).getr();
for(size_t i = 0; i < std::min(n, k); i += flen) {
auto [Ax, Ay] = A.at(i);
auto [Bx, By] = B.at(i);
auto [Cx, Cy] = C.at(i);
auto set_i = [&](size_t i, auto A, auto B, auto C, auto mul) {
auto A0 = lround(A), A1 = lround(C), A2 = lround(B);
auto Ai = A0 + A1 * split() + A2 * splitsplit + uint64_t(base::modmod());
auto Au = montgomery_reduce(u64x4(Ai), mod, imod);
Au = montgomery_mul(Au, mul, mod, imod);
Au = Au >= base::mod() ? Au - base::mod() : Au;
for(size_t j = 0; j < flen; j++) {
res[i + j].setr(typename base::UInt(Au[j]));
}
};
set_i(i, Ax, Bx, Cx, cur);
if(i + n < k) {
set_i(i + n, Ay, By, Cy, montgomery_mul(cur, stepn, mod, imod));
}
cur = montgomery_mul(cur, step4, mod, imod);
}
res.resize(k);
checkpoint("recover mod");
}
void mul(auto &&C, auto const& D, auto &res, size_t k) {
assert(A.size() == C.size());
size_t n = A.size();
if(!n) {
res = {};
return;
}
dot(C, D);
A.ifft();
B.ifft();
C.ifft();
recover_mod(C, res, k);
}
void mul_inplace(auto &&B, auto& res, size_t k) {
mul(B.A, B.B, res, k);
}
void mul(auto const& B, auto& res, size_t k) {
mul(cvector(B.A), B.B, res, k);
}
std::vector<base, big_alloc<base>> operator *= (dft &B) {
std::vector<base, big_alloc<base>> res;
mul_inplace(B, res, 2 * A.size());
return res;
}
std::vector<base, big_alloc<base>> operator *= (dft const& B) {
std::vector<base, big_alloc<base>> res;
mul(B, res, 2 * A.size());
return res;
}
auto operator * (dft const& B) const {
return dft(*this) *= B;
}
point operator [](int i) const {return A.get(i);}
};
template<modint_type base> base dft<base>::factor = 1;
template<modint_type base> base dft<base>::ifactor = 1;
template<modint_type base> bool dft<base>::_init = false;
template<modint_type base> u64x4 dft<base>::mod = {};
template<modint_type base> u64x4 dft<base>::imod = {};
void mul_slow(auto &a, auto const& b, size_t k) {
if(empty(a) || empty(b)) {
a.clear();
} else {
size_t n = std::min(k, size(a));
size_t m = std::min(k, size(b));
a.resize(k);
for(int j = int(k - 1); j >= 0; j--) {
a[j] *= b[0];
for(int i = std::max(j - (int)n, 0) + 1; i < std::min(j + 1, (int)m); i++) {
a[j] += a[j - i] * b[i];
}
}
}
}
size_t com_size(size_t as, size_t bs) {
if(!as || !bs) {
return 0;
}
return std::max(flen, std::bit_ceil(as + bs - 1) / 2);
}
void mul_truncate(auto &a, auto const& b, size_t k) {
using base = std::decay_t<decltype(a[0])>;
if(std::min({k, size(a), size(b)}) < magic) {
mul_slow(a, b, k);
return;
}
auto n = std::max(flen, std::bit_ceil(
std::min(k, size(a)) + std::min(k, size(b)) - 1
) / 2);
auto A = dft<base>(a | std::views::take(k), n);
if(&a == &b) {
A.mul(A, a, k);
} else {
A.mul_inplace(dft<base>(b | std::views::take(k), n), a, k);
}
}
void mul(auto &a, auto const& b) {
size_t N = size(a) + size(b) - 1;
if(std::max(size(a), size(b)) > (1 << 23)) {
using T = std::decay_t<decltype(a[0])>;
// do karatsuba to save memory
auto n = (std::max(size(a), size(b)) + 1) / 2;
auto a0 = to<std::vector<T, big_alloc<T>>>(a | std::views::take(n));
auto a1 = to<std::vector<T, big_alloc<T>>>(a | std::views::drop(n));
