C++ Library for Competitive Programming
/*
* @title 数学/形式的冪級数/多項式補間
*
* verification-helper: PROBLEM https://judge.yosupo.jp/problem/polynomial_interpolation
*/
#include <iostream>
#include <vector>
#include "emthrm/math/convolution/number_theoretic_transform.hpp"
#include "emthrm/math/formal_power_series/formal_power_series.hpp"
#include "emthrm/math/formal_power_series/polynomial_interpolation.hpp"
#include "emthrm/math/modint.hpp"
int main() {
constexpr int MOD = 998244353;
using ModInt = emthrm::MInt<MOD>;
emthrm::FormalPowerSeries<ModInt>::set_mult(
[](const std::vector<ModInt>& a, const std::vector<ModInt>& b)
-> std::vector<ModInt> {
static emthrm::NumberTheoreticTransform<MOD> ntt;
return ntt.convolution(a, b);
});
int n;
std::cin >> n;
std::vector<ModInt> x(n), y(n);
for (int i = 0; i < n; ++i) {
std::cin >> x[i];
}
for (int i = 0; i < n; ++i) {
std::cin >> y[i];
}
const emthrm::FormalPowerSeries<ModInt> c =
emthrm::polynomial_interpolation<emthrm::FormalPowerSeries>(x, y);
for (int i = 0; i < n; ++i) {
std::cout << c[i] << " \n"[i + 1 == n];
}
return 0;
}
#line 1 "test/math/formal_power_series/polynomial_interpolation.test.cpp"
/*
* @title 数学/形式的冪級数/多項式補間
*
* verification-helper: PROBLEM https://judge.yosupo.jp/problem/polynomial_interpolation
*/
#include <iostream>
#include <vector>
#line 1 "include/emthrm/math/convolution/number_theoretic_transform.hpp"
#if __has_include(<atcoder/convolution>)
# include <atcoder/convolution>
# include <atcoder/modint>
#else
# include <algorithm>
# include <bit>
# include <cassert>
# include <iterator>
# include <map>
# include <utility>
#endif
#line 16 "include/emthrm/math/convolution/number_theoretic_transform.hpp"
#line 1 "include/emthrm/math/modint.hpp"
#ifndef ARBITRARY_MODINT
# include <cassert>
#endif
#include <compare>
#line 9 "include/emthrm/math/modint.hpp"
// #include <numeric>
#include <utility>
#line 12 "include/emthrm/math/modint.hpp"
namespace emthrm {
#ifndef ARBITRARY_MODINT
template <unsigned int M>
struct MInt {
unsigned int v;
constexpr MInt() : v(0) {}
constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {}
static constexpr MInt raw(const int x) {
MInt x_;
x_.v = x;
return x_;
}
static constexpr int get_mod() { return M; }
static constexpr void set_mod(const int divisor) {
assert(std::cmp_equal(divisor, M));
}
static void init(const int x) {
inv<true>(x);
fact(x);
fact_inv(x);
}
template <bool MEMOIZES = false>
static MInt inv(const int n) {
// assert(0 <= n && n < M && std::gcd(n, M) == 1);
static std::vector<MInt> inverse{0, 1};
const int prev = inverse.size();
if (n < prev) return inverse[n];
if constexpr (MEMOIZES) {
// "n!" and "M" must be disjoint.
