Felix's Library
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test/formal-power-series/poly/yosupo-Sqrt-of-Formal-Power-Series.test.cpp
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- Last update: 2023-07-31 16:36:45+08:00
- Problem: https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
Depends on
- library/convolution/ntt.hpp
- library/formal-power-series/poly.hpp
library/math/inv-gcd.hpp
- library/math/pow-mod.hpp
- library/math/primitive-root.hpp
- library/math/safe-mod.hpp
- library/misc/type-traits.hpp
- library/modint/modint.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series"
#include <iostream>
#include "../../../library/formal-power-series/poly.hpp"
using namespace std;
using namespace felix;
int main() {
ios::sync_with_stdio(false);
cin.tie(0);
int n;
cin >> n;
Poly<998244353> a(n);
for(int i = 0; i < n; i++) {
cin >> a[i];
}
if(!a.has_sqrt()) {
cout << "-1\n";
} else {
a = a.sqrt();
for(int i = 0; i < n; i++) {
cout << a[i] << " \n"[i == n - 1];
}
}
return 0;
}#line 1 "test/formal-power-series/poly/yosupo-Sqrt-of-Formal-Power-Series.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series"
#include <iostream>
#line 2 "library/formal-power-series/poly.hpp"
#include <vector>
#include <initializer_list>
#include <algorithm>
#include <functional>
#include <cassert>
#line 6 "library/modint/modint.hpp"
#include <type_traits>
#line 3 "library/misc/type-traits.hpp"
#include <numeric>
#line 5 "library/misc/type-traits.hpp"
namespace felix {
namespace internal {
#ifndef _MSC_VER
template<class T> using is_signed_int128 = typename std::conditional<std::is_same<T, __int128_t>::value || std::is_same<T, __int128>::value, std::true_type, std::false_type>::type;
template<class T> using is_unsigned_int128 = typename std::conditional<std::is_same<T, __uint128_t>::value || std::is_same<T, unsigned __int128>::value, std::true_type, std::false_type>::type;
template<class T> using make_unsigned_int128 = typename std::conditional<std::is_same<T, __int128_t>::value, __uint128_t, unsigned __int128>;
template<class T> using is_integral = typename std::conditional<std::is_integral<T>::value || is_signed_int128<T>::value || is_unsigned_int128<T>::value, std::true_type, std::false_type>::type;
template<class T> using is_signed_int = typename std::conditional<(is_integral<T>::value && std::is_signed<T>::value) || is_signed_int128<T>::value, std::true_type, std::false_type>::type;
template<class T> using is_unsigned_int = typename std::conditional<(is_integral<T>::value && std::is_unsigned<T>::value) || is_unsigned_int128<T>::value, std::true_type, std::false_type>::type;
template<class T> using to_unsigned = typename std::conditional<is_signed_int128<T>::value, make_unsigned_int128<T>, typename std::conditional<std::is_signed<T>::value, std::make_unsigned<T>, std::common_type<T>>::type>::type;
#else
template<class T> using is_integral = typename std::is_integral<T>;
template<class T> using is_signed_int = typename std::conditional<is_integral<T>::value && std::is_signed<T>::value, std::true_type, std::false_type>::type;
template<class T> using is_unsigned_int = typename std::conditional<is_integral<T>::value && std::is_unsigned<T>::value, std::true_type, std::false_type>::type;
template<class T> using to_unsigned = typename std::conditional<is_signed_int<T>::value, std::make_unsigned<T>, std::common_type<T>>::type;
#endif
template<class T> using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template<class T> using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template<class T> using to_unsigned_t = typename to_unsigned<T>::type;
