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test/library-checker/system_of_linear_equations.test.cpp
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- Last update: 2022-04-06 15:02:09+09:00
- Problem: https://judge.yosupo.jp/problem/system_of_linear_equations
Depends on
- LU decomposition
(src/algebra/linear/lu.hpp)
- Matrix
(src/algebra/linear/matrix.hpp)
- Modular Arithmetic
(src/algebra/modint.hpp)
- Modular Exponentiation
(src/number_theory/pow_mod.hpp)
- Tonelli-Shanks Algorithm
(src/number_theory/sqrt_mod.hpp)
- Input Stream
(src/utils/io/istream.hpp)
- Output Stream
(src/utils/io/ostream.hpp)
- SFINAE
(src/utils/sfinae.hpp)
Code
#define PROBLEM "https://judge.yosupo.jp/problem/system_of_linear_equations"
#include "src/algebra/linear/lu.hpp"
#include "src/algebra/modint.hpp"
#include "src/utils/io/istream.hpp"
#include "src/utils/io/ostream.hpp"
namespace workspace {
using mint = modint<998244353>;
using mat = matrix<mint>;
void main() {
// start here!
size_t n, m;
cin >> n >> m;
lu_decomposition lu = [&] {
mat a(n, m);
cin >> a;
return a;
}();
auto res = lu.solve([&] {
std::vector<mint> b(n);
cin >> b;
return b;
}());
if (!res.first) {
std::cout << "-1\n";
return;
}
auto ker = lu.kernel();
std::cout << ker.size() << "\n" << res.second << "\n";
for (auto &&v : ker) std::cout << v << "\n";
}
} // namespace workspace
int main() { workspace::main(); }#line 1 "test/library-checker/system_of_linear_equations.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/system_of_linear_equations"
#line 2 "src/algebra/linear/lu.hpp"
/**
* @file lu.hpp
* @brief LU decomposition
*/
#include <numeric>
#line 2 "src/algebra/linear/matrix.hpp"
/**
* @file matrix.hpp
* @brief Matrix
* @date 2021-02-15
*
*
*/
#include <cassert>
#include <valarray>
namespace workspace {
/**
* @brief Fixed size matrix.
*
* @tparam _Scalar
* @tparam _Rows Number of rows
* @tparam _Cols Number of columns
*/
template <class _Scalar, std::size_t _Rows = 0, std::size_t _Cols = _Rows>
class matrix {
public:
_Scalar __data[_Rows][_Cols] = {};
using value_type = _Scalar;
using size_type = std::size_t;
constexpr static matrix eye() {
static_assert(_Rows == _Cols);
matrix __e;
for (size_type __d = 0; __d != _Rows; ++__d) __e.__data[__d][__d] = 1;
return __e;
}
constexpr operator decltype((__data))() { return __data; }
constexpr operator decltype(std::declval<const matrix>().__data)
const&() const {
return __data;
}
constexpr auto begin() { return __data; }
constexpr auto begin() const { return __data; }
constexpr auto end() { return __data + _Rows; }
constexpr auto end() const { return __data + _Rows; }
constexpr size_type rows() const { return _Rows; }
constexpr size_type cols() const { return _Cols; }
constexpr auto transpose() const {
matrix<_Scalar, _Cols, _Rows> __t;
for (size_type __r = 0; __r != _Rows; ++__r)
for (size_type __c = 0; __c != _Cols; ++__c)
__t.__data[__c][__r] = __data[__r][__c];
return __t;
}
constexpr matrix operator+() const { return *this; }
constexpr matrix operator-() const {
matrix __cp = *this;
for (auto& __v : __cp.__data)
for (auto& __e : __v) __e = -__e;
return __cp;
}
