CP-Algorithms Library
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cp-algo/linalg/matrix.hpp
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- Last update: 2025-12-10 23:03:51+01:00
- Include:
#include "cp-algo/linalg/matrix.hpp" - Link: View error logs on GitHub Actions
Depends on
- cp-algo/linalg/vector.hpp
- cp-algo/math/common.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/simd.hpp
Required by
Verified with
Adjugate Matrix (verify/linalg/adj.test.cpp)- Characteristic Polynomial (verify/linalg/characteristic.test.cpp)
- Matrix Determinant (verify/linalg/det.test.cpp)
- Counting Eulerian Circuits (verify/linalg/euler_circs.test.cpp)
Inverse Matrix (verify/linalg/inv.test.cpp)- Matching on General Graph (verify/linalg/matching.test.cpp)
- Pow of Matrix (verify/linalg/pow.test.cpp)
- Pow of Matrix (Frobenius) (verify/linalg/pow_fast.test.cpp)
- Matrix Product (verify/linalg/prod.test.cpp)
- Matrix Product (dynamic modint) (verify/linalg/prod_dynamic_modint.test.cpp)
- Rank of Matrix (verify/linalg/rank.test.cpp)
- Counting Spanning Trees (Directed) (verify/linalg/spanning_directed.test.cpp)
- Counting Spanning Trees (Undirected) (verify/linalg/spanning_undirected.test.cpp)
- System of Linear Equations (verify/linalg/system.test.cpp)
- Matching on General Graph (Tutte matrix) (verify/linalg/tutte.test.cpp)
Code
#ifndef CP_ALGO_LINALG_MATRIX_HPP
#define CP_ALGO_LINALG_MATRIX_HPP
#include "../random/rng.hpp"
#include "../math/common.hpp"
#include "vector.hpp"
#include <iostream>
#include <optional>
#include <cassert>
#include <vector>
#include <array>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
enum gauss_mode {normal, reverse};
template<typename base_t, class _vec_t = std::conditional_t<
math::modint_type<base_t>,
modint_vec<base_t>,
vec<base_t>>>
struct matrix: big_vector<_vec_t> {
using vec_t = _vec_t;
using base = base_t;
using Base = big_vector<vec_t>;
using Base::Base;
matrix(size_t n): Base(n, vec_t(n)) {}
matrix(size_t n, size_t m): Base(n, vec_t(m)) {}
matrix(Base const& t): Base(t) {}
matrix(Base &&t): Base(std::move(t)) {}
template<std::ranges::input_range R>
matrix(R &&r): Base(std::ranges::to<Base>(std::forward<R>(r))) {}
size_t n() const {return size(*this);}
size_t m() const {return n() ? size(row(0)) : 0;}
void resize(size_t n, size_t m) {
Base::resize(n);
for(auto &it: *this) {
it.resize(m);
}
}
auto& row(size_t i) {return (*this)[i];}
auto const& row(size_t i) const {return (*this)[i];}
auto elements() {return *this | std::views::join;}
auto elements() const {return *this | std::views::join;}
matrix operator-() const {
return *this | std::views::transform([](auto x) {return vec_t(-x);});
}
matrix& operator+=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a += b;
}
return *this;
}
matrix& operator -=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a -= b;
}
return *this;
}
matrix operator+(matrix const& t) const {return matrix(*this) += t;}
matrix operator-(matrix const& t) const {return matrix(*this) -= t;}
matrix& operator *=(base t) {for(auto &it: *this) it *= t; return *this;}
matrix operator *(base t) const {return matrix(*this) *= t;}
matrix& operator /=(base t) {return *this *= base(1) / t;}
matrix operator /(base t) const {return matrix(*this) /= t;}
// Make sure the result is matrix, not Base
matrix& operator *=(matrix const& t) {return *this = *this * t;}
void read_transposed() {
for(size_t j = 0; j < m(); j++) {
for(size_t i = 0; i < n(); i++) {
std::cin >> (*this)[i][j];
}
}
}
void read() {
for(auto &it: *this) {
it.read();
}
}
void print() const {
for(auto const& it: *this) {
it.print();
}
}
static matrix block_diagonal(big_vector<matrix> const& blocks) {
size_t n = 0;
for(auto &it: blocks) {
assert(it.n() == it.m());
n += it.n();
}
matrix res(n);
n = 0;
for(auto &it: blocks) {
for(size_t i = 0; i < it.n(); i++) {
std::ranges::copy(it[i], begin(res[n + i]) + n);
}
n += it.n();
}
return res;
}
static matrix random(size_t n, size_t m) {
matrix res(n, m);
std::ranges::generate(res, std::bind(vec_t::random, m));
return res;
}
static matrix random(size_t n) {
return random(n, n);
}
static matrix eye(size_t n) {
matrix res(n);
for(size_t i = 0; i < n; i++) {
res[i][i] = 1;
}
return res;
}
// Concatenate matrices
matrix operator |(matrix const& b) const {
assert(n() == b.n());
matrix res(n(), m()+b.m());
for(size_t i = 0; i < n(); i++) {
res[i] = row(i) | b[i];
}
return res;
}
void assign_submatrix(auto viewx, auto viewy, matrix const& t) {
for(auto [a, b]: std::views::zip(*this | viewx, t)) {
std::ranges::copy(b, begin(a | viewy));
}
}
auto submatrix(auto viewx, auto viewy) const {
return *this | viewx | std::views::transform([viewy](auto const& y) {
return y | viewy;
});
}
matrix T() const {
matrix res(m(), n());
for(size_t i = 0; i < n(); i++) {
for(size_t j = 0; j < m(); j++) {
res[j][i] = row(i)[j];
}
}
return res;
}
matrix operator *(matrix const& b) const {
assert(m() == b.n());
matrix res(n(), b.m());
for(size_t i = 0; i < n(); i++) {
for(size_t j = 0; j < m(); j++) {
res[i].add_scaled(b[j], row(i)[j]);
}
}
return res.normalize();
}
vec_t apply(vec_t const& x) const {
return (matrix(1, x) * *this)[0];
}
matrix pow(uint64_t k) const {
assert(n() == m());
return bpow(*this, k, eye(n()));
}