inverse.resize(n + 1);
for (int i = prev; i <= n; ++i) {
inverse[i] = -inverse[M % i] * raw(M / i);
}
return inverse[n];
}
int u = 1, v = 0;
for (unsigned int a = n, b = M; b;) {
const unsigned int q = a / b;
std::swap(a -= q * b, b);
std::swap(u -= q * v, v);
}
return u;
}
static MInt fact(const int n) {
static std::vector<MInt> factorial{1};
if (const int prev = factorial.size(); n >= prev) {
factorial.resize(n + 1);
for (int i = prev; i <= n; ++i) {
factorial[i] = factorial[i - 1] * i;
}
}
return factorial[n];
}
static MInt fact_inv(const int n) {
static std::vector<MInt> f_inv{1};
if (const int prev = f_inv.size(); n >= prev) {
f_inv.resize(n + 1);
f_inv[n] = inv(fact(n).v);
for (int i = n; i > prev; --i) {
f_inv[i - 1] = f_inv[i] * i;
}
}
return f_inv[n];
}
static MInt nCk(const int n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
fact_inv(n - k) * fact_inv(k));
}
static MInt nPk(const int n, const int k) {
return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
}
static MInt nHk(const int n, const int k) {
return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
}
static MInt large_nCk(long long n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
inv<true>(k);
MInt res = 1;
for (int i = 1; i <= k; ++i) {
res *= inv(i) * n--;
}
return res;
}
constexpr MInt pow(long long exponent) const {
MInt res = 1, tmp = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= tmp;
tmp *= tmp;
}
return res;
}
constexpr MInt& operator+=(const MInt& x) {
if ((v += x.v) >= M) v -= M;
return *this;
}
constexpr MInt& operator-=(const MInt& x) {
if ((v += M - x.v) >= M) v -= M;
return *this;
}
constexpr MInt& operator*=(const MInt& x) {
v = (unsigned long long){v} * x.v % M;
return *this;
}
MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }
constexpr auto operator<=>(const MInt& x) const = default;
constexpr MInt& operator++() {
if (++v == M) [[unlikely]] v = 0;
return *this;
}
constexpr MInt operator++(int) {
const MInt res = *this;
++*this;
return res;
}
constexpr MInt& operator--() {
v = (v == 0 ? M - 1 : v - 1);
return *this;
}
constexpr MInt operator--(int) {
const MInt res = *this;
--*this;
return res;
}
constexpr MInt operator+() const { return *this; }
constexpr MInt operator-() const { return raw(v ? M - v : 0); }
constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; }
constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
MInt operator/(const MInt& x) const { return MInt(*this) /= x; }
friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
return os << x.v;
}
friend std::istream& operator>>(std::istream& is, MInt& x) {
long long v;
is >> v;
x = MInt(v);
return is;
}
};
#else // ARBITRARY_MODINT
template <int ID>
struct MInt {
unsigned int v;
constexpr MInt() : v(0) {}
MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {}
static constexpr MInt raw(const int x) {
MInt x_;
x_.v = x;
return x_;
}
static int get_mod() { return mod(); }
static void set_mod(const unsigned int divisor) { mod() = divisor; }
static void init(const int x) {
inv<true>(x);
fact(x);
fact_inv(x);
}
template <bool MEMOIZES = false>
static MInt inv(const int n) {
// assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1);
static std::vector<MInt> inverse{0, 1};
const int prev = inverse.size();
if (n < prev) return inverse[n];
if constexpr (MEMOIZES) {
// "n!" and "M" must be disjoint.
inverse.resize(n + 1);
for (int i = prev; i <= n; ++i) {
inverse[i] = -inverse[mod() % i] * raw(mod() / i);
}
return inverse[n];
}
int u = 1, v = 0;
for (unsigned int a = n, b = mod(); b;) {
const unsigned int q = a / b;
std::swap(a -= q * b, b);
std::swap(u -= q * v, v);
}
return u;
}
static MInt fact(const int n) {
static std::vector<MInt> factorial{1};
if (const int prev = factorial.size(); n >= prev) {