template <class _Matrix> constexpr matrix& operator+=(const _Matrix& __x) {
auto __m = std::min(_Rows, __x.rows());
auto __n = std::min(_Cols, __x.cols());
for (size_type __r = 0; __r != __m; ++__r)
for (size_type __c = 0; __c != __n; ++__c)
__data[__r][__c] += __x[__r][__c];
return *this;
}
template <class _Matrix>
constexpr matrix operator+(const _Matrix& __x) const {
return matrix(*this) += __x;
}
template <class _Matrix> constexpr matrix& operator-=(const _Matrix& __x) {
auto __m = std::min(_Rows, __x.rows());
auto __n = std::min(_Cols, __x.cols());
for (size_type __r = 0; __r != __m; ++__r)
for (size_type __c = 0; __c != __n; ++__c)
__data[__r][__c] -= __x[__r][__c];
return *this;
}
template <class _Matrix>
constexpr matrix operator-(const _Matrix& __x) const {
return matrix(*this) -= __x;
}
template <class _Scalar2>
constexpr matrix& operator*=(const matrix<_Scalar2, _Cols, _Cols>& __x) {
if (this == &__x) return operator=(operator*(__x));
for (auto& __r : __data) {
_Scalar __tmp[_Cols] = {};
auto __v = *__x.__data;
for (auto& __w : __tmp) {
auto __i = __v++;
for (const auto& __e : __r) __w += __e * *__i, __i += _Cols;
}
auto __w = __tmp;
for (auto& __e : __r) __e = std::move(*__w++);
}
return *this;
}
template <class _Scalar2, size_type _Rows2, size_type _Cols2>
constexpr auto operator*(const matrix<_Scalar2, _Rows2, _Cols2>& __x) const {
matrix<typename std::common_type<_Scalar, _Scalar2>::type, _Rows, _Cols2>
__m;
auto __w = *__m.__data;
for (const auto& __r : __data)
for (auto __v = *__x.__data, __v_end = __v + _Cols2; __v != __v_end;
++__w) {
auto __i = __v++;
for (auto __e = __r; __e != __r + std::min(_Cols, _Rows2); ++__e)
*__w += *__e * *__i, __i += _Cols2;
}
return __m;
}
template <class _Matrix>
constexpr
typename std::enable_if<!std::is_convertible<_Matrix, value_type>::value,
matrix<_Scalar>>::type
operator*(const _Matrix& __x) const {
matrix<_Scalar> __m(_Rows, __x.cols());
for (size_type __r = 0; __r != _Rows; ++__r)
for (size_type __i = 0; __i != __x.cols(); ++__i)
for (size_type __c = 0; __c != std::min(_Cols, __x.rows()); ++__c)
__m[__r][__i] += __data[__r][__c] * __x[__c][__i];
return __m;
}
constexpr matrix& operator*=(const value_type& __x) {
for (auto& __v : __data)
for (auto& __e : __v) __e *= __x;
return *this;
}
constexpr matrix operator*(const value_type& __x) const {
return matrix(*this) *= __x;
}
constexpr matrix& operator/=(const value_type& __x) {
assert(__x != value_type(0));
for (auto& __v : __data)
for (auto& __e : __v) __e /= __x;
return *this;
}
constexpr matrix operator/(const value_type& __x) const {
return matrix(*this) /= __x;
}
template <class _Int> constexpr matrix pow(_Int __e) const {
assert(0 <= __e);
matrix __m = eye();
for (matrix __cp = *this; __e; __cp *= __cp, __e >>= 1)
if (__e & 1) __m *= __cp;
return __m;
}
template <class _Os>
constexpr friend _Os& operator<<(_Os& __os, const matrix& __x) {
for (auto __i = __x.begin(); __i != __x.end(); ++__i, __os << '\n')
for (size_type __c = 0; __c != _Cols; ++__c)
__c ? void(__os << ' ') : (void)0, __os << *(*__i + __c);
return __os;
}
}; // namespace workspace
/**
* @brief Dynamic matrix.