matrix& normalize() {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
template<gauss_mode mode = normal>
void eliminate(size_t i, size_t k) {
auto kinv = base(1) / row(i).normalize()[k];
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).add_scaled(row(i), -row(j).normalize(k) * kinv);
}
}
}
template<gauss_mode mode = normal>
void eliminate(size_t i) {
row(i).normalize();
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
template<gauss_mode mode = normal>
matrix& gauss() {
for(size_t i = 0; i < n(); i++) {
eliminate<mode>(i);
}
return normalize();
}
template<gauss_mode mode = normal>
auto echelonize(size_t lim) {
return gauss<mode>().sort_classify(lim);
}
template<gauss_mode mode = normal>
auto echelonize() {
return echelonize<mode>(m());
}
size_t rank() const {
if(n() > m()) {
return T().rank();
}
return size(matrix(*this).echelonize()[0]);
}
base det() const {
assert(n() == m());
matrix b = *this;
b.echelonize();
base res = 1;
for(size_t i = 0; i < n(); i++) {
res *= b[i][i];
}
return res;
}
std::pair<base, matrix> inv() const {
assert(n() == m());
matrix b = *this | eye(n());
if(size(b.echelonize<reverse>(n())[0]) < n()) {
return {0, {}};
}
base det = 1;
for(size_t i = 0; i < n(); i++) {
det *= b[i][i];
b[i] *= base(1) / b[i][i];
}
return {det, b.submatrix(std::views::all, std::views::drop(n()))};
}
// Can also just run gauss on T() | eye(m)
// but it would be slower :(
auto kernel() const {
auto A = *this;
auto [pivots, free] = A.template echelonize<reverse>();
matrix sols(size(free), m());
for(size_t j = 0; j < size(pivots); j++) {
base scale = base(1) / A[j][pivots[j]];
for(size_t i = 0; i < size(free); i++) {
sols[i][pivots[j]] = A[j][free[i]] * scale;
}
}
for(size_t i = 0; i < size(free); i++) {
sols[i][free[i]] = -1;
}
return sols;
}
// [solution, basis], transposed
std::optional<std::array<matrix, 2>> solve(matrix t) const {
matrix sols = (*this | t).kernel();
if(sols.n() < t.m() || matrix(sols.submatrix(
std::views::drop(sols.n() - t.m()),
std::views::drop(m())
)) != -eye(t.m())) {
return std::nullopt;
} else {
return std::array{
matrix(sols.submatrix(std::views::drop(sols.n() - t.m()), std::views::take(m()))),
matrix(sols.submatrix(std::views::take(sols.n() - t.m()), std::views::take(m())))
};
}
}
// To be called after a gaussian elimination run
// Sorts rows by pivots and classifies
// variables into pivots and free
auto sort_classify(size_t lim) {
size_t rk = 0;
big_vector<size_t> free, pivots;
for(size_t j = 0; j < lim; j++) {
for(size_t i = rk + 1; i < n() && row(rk)[j] == base(0); i++) {
if(row(i)[j] != base(0)) {
std::swap(row(i), row(rk));
row(rk) = -row(rk);
}
}
if(rk < n() && row(rk)[j] != base(0)) {
pivots.push_back(j);
rk++;
} else {
free.push_back(j);
}
}
return std::array{pivots, free};
}
};
template<typename base_t>
auto operator *(base_t t, matrix<base_t> const& A) {return A * t;}
}
#pragma GCC pop_options
#endif // CP_ALGO_LINALG_MATRIX_HPP
#line 1 "cp-algo/linalg/matrix.hpp"
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random {
std::mt19937_64 gen(
std::chrono::steady_clock::now().time_since_epoch().count()
);
uint64_t rng() {
return gen();
}
}
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
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));
}
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;
}
}
#line 1 "cp-algo/linalg/vector.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#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>;
constexpr static Int mod() {
return modint::mod();
}
constexpr static Int remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr 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();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr 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:
constexpr modint& to_modint() {return static_cast<modint&>(*this);}
constexpr 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;}
};
template<typename Int = int>
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/big_alloc.hpp"
#include <map>
#include <deque>
#include <vector>
#include <string>
#include <cstddef>
#line 10 "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, std::size_t Align = 32>
class big_alloc {
static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::popcount(Align) == 1, "Align must be a power of two");
public:
using value_type = T;
template <class U> struct rebind { using other = big_alloc<U, Align>; };
constexpr bool operator==(const big_alloc&) const = default;
constexpr bool operator!=(const big_alloc&) const = default;
big_alloc() noexcept = default;
template <typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if (padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded,
PROT_READ | PROT_WRITE,
MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
madvise(raw, padded, MADV_POPULATE_WRITE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if (!p) return;
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if (padded >= MEGABYTE) { munmap(p, padded); return; }
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
template<typename T>
using big_vector = std::vector<T, big_alloc<T>>;
template<typename T>
using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
template<typename T>
using big_deque = std::deque<T, big_alloc<T>>;