factorial.resize(n + 1);
for (int i = prev; i <= n; ++i) {
factorial[i] = factorial[i - 1] * i;
}
}
return factorial[n];
}
static MInt fact_inv(const int n) {
static std::vector<MInt> f_inv{1};
if (const int prev = f_inv.size(); n >= prev) {
f_inv.resize(n + 1);
f_inv[n] = inv(fact(n).v);
for (int i = n; i > prev; --i) {
f_inv[i - 1] = f_inv[i] * i;
}
}
return f_inv[n];
}
static MInt nCk(const int n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
fact_inv(n - k) * fact_inv(k));
}
static MInt nPk(const int n, const int k) {
return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
}
static MInt nHk(const int n, const int k) {
return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
}
static MInt large_nCk(long long n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
inv<true>(k);
MInt res = 1;
for (int i = 1; i <= k; ++i) {
res *= inv(i) * n--;
}
return res;
}
MInt pow(long long exponent) const {
MInt res = 1, tmp = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= tmp;
tmp *= tmp;
}
return res;
}
MInt& operator+=(const MInt& x) {
if ((v += x.v) >= mod()) v -= mod();
return *this;
}
MInt& operator-=(const MInt& x) {
if ((v += mod() - x.v) >= mod()) v -= mod();
return *this;
}
MInt& operator*=(const MInt& x) {
v = (unsigned long long){v} * x.v % mod();
return *this;
}
MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }
auto operator<=>(const MInt& x) const = default;
MInt& operator++() {
if (++v == mod()) [[unlikely]] v = 0;
return *this;
}
MInt operator++(int) {
const MInt res = *this;
++*this;
return res;
}
MInt& operator--() {
v = (v == 0 ? mod() - 1 : v - 1);
return *this;
}
MInt operator--(int) {
const MInt res = *this;
--*this;
return res;
}
MInt operator+() const { return *this; }
MInt operator-() const { return raw(v ? mod() - v : 0); }
MInt operator+(const MInt& x) const { return MInt(*this) += x; }
MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
MInt operator/(const MInt& x) const { return MInt(*this) /= x; }
friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
return os << x.v;
}
friend std::istream& operator>>(std::istream& is, MInt& x) {
long long v;
is >> v;
x = MInt(v);
return is;
}
private:
static unsigned int& mod() {
static unsigned int divisor = 0;
return divisor;
}
};
#endif // ARBITRARY_MODINT
} // namespace emthrm
#line 18 "include/emthrm/math/convolution/number_theoretic_transform.hpp"
namespace emthrm {
#if __has_include(<atcoder/convolution>)
template <unsigned int T>
struct NumberTheoreticTransform {
using ModInt = MInt<T>;
NumberTheoreticTransform() = default;
template <typename U>
std::vector<ModInt> dft(const std::vector<U>& a);
void idft(std::vector<ModInt>* a);
template <typename U>
std::vector<ModInt> convolution(
const std::vector<U>& a, const std::vector<U>& b) const {
const int a_size = a.size(), b_size = b.size();
std::vector<atcoder::static_modint<T>> c(a_size), d(b_size);
for (int i = 0; i < a_size; ++i) {
c[i] = atcoder::static_modint<T>::raw(ModInt(a[i]).v);
}
for (int i = 0; i < b_size; ++i) {
d[i] = atcoder::static_modint<T>::raw(ModInt(b[i]).v);
}
c = atcoder::convolution(c, d);
const int c_size = c.size();
std::vector<ModInt> res(c_size);
for (int i = 0; i < c_size; ++i) {
res[i] = ModInt::raw(c[i].val());
}
return res;
}
};
#else // __has_include(<atcoder/convolution>)
template <unsigned int T>
struct NumberTheoreticTransform {
using ModInt = MInt<T>;
NumberTheoreticTransform()
: n_max(1 << init().first), root(ModInt::raw(init().second)) {}
template <typename U>
std::vector<ModInt> dft(const std::vector<U>& a) {
std::vector<ModInt> b(std::bit_ceil(a.size()), 0);
std::ranges::copy(a, b.begin());
calc(&b);
return b;
}
void idft(std::vector<ModInt>* a) {