*
* @tparam _Scalar
* @tparam _Rows Number of rows
* @tparam _Cols Number of columns
*/
template <class _Scalar>
class matrix<_Scalar, 0, 0> : public std::valarray<std::valarray<_Scalar>> {
using base = std::valarray<std::valarray<_Scalar>>;
using row_type = typename base::value_type;
public:
using value_type = _Scalar;
using size_type = std::size_t;
using base::operator[];
static matrix eye(size_type __n) {
matrix __e(__n, __n);
for (size_type __d = 0; __d != __n; ++__d) __e[__d][__d] = 1;
return __e;
}
matrix() = default;
matrix(size_type __n) : matrix(__n, __n) {}
matrix(size_type __m, size_type __n) : base(row_type(__n), __m) {}
template <class _Tp, typename = typename std::enable_if<
std::is_constructible<base, _Tp>::value &&
!std::is_constructible<size_type, _Tp>::value>::type>
matrix(_Tp&& __x) : base(__x) {}
matrix(std::initializer_list<row_type> __x) : base(__x) {}
size_type rows() const { return base::size(); }
size_type cols() const { return rows() ? operator[](0).size() : 0; }
matrix transpose() const {
matrix __t(cols(), rows());
for (size_type __r = 0; __r != rows(); ++__r)
for (size_type __c = 0; __c != cols(); ++__c)
__t[__c][__r] = operator[](__r)[__c];
return __t;
}
void resize(size_type __m, size_type __n) {
matrix __t(__m, __n);
if (rows() < __m) __m = rows();
if (cols() < __n) __n = cols();
for (size_type __r = 0; __r != __m; ++__r)
for (size_type __c = 0; __c != __n; ++__c)
__t[__r][__c] = std::move(operator[](__r)[__c]);
base::swap(__t);
}
// binary operators {{
template <class _Matrix, typename = void>
struct is_valarray_based : std::false_type {};
template <class _Matrix>
struct is_valarray_based<
_Matrix,
typename std::enable_if<std::is_same<
row_type, typename std::decay<decltype(
std::declval<_Matrix>()[0])>::type>::value>::type>
: std::true_type {};
template <class _Matrix>
typename std::enable_if<!std::is_convertible<_Matrix, value_type>::value,
matrix&>::type
operator*=(_Matrix&& __x) {
return *this = operator*(std::forward<_Matrix>(__x));
}
template <class _Matrix>
typename std::enable_if<!std::is_convertible<_Matrix, value_type>::value,
matrix>::type
operator*(const _Matrix& __x) const {
matrix __m(rows(), __x.cols());
if constexpr (is_valarray_based<_Matrix>::value)
for (size_type __r = 0; __r != rows(); ++__r)
for (size_type __c = 0; __c != std::min(cols(), __x.rows()); ++__c)
__m[__r] += operator[](__r)[__c] * __x[__c];
else
for (size_type __r = 0; __r != rows(); ++__r)
for (size_type __i = 0; __i != __x.cols(); ++__i)
for (size_type __c = 0; __c != std::min(cols(), __x.rows()); ++__c)
__m[__r][__i] += operator[](__r)[__c] * __x[__c][__i];
return __m;
}
matrix& operator*=(const value_type& __x) {
for (size_type __r = 0; __r != rows(); ++__r)
operator[](__r).operator*=(__x);
return *this;
}
matrix operator*(const value_type& __x) const { return matrix(*this) *= __x; }
friend matrix operator*(const value_type& __x, matrix __i) {
for (size_type __r = 0; __r != __i.rows(); ++__r)
__i.operator[](__r) = __x * __i.operator[](__r);
return __i;
}
matrix& operator/=(const value_type& __x) {
assert(__x != value_type(0));
for (size_type __r = 0; __r != rows(); ++__r)
operator[](__r).operator/=(__x);
return *this;
}
matrix operator/(const value_type& __x) const { return matrix(*this) /= __x; }
// }} binary operators
template <class _Int> matrix pow(_Int __e) const {
assert(0 <= __e);
matrix __m = eye(rows());
for (matrix __cp = *this; __e; __cp *= __cp, __e >>= 1)