template<typename Key, typename Value, typename Compare = std::less<Key>>
using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
using big_string = big_basic_string<char>;
}
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
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 i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using dx4 = simd<double, 4>;
dx4 abs(dx4 a) {
return dx4{
std::abs(a[0]),
std::abs(a[1]),
std::abs(a[2]),
std::abs(a[3])
};
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
auto x_ninv = u64x4(u32x8(low32(x)) * imod);
x += x_ninv * uint64_t(mod);
#endif
return swap_bytes(x);
}
inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t 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
}
inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
}
inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
template<std::size_t Align = 32>
inline bool is_aligned(const auto* p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#pragma GCC pop_options
#line 1 "cp-algo/util/checkpoint.hpp"
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo {
#ifdef CP_ALGO_CHECKPOINT
big_map<big_string, double> checkpoints;
double last;
#endif
template<bool final = false>
void checkpoint([[maybe_unused]] auto const& _msg) {
#ifdef CP_ALGO_CHECKPOINT
big_string msg = _msg;
double now = (double)clock() / CLOCKS_PER_SEC;
double delta = now - last;
last = now;
if(msg.size() && !final) {
checkpoints[msg] += delta;
}
if(final) {
for(auto const& [key, value] : checkpoints) {
std::cerr << key << ": " << value * 1000 << " ms\n";
}
std::cerr << "Total: " << now * 1000 << " ms\n";
}
#endif
}
template<bool final = false>
void checkpoint() {
checkpoint<final>("");
}
}
#line 9 "cp-algo/linalg/vector.hpp"
#include <algorithm>
#include <valarray>
#line 12 "cp-algo/linalg/vector.hpp"
#include <iterator>
#line 14 "cp-algo/linalg/vector.hpp"
#include <ranges>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
template<typename base, class Alloc = big_alloc<base>>
struct vec: std::basic_string<base, std::char_traits<base>, Alloc> {
using Base = std::basic_string<base, std::char_traits<base>, Alloc>;
using Base::Base;
vec(Base const& t): Base(t) {}
vec(Base &&t): Base(std::move(t)) {}
vec(size_t n): Base(n, base()) {}
vec(auto &&r): Base(std::ranges::to<Base>(r)) {}
static vec ei(size_t n, size_t i) {
vec res(n);
res[i] = 1;
return res;
}
auto operator-() const {
return *this | std::views::transform([](auto x) {return -x;});
}
auto operator *(base t) const {
return *this | std::views::transform([t](auto x) {return x * t;});
}
auto operator *=(base t) {
for(auto &it: *this) {
it *= t;
}
return *this;
}
virtual void add_scaled(vec const& b, base scale, size_t i = 0) {
if(scale != base(0)) {
for(; i < size(*this); i++) {
(*this)[i] += scale * b[i];
}
}
}
virtual vec const& normalize() {
return static_cast<vec&>(*this);
}
virtual base normalize(size_t i) {
return (*this)[i];
}
void read() {
for(auto &it: *this) {
std::cin >> it;
}
}
void print() const {
for(auto &it: *this) {
std::cout << it << " ";
}
std::cout << "\n";
}
static vec random(size_t n) {
vec res(n);
std::ranges::generate(res, random::rng);
return res;
}
// Concatenate vectors
vec operator |(vec const& t) const {
return std::views::join(std::array{
std::views::all(*this),
std::views::all(t)
});
}
// Generally, vec shouldn't be modified
// after its pivot index is set
std::pair<size_t, base> find_pivot() {
if(pivot == size_t(-1)) {
pivot = 0;
while(pivot < size(*this) && normalize(pivot) == base(0)) {
pivot++;
}
if(pivot < size(*this)) {
pivot_inv = base(1) / (*this)[pivot];
}
}
return {pivot, pivot_inv};
}
void reduce_by(vec &t) {
auto [pivot, pinv] = t.find_pivot();
if(pivot < size(*this)) {
add_scaled(t, -normalize(pivot) * pinv, pivot);
}
}
private:
size_t pivot = -1;
base pivot_inv;
};
template<math::modint_type base, class Alloc = big_alloc<base>>
struct modint_vec: vec<base, Alloc> {
using Base = vec<base, Alloc>;
using Base::Base;
modint_vec(Base const& t): Base(t) {}
modint_vec(Base &&t): Base(std::move(t)) {}
void add_scaled(Base const& b, base scale, size_t i = 0) override {
static_assert(base::bits >= 64, "Only wide modint types for linalg");
if(scale != base(0)) {
assert(Base::size() == b.size());
size_t n = size(*this);
u64x4 scaler = u64x4() + scale.getr();
if (is_aligned(&(*this)[0]) && is_aligned(&b[0])) // verify we're not in SSO
for(i -= i % 4; i + 3 < n; i += 4) {
auto &ai = vector_cast<u64x4>((*this)[i]);
auto bi = vector_cast<u64x4 const>(b[i]);
#ifdef __AVX2__
ai += u64x4(_mm256_mul_epu32(__m256i(scaler), __m256i(bi)));
#else
ai += scaler * bi;
#endif
}
for(; i < n; i++) {
(*this)[i].add_unsafe(b[i].getr_direct() * scale.getr());
}
if(++counter == 4) {
for(auto &it: *this) {
it.pseudonormalize();
}
counter = 0;
}
}
}
Base const& normalize() override {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
base normalize(size_t i) override {
return (*this)[i].normalize();
}
private:
size_t counter = 0;
};
}
#pragma GCC pop_options
#line 7 "cp-algo/linalg/matrix.hpp"
#include <optional>
#line 10 "cp-algo/linalg/matrix.hpp"
#include <array>
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
enum gauss_mode {normal, reverse};
template<typename base_t, class _vec_t = std::conditional_t<
math::modint_type<base_t>,
modint_vec<base_t>,