assert(std::has_single_bit(a->size()));
calc(a);
std::reverse(std::next(a->begin()), a->end());
const int n = a->size();
const ModInt inv_n = ModInt::inv(n);
for (int i = 0; i < n; ++i) {
(*a)[i] *= inv_n;
}
}
template <typename U>
std::vector<ModInt> convolution(
const std::vector<U>& a, const std::vector<U>& b) {
const int a_size = a.size(), b_size = b.size();
const int c_size = a_size + b_size - 1;
if (std::min(a_size, b_size) <= 60) {
std::vector<ModInt> c(c_size, 0);
if (a_size > b_size) {
for (int i = 0; i < a_size; ++i) {
for (int j = 0; j < b_size; ++j) {
c[i + j] += ModInt(a[i]) * b[j];
}
}
} else {
for (int j = 0; j < b_size; ++j) {
for (int i = 0; i < a_size; ++i) {
c[i + j] += ModInt(b[j]) * a[i];
}
}
}
return c;
}
const int n = std::bit_ceil(static_cast<unsigned int>(c_size));
std::vector<ModInt> c(n, 0), d(n, 0);
std::ranges::copy(a, c.begin());
calc(&c);
std::ranges::copy(b, d.begin());
calc(&d);
for (int i = 0; i < n; ++i) {
c[i] *= d[i];
}
idft(&c);
c.resize(c_size);
return c;
}
private:
static std::pair<int, int> init() {
static const std::map<int, std::pair<int, int>> primes{
{16957441, {14, 102066830}}, // 329
{17006593, {15, 608991743}}, // 26
{19529729, {17, 927947839}}, // 770
{167772161, {25, 243}}, // 3
{469762049, {26, 2187}}, // 3
{645922817, {23, 680782677}}, // 3
{897581057, {23, 126991183}}, // 3
{924844033, {21, 480100938}}, // 5
{935329793, {22, 945616399}}, // 3
{943718401, {22, 39032610}}, // 7
{950009857, {21, 912960248}}, // 7
{962592769, {21, 762567211}}, // 7
{975175681, {21, 973754139}}, // 17
{976224257, {20, 168477898}}, // 3
{985661441, {22, 157780640}}, // 3
{998244353, {23, 15311432}}, // 3
{1004535809, {21, 840453100}}, // 3
{1007681537, {20, 283888334}}, // 3
{1012924417, {21, 428116421}}, // 5
{1045430273, {20, 328125745}}, // 3
{1051721729, {20, 234350985}}, // 6
{1053818881, {20, 309635616}}, // 7
{1224736769, {24, 304180829}}}; // 3
return primes.at(T);
}
const int n_max;
const ModInt root;
std::vector<int> butterfly{0};
std::vector<std::vector<ModInt>> omega{{1}};
void calc(std::vector<ModInt>* a) {
const int n = a->size(), prev_n = butterfly.size();
if (n > prev_n) {
assert(n <= n_max);
butterfly.resize(n);
const int prev_lg = omega.size(), lg = std::countr_zero(a->size());
for (int i = 1; i < prev_n; ++i) {
butterfly[i] <<= lg - prev_lg;
}
for (int i = prev_n; i < n; ++i) {
butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1));
}
omega.resize(lg);
for (int i = prev_lg; i < lg; ++i) {
omega[i].resize(1 << i);
const ModInt tmp = root.pow((ModInt::get_mod() - 1) >> (i + 1));
for (int j = 0; j < (1 << (i - 1)); ++j) {
omega[i][j << 1] = omega[i - 1][j];
omega[i][(j << 1) + 1] = omega[i - 1][j] * tmp;
}
}
}
const int shift =
std::countr_zero(butterfly.size()) - std::countr_zero(a->size());
for (int i = 0; i < n; ++i) {
const int j = butterfly[i] >> shift;
if (i < j) std::swap((*a)[i], (*a)[j]);
}
for (int block = 1, den = 0; block < n; block <<= 1, ++den) {
for (int i = 0; i < n; i += (block << 1)) {
for (int j = 0; j < block; ++j) {
const ModInt tmp = (*a)[i + j + block] * omega[den][j];
(*a)[i + j + block] = (*a)[i + j] - tmp;
(*a)[i + j] += tmp;
}
}
}
}
};
#endif // __has_include(<atcoder/convolution>)
} // namespace emthrm
#line 1 "include/emthrm/math/formal_power_series/formal_power_series.hpp"
#include <algorithm>
#include <cassert>
#include <functional>
#include <initializer_list>
#include <iterator>
#include <numeric>
#line 11 "include/emthrm/math/formal_power_series/formal_power_series.hpp"
namespace emthrm {
template <typename T>
struct FormalPowerSeries {
std::vector<T> coef;
explicit FormalPowerSeries(const int deg = 0) : coef(deg + 1, 0) {}
explicit FormalPowerSeries(const std::vector<T>& coef) : coef(coef) {}