template<class T> struct safely_multipliable {};
template<> struct safely_multipliable<short> { using type = int; };
template<> struct safely_multipliable<unsigned short> { using type = unsigned int; };
template<> struct safely_multipliable<int> { using type = long long; };
template<> struct safely_multipliable<unsigned int> { using type = unsigned long long; };
template<> struct safely_multipliable<long long> { using type = __int128; };
template<> struct safely_multipliable<unsigned long long> { using type = __uint128_t; };
template<class T> using safely_multipliable_t = typename safely_multipliable<T>::type;
} // namespace internal
} // namespace felix
#line 2 "library/math/safe-mod.hpp"
namespace felix {
namespace internal {
template<class T>
constexpr T safe_mod(T x, T m) {
x %= m;
if(x < 0) {
x += m;
}
return x;
}
} // namespace internal
} // namespace felix
#line 3 "library/math/inv-gcd.hpp"
namespace felix {
namespace internal {
template<class T>
constexpr std::pair<T, T> inv_gcd(T a, T b) {
a = safe_mod(a, b);
if(a == 0) {
return {b, 0};
}
T s = b, t = a;
T m0 = 0, m1 = 1;
while(t) {
T u = s / t;
s -= t * u;
m0 -= m1 * u;
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
if(m0 < 0) {
m0 += b / s;
}
return {s, m0};
}
} // namespace internal
} // namespace felix
#line 9 "library/modint/modint.hpp"
namespace felix {
template<int id>
struct modint {
public:
static constexpr int mod() { return (id > 0 ? id : md); }
static constexpr void set_mod(int m) {
if(id > 0 || md == m) {
return;
}
md = m;
fact.resize(1);
inv_fact.resize(1);
invs.resize(1);
}
static constexpr void prepare(int n) {
int sz = (int) fact.size();
if(sz == mod()) {
return;
}
n = 1 << std::__lg(2 * n - 1);
if(n < sz) {
return;
}
if(n < (sz - 1) * 2) {
n = std::min((sz - 1) * 2, mod() - 1);
}
fact.resize(n + 1);
inv_fact.resize(n + 1);
invs.resize(n + 1);
for(int i = sz; i <= n; i++) {
fact[i] = fact[i - 1] * i;
}
auto eg = internal::inv_gcd(fact.back().val(), mod());
assert(eg.first == 1);
inv_fact[n] = eg.second;
for(int i = n - 1; i >= sz; i--) {
inv_fact[i] = inv_fact[i + 1] * (i + 1);
}
for(int i = n; i >= sz; i--) {
invs[i] = inv_fact[i] * fact[i - 1];
}
}
constexpr modint() : v(0) {}
template<class T, internal::is_signed_int_t<T>* = nullptr> constexpr modint(T x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {}
template<class T, internal::is_unsigned_int_t<T>* = nullptr> constexpr modint(T x) : v(x % mod()) {}
constexpr int val() const { return v; }
constexpr modint inv() const {
if(id > 0 && v < std::min(mod() >> 1, 1 << 18)) {
prepare(v);
return invs[v];
} else {
auto eg = internal::inv_gcd(v, mod());
assert(eg.first == 1);
return eg.second;
}
}
constexpr modint& operator+=(const modint& rhs) & {
v += rhs.v;
if(v >= mod()) {
v -= mod();
}
return *this;
}
constexpr modint& operator-=(const modint& rhs) & {
v -= rhs.v;
if(v < 0) {
v += mod();
}
return *this;
}
constexpr modint& operator*=(const modint& rhs) & {
v = 1LL * v * rhs.v % mod();
return *this;
}
constexpr modint& operator/=(const modint& rhs) & {
return *this *= rhs.inv();
}
friend constexpr modint operator+(modint lhs, modint rhs) { return lhs += rhs; }
friend constexpr modint operator-(modint lhs, modint rhs) { return lhs -= rhs; }
friend constexpr modint operator*(modint lhs, modint rhs) { return lhs *= rhs; }
friend constexpr modint operator/(modint lhs, modint rhs) { return lhs /= rhs; }
constexpr modint operator+() const { return *this; }