vec<base_t>>>
struct matrix: big_vector<_vec_t> {
using vec_t = _vec_t;
using base = base_t;
using Base = big_vector<vec_t>;
using Base::Base;
matrix(size_t n): Base(n, vec_t(n)) {}
matrix(size_t n, size_t m): Base(n, vec_t(m)) {}
matrix(Base const& t): Base(t) {}
matrix(Base &&t): Base(std::move(t)) {}
template<std::ranges::input_range R>
matrix(R &&r): Base(std::ranges::to<Base>(std::forward<R>(r))) {}
size_t n() const {return size(*this);}
size_t m() const {return n() ? size(row(0)) : 0;}
void resize(size_t n, size_t m) {
Base::resize(n);
for(auto &it: *this) {
it.resize(m);
}
}
auto& row(size_t i) {return (*this)[i];}
auto const& row(size_t i) const {return (*this)[i];}
auto elements() {return *this | std::views::join;}
auto elements() const {return *this | std::views::join;}
matrix operator-() const {
return *this | std::views::transform([](auto x) {return vec_t(-x);});
}
matrix& operator+=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a += b;
}
return *this;
}
matrix& operator -=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a -= b;
}
return *this;
}
matrix operator+(matrix const& t) const {return matrix(*this) += t;}
matrix operator-(matrix const& t) const {return matrix(*this) -= t;}
matrix& operator *=(base t) {for(auto &it: *this) it *= t; return *this;}
matrix operator *(base t) const {return matrix(*this) *= t;}
matrix& operator /=(base t) {return *this *= base(1) / t;}
matrix operator /(base t) const {return matrix(*this) /= t;}
// Make sure the result is matrix, not Base
matrix& operator *=(matrix const& t) {return *this = *this * t;}
void read_transposed() {
for(size_t j = 0; j < m(); j++) {
for(size_t i = 0; i < n(); i++) {
std::cin >> (*this)[i][j];
}
}
}
void read() {
for(auto &it: *this) {
it.read();
}
}
void print() const {
for(auto const& it: *this) {
it.print();
}
}
static matrix block_diagonal(big_vector<matrix> const& blocks) {
size_t n = 0;
for(auto &it: blocks) {
assert(it.n() == it.m());
n += it.n();
}
matrix res(n);
n = 0;
for(auto &it: blocks) {
for(size_t i = 0; i < it.n(); i++) {
std::ranges::copy(it[i], begin(res[n + i]) + n);
}
n += it.n();
}
return res;
}
static matrix random(size_t n, size_t m) {
matrix res(n, m);
std::ranges::generate(res, std::bind(vec_t::random, m));
return res;
}
static matrix random(size_t n) {
return random(n, n);
}
static matrix eye(size_t n) {
matrix res(n);
for(size_t i = 0; i < n; i++) {
res[i][i] = 1;
}
return res;
}
// Concatenate matrices
matrix operator |(matrix const& b) const {
assert(n() == b.n());
matrix res(n(), m()+b.m());
for(size_t i = 0; i < n(); i++) {
res[i] = row(i) | b[i];
}
return res;
}
void assign_submatrix(auto viewx, auto viewy, matrix const& t) {
for(auto [a, b]: std::views::zip(*this | viewx, t)) {
std::ranges::copy(b, begin(a | viewy));
}
}
auto submatrix(auto viewx, auto viewy) const {
return *this | viewx | std::views::transform([viewy](auto const& y) {
return y | viewy;
});
}
matrix T() const {
matrix res(m(), n());
for(size_t i = 0; i < n(); i++) {
for(size_t j = 0; j < m(); j++) {
res[j][i] = row(i)[j];
}
}
return res;
}
matrix operator *(matrix const& b) const {
assert(m() == b.n());
matrix res(n(), b.m());
for(size_t i = 0; i < n(); i++) {
for(size_t j = 0; j < m(); j++) {
res[i].add_scaled(b[j], row(i)[j]);
}
}
return res.normalize();
}
vec_t apply(vec_t const& x) const {
return (matrix(1, x) * *this)[0];
}
matrix pow(uint64_t k) const {
assert(n() == m());
return bpow(*this, k, eye(n()));
}
matrix& normalize() {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
template<gauss_mode mode = normal>
void eliminate(size_t i, size_t k) {
auto kinv = base(1) / row(i).normalize()[k];
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).add_scaled(row(i), -row(j).normalize(k) * kinv);
}
}
}
template<gauss_mode mode = normal>
void eliminate(size_t i) {
row(i).normalize();
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
template<gauss_mode mode = normal>
matrix& gauss() {
for(size_t i = 0; i < n(); i++) {
eliminate<mode>(i);
}
return normalize();
}
template<gauss_mode mode = normal>
auto echelonize(size_t lim) {
return gauss<mode>().sort_classify(lim);
}
template<gauss_mode mode = normal>
auto echelonize() {
return echelonize<mode>(m());
}
size_t rank() const {
if(n() > m()) {
return T().rank();
}
return size(matrix(*this).echelonize()[0]);
}
base det() const {
assert(n() == m());
matrix b = *this;
b.echelonize();
base res = 1;
for(size_t i = 0; i < n(); i++) {
res *= b[i][i];
}
return res;
}
std::pair<base, matrix> inv() const {
assert(n() == m());
matrix b = *this | eye(n());
if(size(b.echelonize<reverse>(n())[0]) < n()) {
return {0, {}};
}
base det = 1;
for(size_t i = 0; i < n(); i++) {
det *= b[i][i];
b[i] *= base(1) / b[i][i];
}
return {det, b.submatrix(std::views::all, std::views::drop(n()))};
}
// Can also just run gauss on T() | eye(m)
// but it would be slower :(
auto kernel() const {
auto A = *this;
auto [pivots, free] = A.template echelonize<reverse>();
matrix sols(size(free), m());
for(size_t j = 0; j < size(pivots); j++) {
base scale = base(1) / A[j][pivots[j]];
for(size_t i = 0; i < size(free); i++) {
sols[i][pivots[j]] = A[j][free[i]] * scale;
}
}
for(size_t i = 0; i < size(free); i++) {
sols[i][free[i]] = -1;
}
return sols;
}
// [solution, basis], transposed