FormalPowerSeries(const std::initializer_list<T> init)
: coef(init.begin(), init.end()) {}
template <typename InputIter>
explicit FormalPowerSeries(const InputIter first, const InputIter last)
: coef(first, last) {}
inline const T& operator[](const int term) const { return coef[term]; }
inline T& operator[](const int term) { return coef[term]; }
using Mult = std::function<std::vector<T>(const std::vector<T>&,
const std::vector<T>&)>;
using Sqrt = std::function<bool(const T&, T*)>;
static void set_mult(const Mult mult) { get_mult() = mult; }
static void set_sqrt(const Sqrt sqrt) { get_sqrt() = sqrt; }
void resize(const int deg) { coef.resize(deg + 1, 0); }
void shrink() {
while (coef.size() > 1 && coef.back() == 0) coef.pop_back();
}
int degree() const { return std::ssize(coef) - 1; }
FormalPowerSeries& operator=(const std::vector<T>& coef_) {
coef = coef_;
return *this;
}
FormalPowerSeries& operator=(const FormalPowerSeries& x) = default;
FormalPowerSeries& operator+=(const FormalPowerSeries& x) {
const int deg_x = x.degree();
if (deg_x > degree()) resize(deg_x);
for (int i = 0; i <= deg_x; ++i) {
coef[i] += x[i];
}
return *this;
}
FormalPowerSeries& operator-=(const FormalPowerSeries& x) {
const int deg_x = x.degree();
if (deg_x > degree()) resize(deg_x);
for (int i = 0; i <= deg_x; ++i) {
coef[i] -= x[i];
}
return *this;
}
FormalPowerSeries& operator*=(const T x) {
for (T& e : coef) e *= x;
return *this;
}
FormalPowerSeries& operator*=(const FormalPowerSeries& x) {
return *this = get_mult()(coef, x.coef);
}
FormalPowerSeries& operator/=(const T x) {
assert(x != 0);
return *this *= static_cast<T>(1) / x;
}
FormalPowerSeries& operator/=(const FormalPowerSeries& x) {
const int n = degree() - x.degree() + 1;
if (n <= 0) return *this = FormalPowerSeries();
const std::vector<T> tmp = get_mult()(
std::vector<T>(coef.rbegin(), std::next(coef.rbegin(), n)),
FormalPowerSeries(
x.coef.rbegin(),
std::next(x.coef.rbegin(), std::min(x.degree() + 1, n)))
.inv(n - 1).coef);
return *this = FormalPowerSeries(std::prev(tmp.rend(), n), tmp.rend());
}
FormalPowerSeries& operator%=(const FormalPowerSeries& x) {
if (x.degree() == 0) return *this = FormalPowerSeries{0};
*this -= *this / x * x;
resize(x.degree() - 1);
return *this;
}
FormalPowerSeries& operator<<=(const int n) {
coef.insert(coef.begin(), n, 0);
return *this;
}
FormalPowerSeries& operator>>=(const int n) {
if (degree() < n) return *this = FormalPowerSeries();
coef.erase(coef.begin(), coef.begin() + n);
return *this;
}
bool operator==(FormalPowerSeries x) const {
x.shrink();
FormalPowerSeries y = *this;
y.shrink();
return x.coef == y.coef;
}
FormalPowerSeries operator+() const { return *this; }
FormalPowerSeries operator-() const {
FormalPowerSeries res = *this;
for (T& e : res.coef) e = -e;
return res;
}
FormalPowerSeries operator+(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) += x;
}
FormalPowerSeries operator-(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) -= x;
}
FormalPowerSeries operator*(const T x) const {
return FormalPowerSeries(*this) *= x;
}
FormalPowerSeries operator*(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) *= x;
}
FormalPowerSeries operator/(const T x) const {
return FormalPowerSeries(*this) /= x;
}
FormalPowerSeries operator/(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) /= x;
}
FormalPowerSeries operator%(const FormalPowerSeries& x) const {
return FormalPowerSeries(*this) %= x;
}
FormalPowerSeries operator<<(const int n) const {
return FormalPowerSeries(*this) <<= n;
}
FormalPowerSeries operator>>(const int n) const {
return FormalPowerSeries(*this) >>= n;
}
T horner(const T x) const {
return std::accumulate(
coef.rbegin(), coef.rend(), static_cast<T>(0),