constexpr modint operator-() const { return modint() - *this; }
constexpr bool operator==(const modint& rhs) const { return v == rhs.v; }
constexpr bool operator!=(const modint& rhs) const { return v != rhs.v; }
constexpr modint pow(long long p) const {
modint a(*this), res(1);
if(p < 0) {
a = a.inv();
p = -p;
}
while(p) {
if(p & 1) {
res *= a;
}
a *= a;
p >>= 1;
}
return res;
}
constexpr bool has_sqrt() const {
if(mod() == 2 || v == 0) {
return true;
}
if(pow((mod() - 1) / 2).val() != 1) {
return false;
}
return true;
}
constexpr modint sqrt() const {
if(mod() == 2 || v < 2) {
return *this;
}
assert(pow((mod() - 1) / 2).val() == 1);
modint b = 1;
while(b.pow((mod() - 1) >> 1).val() == 1) {
b += 1;
}
int m = mod() - 1, e = __builtin_ctz(m);
m >>= e;
modint x = modint(*this).pow((m - 1) >> 1);
modint y = modint(*this) * x * x;
x *= v;
modint z = b.pow(m);
while(y.val() != 1) {
int j = 0;
modint t = y;
while(t.val() != 1) {
t *= t;
j++;
}
z = z.pow(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return x;
}
friend std::istream& operator>>(std::istream& in, modint& num) {
long long x;
in >> x;
num = modint<id>(x);
return in;
}
friend std::ostream& operator<<(std::ostream& out, const modint& num) {
return out << num.val();
}
public:
static std::vector<modint> fact, inv_fact, invs;
private:
int v;
static int md;
};
template<int id> int modint<id>::md = 998244353;
template<int id> std::vector<modint<id>> modint<id>::fact = {1};
template<int id> std::vector<modint<id>> modint<id>::inv_fact = {1};
template<int id> std::vector<modint<id>> modint<id>::invs = {0};
using modint998244353 = modint<998244353>;
using modint1000000007 = modint<1000000007>;
namespace internal {
template<class T> struct is_modint : public std::false_type {};
template<int id> struct is_modint<modint<id>> : public std::true_type {};
template<class T, class ENABLE = void> struct is_static_modint : public std::false_type {};
template<int id> struct is_static_modint<modint<id>, std::enable_if_t<(id > 0)>> : public std::true_type {};
template<class T> using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
template<class T, class ENABLE = void> struct is_dynamic_modint : public std::false_type {};
template<int id> struct is_dynamic_modint<modint<id>, std::enable_if_t<(id <= 0)>> : public std::true_type {};
template<class T> using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace felix
#line 3 "library/convolution/ntt.hpp"
#include <array>
#line 4 "library/math/pow-mod.hpp"
namespace felix {
namespace internal {
template<class T>
constexpr T pow_mod_constexpr(T x, long long n, T m) {
using U = safely_multipliable_t<T>;
if(m == 1) {
return 0;
}
U r = 1, y = safe_mod(x, m);
while(n) {
if(n & 1) {
r = (r * y) % m;
}
y = (y * y) % m;
n >>= 1;
}
return r;
}
} // namespace internal
} // namespace felix
#line 4 "library/math/primitive-root.hpp"
namespace felix {
namespace internal {
constexpr int primitive_root_constexpr(int m) {
if(m == 998244353) return 3;
if(m == 167772161) return 3;
if(m == 469762049) return 3;
if(m == 754974721) return 11;
if(m == 2) return 1;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
x >>= __builtin_ctz(x);
for(int i = 3; 1LL * i * i <= x; i += 2) {
if(x % i == 0) {
divs[cnt++] = i;
while(x % i == 0) {
x /= i;
}
}
}
if(x > 1) {
divs[cnt++] = x;
}
for(int g = 2;; g++) {
bool ok = true;
for(int i = 0; i < cnt; i++) {
if(pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if(ok) {
return g;
}
}
assert(false);
}
} // namespace internal
} // namespace felix