std::optional<std::array<matrix, 2>> solve(matrix t) const {
matrix sols = (*this | t).kernel();
if(sols.n() < t.m() || matrix(sols.submatrix(
std::views::drop(sols.n() - t.m()),
std::views::drop(m())
)) != -eye(t.m())) {
return std::nullopt;
} else {
return std::array{
matrix(sols.submatrix(std::views::drop(sols.n() - t.m()), std::views::take(m()))),
matrix(sols.submatrix(std::views::take(sols.n() - t.m()), std::views::take(m())))
};
}
}
// To be called after a gaussian elimination run
// Sorts rows by pivots and classifies
// variables into pivots and free
auto sort_classify(size_t lim) {
size_t rk = 0;
big_vector<size_t> free, pivots;
for(size_t j = 0; j < lim; j++) {
for(size_t i = rk + 1; i < n() && row(rk)[j] == base(0); i++) {
if(row(i)[j] != base(0)) {
std::swap(row(i), row(rk));
row(rk) = -row(rk);
}
}
if(rk < n() && row(rk)[j] != base(0)) {
pivots.push_back(j);
rk++;
} else {
free.push_back(j);
}
}
return std::array{pivots, free};
}
};
template<typename base_t>
auto operator *(base_t t, matrix<base_t> const& A) {return A * t;}
}
#pragma GCC pop_options
#ifndef CP_ALGO_LINALG_MATRIX_HPP
#define CP_ALGO_LINALG_MATRIX_HPP
#include "../random/rng.hpp"
#include "../math/common.hpp"
#include "vector.hpp"
#include <iostream>
#include <optional>
#include <cassert>
#include <vector>
#include <array>
CP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo::linalg{enum gauss_mode{normal,reverse};template<typename base_t,class _vec_t=std::conditional_t<math::modint_type<base_t>,modint_vec<base_t>,vec<base_t>>>struct matrix:big_vector<_vec_t>{using vec_t=_vec_t;using base=base_t;using Base=big_vector<vec_t>;using Base::Base;matrix(size_t n):Base(n,vec_t(n)){}matrix(size_t n,size_t m):Base(n,vec_t(m)){}matrix(Base const&t):Base(t){}matrix(Base&&t):Base(std::move(t)){}template<std::ranges::input_range R>matrix(R&&r):Base(std::ranges::to<Base>(std::forward<R>(r))){}size_t n()const{return size(*this);}size_t m()const{return n()?size(row(0)):0;}void resize(size_t n,size_t m){Base::resize(n);for(auto&it:*this){it.resize(m);}}auto&row(size_t i){return(*this)[i];}auto const&row(size_t i)const{return(*this)[i];}auto elements(){return*this|std::views::join;}auto elements()const{return*this|std::views::join;}matrix operator-()const{return*this|std::views::transform([](auto x){return vec_t(-x);});}matrix&operator+=(matrix const&t){for(auto[a,b]:std::views::zip(elements(),t.elements())){a+=b;}return*this;}matrix&operator-=(matrix const&t){for(auto[a,b]:std::views::zip(elements(),t.elements())){a-=b;}return*this;}matrix operator+(matrix const&t)const{return matrix(*this)+=t;}matrix operator-(matrix const&t)const{return matrix(*this)-=t;}matrix&operator*=(base t){for(auto&it:*this)it*=t;return*this;}matrix operator*(base t)const{return matrix(*this)*=t;}matrix&operator/=(base t){return*this*=base(1)/t;}matrix operator/(base t)const{return matrix(*this)/=t;}matrix&operator*=(matrix const&t){return*this=*this*t;}void read_transposed(){for(size_t j=0;j<m();j++){for(size_t i=0;i<n();i++){std::cin>>(*this)[i][j];}}}void read(){for(auto&it:*this){it.read();}}void print()const{for(auto const&it:*this){it.print();}}static matrix block_diagonal(big_vector<matrix>const&blocks){size_t n=0;for(auto&it:blocks){assert(it.n()==it.m());n+=it.n();}matrix res(n);n=0;for(auto&it:blocks){for(size_t i=0;i<it.n();i++){std::ranges::copy(it[i],begin(res[n+i])+n);}n+=it.n();}return res;}static matrix random(size_t n,size_t m){matrix res(n,m);std::ranges::generate(res,std::bind(vec_t::random,m));return res;}static matrix random(size_t n){return random(n,n);}static matrix eye(size_t n){matrix res(n);for(size_t i=0;i<n;i++){res[i][i]=1;}return res;}matrix operator|(matrix const&b)const{assert(n()==b.n());matrix res(n(),m()+b.m());for(size_t i=0;i<n();i++){res[i]=row(i)|b[i];}return res;}void assign_submatrix(auto viewx,auto viewy,matrix const&t){for(auto[a,b]:std::views::zip(*this|viewx,t)){std::ranges::copy(b,begin(a|viewy));}}auto submatrix(auto viewx,auto viewy)const{return*this|viewx|std::views::transform([viewy](auto const&y){return y|viewy;});}matrix T()const{matrix res(m(),n());for(size_t i=0;i<n();i++){for(size_t j=0;j<m();j++){res[j][i]=row(i)[j];}}return res;}matrix operator*(matrix const&b)const{assert(m()==b.n());matrix res(n(),b.m());for(size_t i=0;i<n();i++){for(size_t j=0;j<m();j++){res[i].add_scaled(b[j],row(i)[j]);}}return res.normalize();}vec_t apply(vec_t const&x)const{return(matrix(1,x)**this)[0];}matrix pow(uint64_t k)const{assert(n()==m());return bpow(*this,k,eye(n()));}matrix&normalize(){for(auto&it:*this){it.normalize();}return*this;}template<gauss_mode mode=normal>void eliminate(size_t i,size_t k){auto kinv=base(1)/row(i).normalize()[k];for(size_t j=(mode==normal)*i;j<n();j++){if(j!=i){row(j).add_scaled(row(i),-row(j).normalize(k)*kinv);}}}template<gauss_mode mode=normal>void eliminate(size_t i){row(i).normalize();for(size_t j=(mode==normal)*i;j<n();j++){if(j!