[x](const T l, const T r) -> T { return l * x + r; });
}
FormalPowerSeries differential() const {
const int deg = degree();
assert(deg >= 0);
FormalPowerSeries res(std::max(deg - 1, 0));
for (int i = 1; i <= deg; ++i) {
res[i - 1] = coef[i] * i;
}
return res;
}
FormalPowerSeries exp(const int deg) const {
assert(coef[0] == 0);
const int n = coef.size();
const FormalPowerSeries one{1};
FormalPowerSeries res = one;
for (int i = 1; i <= deg; i <<= 1) {
res *= FormalPowerSeries(coef.begin(),
std::next(coef.begin(), std::min(n, i << 1)))
- res.log((i << 1) - 1) + one;
res.coef.resize(i << 1);
}
res.resize(deg);
return res;
}
FormalPowerSeries exp() const { return exp(degree()); }
FormalPowerSeries inv(const int deg) const {
assert(coef[0] != 0);
const int n = coef.size();
FormalPowerSeries res{static_cast<T>(1) / coef[0]};
for (int i = 1; i <= deg; i <<= 1) {
res = res + res - res * res * FormalPowerSeries(
coef.begin(), std::next(coef.begin(), std::min(n, i << 1)));
res.coef.resize(i << 1);
}
res.resize(deg);
return res;
}
FormalPowerSeries inv() const { return inv(degree()); }
FormalPowerSeries log(const int deg) const {
assert(coef[0] == 1);
FormalPowerSeries integrand = differential() * inv(deg - 1);
integrand.resize(deg);
for (int i = deg; i > 0; --i) {
integrand[i] = integrand[i - 1] / i;
}
integrand[0] = 0;
return integrand;
}
FormalPowerSeries log() const { return log(degree()); }
FormalPowerSeries pow(long long exponent, const int deg) const {
const int n = coef.size();
if (exponent == 0) {
FormalPowerSeries res(deg);
if (deg != -1) [[unlikely]] res[0] = 1;
return res;
}
assert(deg >= 0);
for (int i = 0; i < n; ++i) {
if (coef[i] == 0) continue;
if (i > deg / exponent) break;
const long long shift = exponent * i;
T tmp = 1, base = coef[i];
for (long long e = exponent; e > 0; e >>= 1) {
if (e & 1) tmp *= base;
base *= base;
}
const FormalPowerSeries res = ((*this >> i) / coef[i]).log(deg - shift);
return ((res * exponent).exp(deg - shift) * tmp) << shift;
}
return FormalPowerSeries(deg);
}
FormalPowerSeries pow(const long long exponent) const {
return pow(exponent, degree());
}
FormalPowerSeries mod_pow(long long exponent,
const FormalPowerSeries& md) const {
const int deg = md.degree() - 1;
if (deg < 0) [[unlikely]] return FormalPowerSeries(-1);
const FormalPowerSeries inv_rev_md =
FormalPowerSeries(md.coef.rbegin(), md.coef.rend()).inv();
const auto mod_mult = [&md, &inv_rev_md, deg](
FormalPowerSeries* multiplicand, const FormalPowerSeries& multiplier)
-> void {
*multiplicand *= multiplier;
if (deg < multiplicand->degree()) {
const int n = multiplicand->degree() - deg;
const FormalPowerSeries quotient =
FormalPowerSeries(multiplicand->coef.rbegin(),
std::next(multiplicand->coef.rbegin(), n))
* FormalPowerSeries(
inv_rev_md.coef.begin(),
std::next(inv_rev_md.coef.begin(), std::min(deg + 2, n)));
*multiplicand -=
FormalPowerSeries(std::prev(quotient.coef.rend(), n),
quotient.coef.rend()) * md;
multiplicand->resize(deg);
}
multiplicand->shrink();
};
FormalPowerSeries res{1}, base = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) mod_mult(&res, base);
mod_mult(&base, base);
}
return res;
}
FormalPowerSeries sqrt(const int deg) const {
const int n = coef.size();
if (coef[0] == 0) {
for (int i = 1; i < n; ++i) {
if (coef[i] == 0) continue;
if (i & 1) return FormalPowerSeries(-1);
const int shift = i >> 1;
if (deg < shift) break;
FormalPowerSeries res = (*this >> i).sqrt(deg - shift);
if (res.coef.empty()) return FormalPowerSeries(-1);
res <<= shift;
res.resize(deg);
return res;
}
return FormalPowerSeries(deg);
}
T s;
if (!get_sqrt()(coef.front(), &s)) return FormalPowerSeries(-1);
FormalPowerSeries res{s};
const T half = static_cast<T>(1) / 2;
for (int i = 1; i <= deg; i <<= 1) {