#line 11 "library/convolution/ntt.hpp"
namespace felix {
namespace internal {
template<int mod>
struct NTT_prepare {
using mint = modint<mod>;
static constexpr int primitive_root = primitive_root_constexpr(mod);
static constexpr int level = __builtin_ctz(mod - 1);
std::array<mint, level + 1> root, iroot;
std::array<mint, std::max(0, level - 2 + 1)> rate2, irate2;
std::array<mint, std::max(0, level - 3 + 1)> rate3, irate3;
constexpr NTT_prepare() {
root[level] = mint(primitive_root).pow((mod - 1) >> level);
iroot[level] = root[level].inv();
for(int i = level - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
mint prod = 1, iprod = 1;
for(int i = 0; i <= level - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
mint prod = 1, iprod = 1;
for(int i = 0; i <= level - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
};
template<int mod>
struct NTT {
using mint = modint<mod>;
static NTT_prepare<mod> info;
static void NTT4(std::vector<mint>& a) {
int n = (int) a.size();
int h = __builtin_ctz(n);
int len = 0;
while(len < h) {
if(h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
for(int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for(int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if(s + 1 != (1 << len)) {
rot *= info.rate2[__builtin_ctz(~(unsigned int) s)];
}
}
len++;
} else {
int p = 1 << (h - len - 2);
mint rot = 1, imag = info.root[2];
for(int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h - len);
for(int i = 0; i < p; i++) {
auto mod2 = 1ULL * mod * mod;
auto a0 = 1ULL * a[i + offset].val();
auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
}
if(s + 1 != (1 << len))
rot *= info.rate3[__builtin_ctz(~(unsigned int) s)];
}
len += 2;
}
}
}
static void iNTT4(std::vector<mint>& a) {
int n = (int) a.size();
int h = __builtin_ctz(n);
int len = h;
while(len) {
if(len == 1) {
int p = 1 << (h - len);
mint irot = 1;
for(int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for(int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] = 1ULL * (mod + l.val() - r.val()) * irot.val();
}
if(s + 1 != (1 << (len - 1))) {
irot *= info.irate2[__builtin_ctz(~(unsigned int) s)];
}
}
len--;
} else {
int p = 1 << (h - len);
mint irot = 1, iimag = info.iroot[2];
for(int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for(int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].val();
auto a1 = 1ULL * a[i + offset + 1 * p].val();
auto a2 = 1ULL * a[i + offset + 2 * p].val();
auto a3 = 1ULL * a[i + offset + 3 * p].val();
auto a2na3iimag = 1ULL * mint((mod + a2 - a3) * iimag.val()).val();
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] = (a0 + (mod - a1) + a2na3iimag) * irot.val();
a[i + offset + 2 * p] = (a0 + a1 + (mod - a2) + (mod - a3)) * irot2.val();
a[i + offset + 3 * p] = (a0 + (mod - a1) + (mod - a2na3iimag)) * irot3.val();
}
if(s + 1 != (1 << (len - 2))) {
irot *= info.irate3[__builtin_ctz(~(unsigned int) s)];
}
}
len -= 2;
}
}
}
};
template<int mod> NTT_prepare<mod> NTT<mod>::info;
template<class T>
std::vector<T> convolution_naive(const std::vector<T>& a, const std::vector<T>& b) {
int n = (int) a.size(), m = (int) b.size();
std::vector<T> ans(n + m - 1);
if(n >= m) {
for(int i = 0; i < n; i++) {
for(int j = 0; j < m; j++) {
ans[i + j] += a[i] * b[j];
}
}
} else {
for(int j = 0; j < m; j++) {
for(int i = 0; i < n; i++) {
ans[i + j] += a[i] * b[j];
}
}
}
return ans;
}
template<class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_ntt(std::vector<mint> a, std::vector<mint> b) {
int n = (int) a.size(), m = (int) b.size();