=i){row(j).reduce_by(row(i));}}}template<gauss_mode mode=normal>matrix&gauss(){for(size_t i=0;i<n();i++){eliminate<mode>(i);}return normalize();}template<gauss_mode mode=normal>auto echelonize(size_t lim){return gauss<mode>().sort_classify(lim);}template<gauss_mode mode=normal>auto echelonize(){return echelonize<mode>(m());}size_t rank()const{if(n()>m()){return T().rank();}return size(matrix(*this).echelonize()[0]);}base det()const{assert(n()==m());matrix b=*this;b.echelonize();base res=1;for(size_t i=0;i<n();i++){res*=b[i][i];}return res;}std::pair<base,matrix>inv()const{assert(n()==m());matrix b=*this|eye(n());if(size(b.echelonize<reverse>(n())[0])<n()){return{0,{}};}base det=1;for(size_t i=0;i<n();i++){det*=b[i][i];b[i]*=base(1)/b[i][i];}return{det,b.submatrix(std::views::all,std::views::drop(n()))};}auto kernel()const{auto A=*this;auto[pivots,free]=A.template echelonize<reverse>();matrix sols(size(free),m());for(size_t j=0;j<size(pivots);j++){base scale=base(1)/A[j][pivots[j]];for(size_t i=0;i<size(free);i++){sols[i][pivots[j]]=A[j][free[i]]*scale;}}for(size_t i=0;i<size(free);i++){sols[i][free[i]]=-1;}return sols;}std::optional<std::array<matrix,2>>solve(matrix t)const{matrix sols=(*this|t).kernel();if(sols.n()<t.m()||matrix(sols.submatrix(std::views::drop(sols.n()-t.m()),std::views::drop(m())))!=-eye(t.m())){return std::nullopt;}else{return std::array{matrix(sols.submatrix(std::views::drop(sols.n()-t.m()),std::views::take(m()))),matrix(sols.submatrix(std::views::take(sols.n()-t.m()),std::views::take(m())))};}}auto sort_classify(size_t lim){size_t rk=0;big_vector<size_t>free,pivots;for(size_t j=0;j<lim;j++){for(size_t i=rk+1;i<n()&&row(rk)[j]==base(0);i++){if(row(i)[j]!=base(0)){std::swap(row(i),row(rk));row(rk)=-row(rk);}}if(rk<n()&&row(rk)[j]!=base(0)){pivots.push_back(j);rk++;}else{free.push_back(j);}}return std::array{pivots,free};}};template<typename base_t>auto operator*(base_t t,matrix<base_t>const&A){return A*t;}}
#pragma GCC pop_options
#endif
#line 1 "cp-algo/linalg/matrix.hpp"
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random{std::mt19937_64 gen(std::chrono::steady_clock::now().time_since_epoch().count());uint64_t rng(){return gen();}}
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
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;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));}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;}}
#line 1 "cp-algo/linalg/vector.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#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>;constexpr static Int mod(){return modint::mod();}constexpr static Int remod(){return modint::remod();}constexpr static UInt2 modmod(){return UInt2(mod())*mod();}constexpr modint_base()=default;constexpr 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;}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();}constexpr void setr(UInt rr){r=rr;}constexpr UInt getr()const{return r;}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:constexpr modint&to_modint(){return static_cast<modint&>(*this);}constexpr 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;}};template<typename Int=int>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);}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/big_alloc.hpp"
#include <map>
#include <deque>
#include <vector>
#include <string>
#include <cstddef>
#line 10 "cp-algo/util/big_alloc.hpp"
#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,std::size_t Align=32>class big_alloc{static_assert(Align>=alignof(void*),"Align must be at least pointer-size");static_assert(std::popcount(Align)==1,"Align must be a power of two");public:using value_type=T;template<class U>struct rebind{using other=big_alloc<U,Align>;};constexpr bool operator==(const big_alloc&)const=default;constexpr bool operator!=(const big_alloc&)const=default;big_alloc()noexcept=default;template<typename U,std::size_t A>big_alloc(const big_alloc<U,A>&)noexcept{}[[nodiscard]]T*allocate(std::size_t n){std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){void*raw=mmap(nullptr,padded,PROT_READ|PROT_WRITE,MAP_PRIVATE|MAP_ANONYMOUS,-1,0);madvise(raw,padded,MADV_HUGEPAGE);madvise(raw,padded,MADV_POPULATE_WRITE);return static_cast<T*>(raw);}
#endif
return static_cast<T*>(::operator new(padded,std::align_val_t(align)));}void deallocate(T*p,std::size_t n)noexcept{if(!p)return;std::size_t padded=round_up(n*sizeof(T));std::size_t align=std::max<std::size_t>(alignof(T),Align);
#if CP_ALGO_USE_MMAP
if(padded>=MEGABYTE){munmap(p,padded);return;}
#endif
::operator delete(p,padded,std::align_val_t(align));}private:static constexpr std::size_t MEGABYTE=1<<20;static constexpr std::size_t round_up(std::size_t x)noexcept{return(x+Align-1)/Align*Align;}};template<typename T>using big_vector=std::vector<T,big_alloc<T>>;template<typename T>using big_basic_string=std::basic_string<T,std::char_traits<T>,big_alloc<T>>;template<typename T>using big_deque=std::deque<T,big_alloc<T>>;template<typename Key,typename Value,typename Compare=std::less<Key>>using big_map=std::map<Key,Value,Compare,big_alloc<std::pair<const Key,Value>>>;using big_string=big_basic_string<char>;}
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options")\CP_ALGO_SIMD_AVX2_TARGETCP_ALGO_SIMD_PRAGMA_PUSHnamespace 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 i16x4=simd<int16_t,4>;using u8x32=simd<uint8_t,32>;using dx4=simd<double,4>;dx4 abs(dx4 a){return dx4{std::abs(a[0]),std::abs(a[1]),std::abs(a[2]),std::abs(a[3])};}static constexpr dx4 magic=dx4()+(3ULL<<51);inline i64x4 lround(dx4 x){return i64x4(x+magic)-i64x4(magic);}inline dx4 to_double(i64x4 x){return dx4(x+i64x4(magic))-magic;}inline dx4 round(dx4 a){return dx4{std::nearbyint(a[0]),std::nearbyint(a[1]),std::nearbyint(a[2]),std::nearbyint(a[3])};}inline u64x4 low32(u64x4 x){return x&uint32_t(-1);}inline auto swap_bytes(auto x){return decltype(x)(__builtin_shufflevector(u32x8(x),u32x8(x),1,0,3,2,5,4,7,6));}inline u64x4 montgomery_reduce(u64x4 x,uint32_t mod,uint32_t imod){