auto b0 = to<std::vector<T, big_alloc<T>>>(b | std::views::take(n));
auto b1 = to<std::vector<T, big_alloc<T>>>(b | std::views::drop(n));
a0.resize(n); a1.resize(n);
b0.resize(n); b1.resize(n);
auto a01 = to<std::vector<T, big_alloc<T>>>(std::views::zip_transform(std::plus{}, a0, a1));
auto b01 = to<std::vector<T, big_alloc<T>>>(std::views::zip_transform(std::plus{}, b0, b1));
checkpoint("karatsuba split");
mul(a0, b0);
mul(a1, b1);
mul(a01, b01);
a.assign(4 * n, 0);
for(auto [i, ai]: a0 | std::views::enumerate) {
a[i] += ai;
a[i + n] -= ai;
}
for(auto [i, ai]: a1 | std::views::enumerate) {
a[i + n] -= ai;
a[i + 2 * n] += ai;
}
for(auto [i, ai]: a01 | std::views::enumerate) {
a[i + n] += ai;
}
a.resize(N);
checkpoint("karatsuba join");
} else if(size(a)) {
mul_truncate(a, b, N);
}
}
}
#line 6 "cp-algo/math/poly/impl/euclid.hpp"
#include <algorithm>
#include <numeric>
#line 9 "cp-algo/math/poly/impl/euclid.hpp"
#include <vector>
#line 11 "cp-algo/math/poly/impl/euclid.hpp"
#include <list>
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
template<typename poly>
using gcd_result = std::pair<
std::list<std::decay_t<poly>>,
linfrac<std::decay_t<poly>>>;
template<typename poly>
gcd_result<poly> half_gcd(poly &&A, poly &&B) {
assert(A.deg() >= B.deg());
size_t m = size(A.a) / 2;
if(B.deg() < (int)m) {
return {};
}
auto [ai, R] = A.divmod(B);
std::tie(A, B) = {B, R};
std::list a = {ai};
auto T = -linfrac(ai).adj();
auto advance = [&](size_t k) {
auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
a.splice(end(a), ak);
T.prepend(Tk);
return Tk;
};
advance(m).apply(A, B);
if constexpr (std::is_reference_v<poly>) {
advance(2 * m - A.deg()).apply(A, B);
} else {
advance(2 * m - A.deg());
}
return {std::move(a), std::move(T)};
}
template<typename poly>
gcd_result<poly> full_gcd(poly &&A, poly &&B) {
using poly_t = std::decay_t<poly>;
std::list<poly_t> ak;
std::vector<linfrac<poly_t>> trs;
while(!B.is_zero()) {
auto [a0, R] = A.divmod(B);
ak.push_back(a0);
trs.push_back(-linfrac(a0).adj());
std::tie(A, B) = {B, R};
auto [a, Tr] = half_gcd(A, B);
ak.splice(end(ak), a);
trs.push_back(Tr);
}
return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
}
// computes product of linfrac on [L, R)
auto convergent(auto L, auto R) {
using poly = decltype(L)::value_type;
if(R == next(L)) {
return linfrac(*L);
} else {
int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
auto M = L;
for(int c = M->deg(); 2 * c <= s; M++) {
c += next(M)->deg();
}
return convergent(L, M) * convergent(M, R);
}
}
template<typename poly>
poly min_rec(poly const& p, size_t d) {
auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
if(R2.is_zero()) {
return poly(1);
}
auto [a, Tr] = full_gcd(R1, R2);
a.emplace_back();
auto pref = begin(a);
for(int delta = (int)d - a.front().deg(); delta >= 0; pref++) {
delta -= pref->deg() + next(pref)->deg();
}
return convergent(begin(a), pref).a;
}
template<typename poly>
std::optional<poly> inv_mod(poly p, poly q) {
assert(!q.is_zero());
auto [a, Tr] = full_gcd(q, p);
if(q.deg() != 0) {
return std::nullopt;
}
return Tr.b / q[0];
}
}