res = (FormalPowerSeries(coef.begin(),
std::next(coef.begin(), std::min(n, i << 1)))
* res.inv((i << 1) - 1) + res) * half;
}
res.resize(deg);
return res;
}
FormalPowerSeries sqrt() const { return sqrt(degree()); }
FormalPowerSeries translate(const T c) const {
const int n = coef.size();
std::vector<T> fact(n, 1), inv_fact(n, 1);
for (int i = 1; i < n; ++i) {
fact[i] = fact[i - 1] * i;
}
inv_fact[n - 1] = static_cast<T>(1) / fact[n - 1];
for (int i = n - 1; i > 0; --i) {
inv_fact[i - 1] = inv_fact[i] * i;
}
std::vector<T> g(n), ex(n);
for (int i = 0; i < n; ++i) {
g[i] = coef[i] * fact[i];
}
std::reverse(g.begin(), g.end());
T pow_c = 1;
for (int i = 0; i < n; ++i) {
ex[i] = pow_c * inv_fact[i];
pow_c *= c;
}
const std::vector<T> conv = get_mult()(g, ex);
FormalPowerSeries res(n - 1);
for (int i = 0; i < n; ++i) {
res[i] = conv[n - 1 - i] * inv_fact[i];
}
return res;
}
private:
static Mult& get_mult() {
static Mult mult = [](const std::vector<T>& a, const std::vector<T>& b)
-> std::vector<T> {
const int n = a.size(), m = b.size();
std::vector<T> res(n + m - 1, 0);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < m; ++j) {
res[i + j] += a[i] * b[j];
}
}
return res;
};
return mult;
}
static Sqrt& get_sqrt() {
static Sqrt sqrt = [](const T&, T*) -> bool { return false; };
return sqrt;
}
};
} // namespace emthrm
#line 1 "include/emthrm/math/formal_power_series/polynomial_interpolation.hpp"
#line 5 "include/emthrm/math/formal_power_series/polynomial_interpolation.hpp"
#line 1 "include/emthrm/math/formal_power_series/multipoint_evaluation.hpp"
#line 7 "include/emthrm/math/formal_power_series/multipoint_evaluation.hpp"
namespace emthrm {
template <template <typename> class C, typename T>
struct MultipointEvaluation {
std::vector<T> f_x;
std::vector<C<T>> subproduct_tree;
explicit MultipointEvaluation(const std::vector<T> &xs)
: f_x(xs.size()), subproduct_tree(xs.size() << 1), n(xs.size()) {
std::transform(xs.begin(), xs.end(), std::next(subproduct_tree.begin(), n),
[](const T& x) -> C<T> { return C<T>{-x, 1}; });
for (int i = n - 1; i > 0; --i) {
subproduct_tree[i] =
subproduct_tree[i << 1] * subproduct_tree[(i << 1) + 1];
}
}
void build(const C<T>& f) { dfs(f, 1); }
private:
const int n;
void dfs(C<T> f, int node) {
f %= subproduct_tree[node];
if (node < n) {
dfs(f, node << 1);
dfs(f, (node << 1) + 1);
} else {
f_x[node - n] = f[0];
}
}
};
} // namespace emthrm
#line 7 "include/emthrm/math/formal_power_series/polynomial_interpolation.hpp"
namespace emthrm {
template <template <typename> class C, typename T>
C<T> polynomial_interpolation(const std::vector<T>& x,
const std::vector<T>& y) {
MultipointEvaluation<C, T> m(x);
m.build(m.subproduct_tree[1].differential());
const int n = x.size();
const auto f = [&y, &m, n](auto f, const int node) -> C<T> {
return node >= n ? C<T>{y[node - n] / m.f_x[node - n]} :
f(f, node << 1) * m.subproduct_tree[(node << 1) + 1]
+ f(f, (node << 1) + 1) * m.subproduct_tree[node << 1];
};
return f(f, 1);
}
} // namespace emthrm
#line 14 "test/math/formal_power_series/polynomial_interpolation.test.cpp"
int main() {
constexpr int MOD = 998244353;
using ModInt = emthrm::MInt<MOD>;
emthrm::FormalPowerSeries<ModInt>::set_mult(
[](const std::vector<ModInt>& a, const std::vector<ModInt>& b)
-> std::vector<ModInt> {
static emthrm::NumberTheoreticTransform<MOD> ntt;
return ntt.convolution(a, b);
});
int n;
std::cin >> n;
std::vector<ModInt> x(n), y(n);
for (int i = 0; i < n; ++i) {
std::cin >> x[i];
}
for (int i = 0; i < n; ++i) {
std::cin >> y[i];
}
const emthrm::FormalPowerSeries<ModInt> c =
emthrm::polynomial_interpolation<emthrm::FormalPowerSeries>(x, y);
for (int i = 0; i < n; ++i) {
std::cout << c[i] << " \n"[i + 1 == n];
}
return 0;
}