#ifdef __AVX2__
auto x_ninv=u64x4(_mm256_mul_epu32(__m256i(x),__m256i()+imod));x+=u64x4(_mm256_mul_epu32(__m256i(x_ninv),__m256i()+mod));
#else
auto x_ninv=u64x4(u32x8(low32(x))*imod);x+=x_ninv*uint64_t(mod);
#endif
return swap_bytes(x);}inline u64x4 montgomery_mul(u64x4 x,u64x4 y,uint32_t mod,uint32_t 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
}inline u32x8 montgomery_mul(u32x8 x,u32x8 y,uint32_t mod,uint32_t imod){return u32x8(montgomery_mul(u64x4(x),u64x4(y),mod,imod))|u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)),u64x4(swap_bytes(y)),mod,imod)));}inline dx4 rotate_right(dx4 x){static constexpr u64x4 shuffler={3,0,1,2};return __builtin_shuffle(x,shuffler);}template<std::size_t Align=32>inline bool is_aligned(const auto*p)noexcept{return(reinterpret_cast<std::uintptr_t>(p)%Align)==0;}template<class Target>inline Target&vector_cast(auto&&p){return*reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));}}
#pragma GCC pop_options
#line 1 "cp-algo/util/checkpoint.hpp"
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo{
#ifdef CP_ALGO_CHECKPOINT
big_map<big_string,double>checkpoints;double last;
#endif
template<bool final=false>void checkpoint([[maybe_unused]]auto const&_msg){
#ifdef CP_ALGO_CHECKPOINT
big_string msg=_msg;double now=(double)clock()/CLOCKS_PER_SEC;double delta=now-last;last=now;if(msg.size()&&!final){checkpoints[msg]+=delta;}if(final){for(auto const&[key,value]:checkpoints){std::cerr<<key<<": "<<value*1000<<" ms\n";}std::cerr<<"Total: "<<now*1000<<" ms\n";}
#endif
}template<bool final=false>void checkpoint(){checkpoint<final>("");}}
#line 9 "cp-algo/linalg/vector.hpp"
#include <algorithm>
#include <valarray>
#line 12 "cp-algo/linalg/vector.hpp"
#include <iterator>
#line 14 "cp-algo/linalg/vector.hpp"
#include <ranges>
CP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo::linalg{template<typename base,class Alloc=big_alloc<base>>struct vec:std::basic_string<base,std::char_traits<base>,Alloc>{using Base=std::basic_string<base,std::char_traits<base>,Alloc>;using Base::Base;vec(Base const&t):Base(t){}vec(Base&&t):Base(std::move(t)){}vec(size_t n):Base(n,base()){}vec(auto&&r):Base(std::ranges::to<Base>(r)){}static vec ei(size_t n,size_t i){vec res(n);res[i]=1;return res;}auto operator-()const{return*this|std::views::transform([](auto x){return-x;});}auto operator*(base t)const{return*this|std::views::transform([t](auto x){return x*t;});}auto operator*=(base t){for(auto&it:*this){it*=t;}return*this;}virtual void add_scaled(vec const&b,base scale,size_t i=0){if(scale!=base(0)){for(;i<size(*this);i++){(*this)[i]+=scale*b[i];}}}virtual vec const&normalize(){return static_cast<vec&>(*this);}virtual base normalize(size_t i){return(*this)[i];}void read(){for(auto&it:*this){std::cin>>it;}}void print()const{for(auto&it:*this){std::cout<<it<<" ";}std::cout<<"\n";}static vec random(size_t n){vec res(n);std::ranges::generate(res,random::rng);return res;}vec operator|(vec const&t)const{return std::views::join(std::array{std::views::all(*this),std::views::all(t)});}std::pair<size_t,base>find_pivot(){if(pivot==size_t(-1)){pivot=0;while(pivot<size(*this)&&normalize(pivot)==base(0)){pivot++;}if(pivot<size(*this)){pivot_inv=base(1)/(*this)[pivot];}}return{pivot,pivot_inv};}void reduce_by(vec&t){auto[pivot,pinv]=t.find_pivot();if(pivot<size(*this)){add_scaled(t,-normalize(pivot)*pinv,pivot);}}private:size_t pivot=-1;base pivot_inv;};template<math::modint_type base,class Alloc=big_alloc<base>>struct modint_vec:vec<base,Alloc>{using Base=vec<base,Alloc>;using Base::Base;modint_vec(Base const&t):Base(t){}modint_vec(Base&&t):Base(std::move(t)){}void add_scaled(Base const&b,base scale,size_t i=0)override{static_assert(base::bits>=64,"Only wide modint types for linalg");if(scale!=base(0)){assert(Base::size()==b.size());size_t n=size(*this);u64x4 scaler=u64x4()+scale.getr();if(is_aligned(&(*this)[0])&&is_aligned(&b[0]))for(i-=i%4;i+3<n;i+=4){auto&ai=vector_cast<u64x4>((*this)[i]);auto bi=vector_cast<u64x4 const>(b[i]);
#ifdef __AVX2__
ai+=u64x4(_mm256_mul_epu32(__m256i(scaler),__m256i(bi)));
#else
ai+=scaler*bi;
#endif
}for(;i<n;i++){(*this)[i].add_unsafe(b[i].getr_direct()*scale.getr());}if(++counter==4){for(auto&it:*this){it.pseudonormalize();}counter=0;}}}Base const&normalize()override{for(auto&it:*this){it.normalize();}return*this;}base normalize(size_t i)override{return(*this)[i].normalize();}private:size_t counter=0;};}
#pragma GCC pop_options
#line 7 "cp-algo/linalg/matrix.hpp"
#include <optional>
#line 10 "cp-algo/linalg/matrix.hpp"
#include <array>