int sz = 1 << std::__lg(2 * (n + m - 1) - 1);
a.resize(sz);
b.resize(sz);
NTT<mint::mod()>::NTT4(a);
NTT<mint::mod()>::NTT4(b);
for(int i = 0; i < sz; i++) {
a[i] *= b[i];
}
NTT<mint::mod()>::iNTT4(a);
a.resize(n + m - 1);
mint iz = mint(sz).inv();
for(int i = 0; i < n + m - 1; i++) {
a[i] *= iz;
}
return a;
}
} // namespace internal
template<class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint>&& a, std::vector<mint>&& b) {
int n = (int) a.size(), m = (int) b.size();
if(n == 0 || m == 0) {
return {};
}
int sz = 1 << std::__lg(2 * (n + m - 1) - 1);
assert((mint::mod() - 1) % sz == 0);
if(std::min(n, m) < 128) {
return internal::convolution_naive(a, b);
}
return internal::convolution_ntt(a, b);
}
template<class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(const std::vector<mint>& a, const std::vector<mint>& b) {
int n = (int) a.size(), m = (int) b.size();
if(n == 0 || m == 0) {
return {};
}
int sz = 1 << std::__lg(2 * (n + m - 1) - 1);
assert((mint::mod() - 1) % sz == 0);
if(std::min(n, m) < 128) {
return internal::convolution_naive(a, b);
}
return internal::convolution_ntt(a, b);
}
template<int mod, class T, std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
using mint = modint<mod>;
int n = (int) a.size(), m = (int) b.size();
if(n == 0 || m == 0) {
return {};
}
int sz = 1 << std::__lg(2 * (n + m - 1) - 1);
assert((mod - 1) % sz == 0);
std::vector<mint> a2(a.begin(), a.end());
std::vector<mint> b2(b.begin(), b.end());
auto c2 = convolution(std::move(a2), std::move(b2));
std::vector<T> c(n + m - 1);
for(int i = 0; i < n + m - 1; i++) {
c[i] = c2[i].val();
}
return c;
}
template<class T>
std::vector<__uint128_t> convolution_u128(const std::vector<T>& a, const std::vector<T>& b) {
static constexpr int m0 = 167772161; // 2^25
static constexpr int m1 = 469762049; // 2^26
static constexpr int m2 = 754974721; // 2^24
static constexpr int r01 = internal::inv_gcd(m0, m1).second;
static constexpr int r02 = internal::inv_gcd(m0, m2).second;
static constexpr int r12 = internal::inv_gcd(m1, m2).second;
static constexpr int r02r12 = 1LL * r02 * r12 % m2;
static constexpr long long w1 = m0;
static constexpr long long w2 = 1LL * m0 * m1;
int n = (int) a.size(), m = (int) b.size();
if(n == 0 || m == 0) {
return {};
}
std::vector<__uint128_t> c(n + m - 1);
if(std::min(n, m) < 128) {
std::vector<__uint128_t> a2(a.begin(), a.end());
std::vector<__uint128_t> b2(b.begin(), b.end());
return internal::convolution_naive(a2, b2);
}
static constexpr int MAX_AB_BIT = 24;
static_assert(m0 % (1ULL << MAX_AB_BIT) == 1, "m0 isn't enough to support an array length of 2^24.");
static_assert(m1 % (1ULL << MAX_AB_BIT) == 1, "m1 isn't enough to support an array length of 2^24.");
static_assert(m2 % (1ULL << MAX_AB_BIT) == 1, "m2 isn't enough to support an array length of 2^24.");
assert(n + m - 1 <= (1 << MAX_AB_BIT));
auto c0 = convolution<m0>(a, b);
auto c1 = convolution<m1>(a, b);
auto c2 = convolution<m2>(a, b);
for(int i = 0; i < n + m - 1; i++) {
long long n1 = c1[i], n2 = c2[i];
long long x = c0[i];
long long y = (n1 + m1 - x) * r01 % m1;
long long z = ((n2 + m2 - x) * r02r12 + (m2 - y) * r12) % m2;
c[i] = x + y * w1 + __uint128_t(z) * w2;
}
return c;
}
template<class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_large(const std::vector<mint>& a, const std::vector<mint>& b) {
static constexpr int max_size = (mint::mod() - 1) & -(mint::mod() - 1);
static constexpr int half_size = max_size >> 1;
static constexpr int inv_max_size = internal::inv_gcd(max_size, mint::mod()).second;