CP_ALGO_SIMD_PRAGMA_PUSHnamespace cp_algo::linalg{enum gauss_mode{normal,reverse};template<typename base_t,class _vec_t=std::conditional_t<math::modint_type<base_t>,modint_vec<base_t>,vec<base_t>>>struct matrix:big_vector<_vec_t>{using vec_t=_vec_t;using base=base_t;using Base=big_vector<vec_t>;using Base::Base;matrix(size_t n):Base(n,vec_t(n)){}matrix(size_t n,size_t m):Base(n,vec_t(m)){}matrix(Base const&t):Base(t){}matrix(Base&&t):Base(std::move(t)){}template<std::ranges::input_range R>matrix(R&&r):Base(std::ranges::to<Base>(std::forward<R>(r))){}size_t n()const{return size(*this);}size_t m()const{return n()?size(row(0)):0;}void resize(size_t n,size_t m){Base::resize(n);for(auto&it:*this){it.resize(m);}}auto&row(size_t i){return(*this)[i];}auto const&row(size_t i)const{return(*this)[i];}auto elements(){return*this|std::views::join;}auto elements()const{return*this|std::views::join;}matrix operator-()const{return*this|std::views::transform([](auto x){return vec_t(-x);});}matrix&operator+=(matrix const&t){for(auto[a,b]:std::views::zip(elements(),t.elements())){a+=b;}return*this;}matrix&operator-=(matrix const&t){for(auto[a,b]:std::views::zip(elements(),t.elements())){a-=b;}return*this;}matrix operator+(matrix const&t)const{return matrix(*this)+=t;}matrix operator-(matrix const&t)const{return matrix(*this)-=t;}matrix&operator*=(base t){for(auto&it:*this)it*=t;return*this;}matrix operator*(base t)const{return matrix(*this)*=t;}matrix&operator/=(base t){return*this*=base(1)/t;}matrix operator/(base t)const{return matrix(*this)/=t;}matrix&operator*=(matrix const&t){return*this=*this*t;}void read_transposed(){for(size_t j=0;j<m();j++){for(size_t i=0;i<n();i++){std::cin>>(*this)[i][j];}}}void read(){for(auto&it:*this){it.read();}}void print()const{for(auto const&it:*this){it.print();}}static matrix block_diagonal(big_vector<matrix>const&blocks){size_t n=0;for(auto&it:blocks){assert(it.n()==it.m());n+=it.n();}matrix res(n);n=0;for(auto&it:blocks){for(size_t i=0;i<it.n();i++){std::ranges::copy(it[i],begin(res[n+i])+n);}n+=it.n();}return res;}static matrix random(size_t n,size_t m){matrix res(n,m);std::ranges::generate(res,std::bind(vec_t::random,m));return res;}static matrix random(size_t n){return random(n,n);}static matrix eye(size_t n){matrix res(n);for(size_t i=0;i<n;i++){res[i][i]=1;}return res;}matrix operator|(matrix const&b)const{assert(n()==b.n());matrix res(n(),m()+b.m());for(size_t i=0;i<n();i++){res[i]=row(i)|b[i];}return res;}void assign_submatrix(auto viewx,auto viewy,matrix const&t){for(auto[a,b]:std::views::zip(*this|viewx,t)){std::ranges::copy(b,begin(a|viewy));}}auto submatrix(auto viewx,auto viewy)const{return*this|viewx|std::views::transform([viewy](auto const&y){return y|viewy;});}matrix T()const{matrix res(m(),n());for(size_t i=0;i<n();i++){for(size_t j=0;j<m();j++){res[j][i]=row(i)[j];}}return res;}matrix operator*(matrix const&b)const{assert(m()==b.n());matrix res(n(),b.m());for(size_t i=0;i<n();i++){for(size_t j=0;j<m();j++){res[i].add_scaled(b[j],row(i)[j]);}}return res.normalize();}vec_t apply(vec_t const&x)const{return(matrix(1,x)**this)[0];}matrix pow(uint64_t k)const{assert(n()==m());return bpow(*this,k,eye(n()));}matrix&normalize(){for(auto&it:*this){it.normalize();}return*this;}template<gauss_mode mode=normal>void eliminate(size_t i,size_t k){auto kinv=base(1)/row(i).normalize()[k];for(size_t j=(mode==normal)*i;j<n();j++){if(j!=i){row(j).add_scaled(row(i),-row(j).normalize(k)*kinv);}}}template<gauss_mode mode=normal>void eliminate(size_t i){row(i).normalize();for(size_t j=(mode==normal)*i;j<n();j++){if(j!=i){row(j).reduce_by(row(i));}}}template<gauss_mode mode=normal>matrix&gauss(){for(size_t i=0;i<n();i++){eliminate<mode>(i);}return normalize();}template<gauss_mode mode=normal>auto echelonize(size_t lim){return gauss<mode>().sort_classify(lim);}template<gauss_mode mode=normal>auto echelonize(){return echelonize<mode>(m());}size_t rank()const{if(n()>m()){return T().rank();}return size(matrix(*this).echelonize()[0]);}base det()const{assert(n()==m());matrix b=*this;b.echelonize();base res=1;for(size_t i=0;i<n();i++){res*=b[i][i];}return res;}std::pair<base,matrix>inv()const{assert(n()==m());matrix b=*this|eye(n());if(size(b.echelonize<reverse>(n())[0])<n()){return{0,{}};}base det=1;for(size_t i=0;i<n();i++){det*=b[i][i];b[i]*=base(1)/b[i][i];}return{det,b.submatrix(std::views::all,std::views::drop(n()))};}auto kernel()const{auto A=*this;auto[pivots,free]=A.template echelonize<reverse>();matrix sols(size(free),m());for(size_t j=0;j<size(pivots);j++){base scale=base(1)/A[j][pivots[j]];for(size_t i=0;i<size(free);i++){sols[i][pivots[j]]=A[j][free[i]]*scale;}}for(size_t i=0;i<size(free);i++){sols[i][free[i]]=-1;}return sols;}std::optional<std::array<matrix,2>>solve(matrix t)const{matrix sols=(*this|t).kernel();if(sols.n()<t.m()||matrix(sols.submatrix(std::views::drop(sols.n()-t.m()),std::views::drop(m())))!=-eye(t.m())){return std::nullopt;}else{return std::array{matrix(sols.submatrix(std::views::drop(sols.n()-t.m()),std::views::take(m()))),matrix(sols.submatrix(std::views::take(sols.n()-t.m()),std::views::take(m())))};}}auto sort_classify(size_t lim){size_t rk=0;big_vector<size_t>free,pivots;for(size_t j=0;j<lim;j++){for(size_t i=rk+1;i<n()&&row(rk)[j]==base(0);i++){if(row(i)[j]!=base(0)){std::swap(row(i),row(rk));row(rk)=-row(rk);}}if(rk<n()&&row(rk)[j]!=base(0)){pivots.push_back(j);rk++;}else{free.push_back(j);}}return std::array{pivots,free};}};template<typename base_t>auto operator*(base_t t,matrix<base_t>const&A){return A*t;}}
#pragma GCC pop_options