const int n = (int) a.size(), m = (int) b.size();
if(n == 0 || m == 0) {
return {};
}
if(std::min(n, m) < 128 || n + m - 1 <= max_size) {
return internal::convolution_naive(a, b);
}
const int dn = (n + half_size - 1) / half_size;
const int dm = (m + half_size - 1) / half_size;
std::vector<std::vector<mint>> as(dn), bs(dm);
for(int i = 0; i < dn; ++i) {
const int offset = half_size * i;
as[i] = std::vector<mint>(a.begin() + offset, a.begin() + std::min(n, offset + half_size));
as[i].resize(max_size);
internal::NTT<mint::mod()>::NTT4(as[i]);
}
for(int j = 0; j < dm; ++j) {
const int offset = half_size * j;
bs[j] = std::vector<mint>(b.begin() + offset, b.begin() + std::min(m, offset + half_size));
bs[j].resize(max_size);
internal::NTT<mint::mod()>::NTT4(bs[j]);
}
std::vector<std::vector<mint>> cs(dn + dm - 1, std::vector<mint>(max_size));
for(int i = 0; i < dn; ++i) {
for(int j = 0; j < dm; ++j) {
for(int k = 0; k < max_size; ++k) {
cs[i + j][k] += as[i][k] * bs[j][k];
}
}
}
std::vector<mint> c(n + m - 1);
for(int i = 0; i < dn + dm - 1; ++i) {
internal::NTT<mint::mod()>::iNTT4(cs[i]);
const int offset = half_size * i;
const int jmax = std::min(n + m - 1 - offset, max_size);
for(int j = 0; j < jmax; ++j) {
c[offset + j] += cs[i][j] * inv_max_size;
}
}
return c;
}
} // namespace felix
#line 9 "library/formal-power-series/poly.hpp"
namespace felix {
template<int mod>
struct Poly {
using mint = modint<mod>;
public:
Poly() {}
explicit Poly(int n) : a(n) {}
explicit Poly(const std::vector<mint>& a) : a(a) {}
Poly(const std::initializer_list<mint>& a) : a(a) {}
template<class F>
explicit Poly(int n, F f) : a(n) {
for(int i = 0; i < n; i++) {
a[i] = f(i);
}
}
constexpr int size() const { return (int) a.size(); }
constexpr void resize(int n) { a.resize(n); }
constexpr void shrink() {
while(size() && a.back() == 0) {
a.pop_back();
}
}
constexpr mint at(int idx) const {
if(idx >= 0 && idx < size()) {
return a[idx];
} else {
return 0;
}
}
constexpr mint& operator[](int idx) { return a[idx]; }
constexpr friend Poly operator+(const Poly& a, const Poly& b) {
Poly c(std::max(a.size(), b.size()));
for(int i = 0; i < c.size(); i++) {
c[i] = a.at(i) + b.at(i);
}
return c;
}
constexpr friend Poly operator-(const Poly& a, const Poly& b) {
Poly c(std::max(a.size(), b.size()));
for(int i = 0; i < c.size(); i++) {
c[i] = a.at(i) - b.at(i);
}
return c;
}
constexpr friend Poly operator*(Poly a, Poly b) {
return Poly(convolution(a.a, b.a));
}
constexpr friend Poly operator*(mint a, Poly b) {
for(int i = 0; i < b.size(); i++) {
b[i] *= a;
}
return b;
}
constexpr friend Poly operator*(Poly a, mint b) {
for(int i = 0; i < a.size(); i++) {
a[i] *= b;
}
return a;
}
constexpr Poly& operator+=(Poly b) { return (*this) = (*this) + b; }
constexpr Poly& operator-=(Poly b) { return (*this) = (*this) - b; }
constexpr Poly& operator*=(Poly b) { return (*this) = (*this) * b; }
constexpr Poly& operator*=(mint b) { return (*this) = (*this) * b; }
constexpr Poly mulxk(int k) const {
auto b = a;
b.insert(b.begin(), k, mint(0));
return Poly(b);
}
constexpr Poly modxk(int k) const {
k = std::min(k, size());
return Poly(std::vector<mint>(a.begin(), a.begin() + k));
}
constexpr Poly divxk(int k) const {
if(size() <= k) {
return Poly();
}
return Poly(std::vector<mint>(a.begin() + k, a.end()));
}
constexpr Poly deriv() const {
if(a.empty()) {
return Poly();
}
Poly c(size() - 1);
for(int i = 0; i < size() - 1; ++i) {
c[i] = (i + 1) * a[i + 1];
}
return c;
}
constexpr Poly integr() const {
Poly c(size() + 1);
mint::prepare(size());
for(int i = 0; i < size(); ++i) {
c[i + 1] = a[i] / mint(i + 1);
}
return c;
}
constexpr Poly inv(int m = -1) const {
if(m == -1) {
m = size();
}
Poly x{a[0].inv()};
int k = 1;
while(k < m) {
k *= 2;
x = (x * (Poly{mint(2)} - modxk(k) * x)).modxk(k);
}
return x.modxk(m);
}
constexpr Poly log(int m = -1) const {
if(m == -1) {
m = size();
}
return (deriv() * inv(m)).integr().modxk(m);
}
constexpr Poly exp(int m = -1) const {
if(m == -1) {
m = size();
}
Poly x{mint(1)};
int k = 1;
while(k < m) {
k *= 2;
x = (x * (Poly{mint(1)} - x.log(k) + modxk(k))).modxk(k);
}
return x.modxk(m);
}
constexpr Poly pow(long long k, int m = -1) const {
if(m == -1) {
m = size();
}
if(k == 0) {
Poly b(m);
b[0] = 1;
return b;
}
int s = 0, sz = size();
while(s < sz && a[s].val() == 0) {
s++;
}
if(s == sz) {
return *this;
}
if(m > 0 && s >= (sz + k - 1) / k) {
return Poly(m);
}
if(s * k >= m) {
return Poly(m);
}
return (((divxk(s) * a[s].inv()).log(m) * mint(k)).exp(m) * a[s].pow(k)).mulxk(s * k).modxk(m);
}
constexpr bool has_sqrt() const {
if(size() == 0) {
return true;
}
int x = 0;
while(x < size() && a[x].val() == 0) {
x++;
}
if(x == size()) {
return true;
}
if(x % 2 == 1) {
return false;
}
mint y = a[x];
return (y == 0 || y.pow((mod - 1) / 2) == 1);
}
constexpr Poly sqrt(int m = -1) const {
if(m == -1) {
m = size();
}
if(size() == 0) {
return Poly();
}
int x = 0;
while(x < size() && a[x].val() == 0) {
x++;
}
if(x == size()) {
return Poly(size());
}
Poly f = divxk(x);
Poly g({mint(f[0]).sqrt()});
mint inv2 = mint(1) / 2;
for(int i = 1; i < m; i *= 2) {
g = (g + f.modxk(i * 2) * g.inv(i * 2)) * inv2;
}
return g.modxk(m).mulxk(x / 2);
}
constexpr Poly shift(mint c) const {
int n = size();
mint::prepare(n);
Poly b(*this);
for(int i = 0; i < n; i++) {
b[i] *= mint::fact[i];
}
std::reverse(b.a.begin(), b.a.end());
Poly exp_cx(std::vector<mint>(n, mint(1)));
for(int i = 1; i < n; i++) {
exp_cx[i] = exp_cx[i - 1] * c / i;
}
b = (b * exp_cx).modxk(n);
std::reverse(b.a.begin(), b.a.end());
for(int i = 0; i < n; i++) {
b[i] *= mint::inv_fact[i];
}
return b;
}
constexpr Poly mulT(Poly b) const {
if(b.size() == 0) {
return Poly();
}
int n = b.size();
std::reverse(b.a.begin(), b.a.end());
return ((*this) * b).divxk(n - 1);
}
std::vector<mint> eval(std::vector<mint> x) const {
if(size() == 0) {
return std::vector<mint>(x.size(), mint(0));
}
const int n = std::max((int) x.size(), size());
std::vector<Poly> q(4 * n);
std::vector<mint> ans(x.size());
x.resize(n);
std::function<void(int, int, int)> build = [&](int p, int l, int r) {
if(r - l == 1) {
q[p] = Poly{1, -x[l]};
} else {
int m = (l + r) / 2;
build(2 * p, l, m);
build(2 * p + 1, m, r);
q[p] = q[2 * p] * q[2 * p + 1];
}
};
build(1, 0, n);
std::function<void(int, int, int, const Poly&)> work = [&](int p, int l, int r, const Poly& num) {
if(r - l == 1) {
if(l < (int) ans.size()) {
ans[l] = num.at(0);
}
} else {
int m = (l + r) / 2;
work(2 * p, l, m, num.mulT(q[2 * p + 1]).modxk(m - l));
work(2 * p + 1, m, r, num.mulT(q[2 * p]).modxk(r - m));
}
};
work(1, 0, n, mulT(q[1].inv(n)));
return ans;
}
private:
std::vector<mint> a;
};
} // namespace felix
#line 5 "test/formal-power-series/poly/yosupo-Sqrt-of-Formal-Power-Series.test.cpp"
using namespace std;
using namespace felix;
int main() {
ios::sync_with_stdio(false);
cin.tie(0);
int n;
cin >> n;
Poly<998244353> a(n);
for(int i = 0; i < n; i++) {
cin >> a[i];
}
if(!a.has_sqrt()) {
cout << "-1\n";
} else {
a = a.sqrt();
for(int i = 0; i < n; i++) {
cout << a[i] << " \n"[i == n - 1];
}
}
return 0;
}