if (__e & 1) __m *= __cp;
return __m;
}
// template <class _Is> friend _Is& operator>>(_Is& __is, matrix& __x) {
// for (size_type __r = 0; __r != __x.rows(); ++__r)
// for (size_type __c = 0; __c != __x.cols(); ++__c)
// __is >> __x.operator[](__r).operator[](__c);
// return __is;
// }
template <class _Os> friend _Os& operator<<(_Os& __os, const matrix& __x) {
for (size_type __r = 0; __r != __x.rows(); ++__r, __os << '\n')
for (size_type __c = 0; __c != __x.cols(); ++__c)
__c ? void(__os << ' ') : (void)0,
__os << __x.operator[](__r).operator[](__c);
return __os;
}
};
} // namespace workspace
#line 11 "src/algebra/linear/lu.hpp"
namespace workspace {
template <class _Matrix> class lu_decomposition : public _Matrix {
public:
using value_type = typename _Matrix::value_type;
using size_type = typename _Matrix::size_type;
lu_decomposition() = default;
lu_decomposition(const _Matrix &__x) : _Matrix(__x) { run(); }
lu_decomposition(_Matrix &&__x) : _Matrix(std::move(__x)) { run(); }
protected:
std::vector<size_type> __perm, __pivots;
bool sgn;
void run() {
__perm.resize(_Matrix::rows());
std::iota(__perm.begin(), __perm.end(), 0);
sgn = false;
__pivots.clear();
for (size_type __c = 0;
__c != _Matrix::cols() && __pivots.size() != _Matrix::rows(); ++__c) {
auto __max = (*this)[__pivots.size()][__c];
auto __pos = __pivots.size();
if constexpr (std::is_floating_point<
value_type>::value) { // Find the biggest absolute
// value in the column.
for (size_type __r = __pivots.size() + 1; __r != _Matrix::rows(); ++__r)
if (std::abs(__max) < std::abs((*this)[__r][__c]))
__max = (*this)[__pos = __r][__c];
}
else if (__max ==
static_cast<value_type>(
0)) // Find the first non-zero element in the column.
for (size_type __r = __pivots.size() + 1; __r != _Matrix::rows(); ++__r)
if ((__max = (*this)[__r][__c]) != static_cast<value_type>(0)) {
__pos = __r;
break;
}
if (__pos != __pivots.size()) { // Swap 2 rows.
sgn = !sgn;
std::swap(__perm[__pos], __perm[__pivots.size()]);
std::swap((*this)[__pos], (*this)[__pivots.size()]);
}
if (__max != static_cast<value_type>(0)) { // Forward elimination
for (size_type __r = __pivots.size() + 1; __r != _Matrix::rows();
++__r) {
auto __m = (*this)[__r][__c] / __max;
(*this)[__r][__c] = 0;
(*this)[__r][__pivots.size()] = __m;
for (size_type __i = __c + 1; __i != _Matrix::cols(); ++__i)
(*this)[__r][__i] -= (*this)[__pivots.size()][__i] * __m;
}
__pivots.emplace_back(__c);
}
}
}
public:
size_type rank() const { return __pivots.size(); }
value_type det() const {
assert(_Matrix::rows() == _Matrix::cols());
value_type __d = sgn ? -1 : 1;
for (size_type __i = 0; __i != _Matrix::rows(); ++__i)
__d *= (*this)[__i][__i];
return __d;
}
_Matrix lower() const;
_Matrix upper() const;
_Matrix inverse() const {
assert(_Matrix::rows() == _Matrix::cols());
_Matrix __inv;
// add solve(e_i) to __inv for i=0, ..., rows()
return __inv;
}
// O(dim(ker) * size)
_Matrix kernel() const {
_Matrix __ker(_Matrix::cols() - rank(), _Matrix::cols());
for (size_type __c = 0, __i = 0; __c != _Matrix::cols(); ++__c) {
if (__i != __pivots.size() && __pivots[__i] == __c) {
++__i;
continue;
}
auto &__v = __ker[__c - __i];
__v[__c] = 1;
for (size_type __r = 0; __r != __i; ++__r) __v[__r] = -(*this)[__r][__c];
for (size_type __j = __i; __j--;) {
auto __x = __v[__j] / (*this)[__j][__pivots[__j]];
__v[__j] = 0;
__v[__pivots[__j]] = __x;
for (size_type __r = 0; __r != __j; ++__r)
__v[__r] -= (*this)[__r][__pivots[__j]] * __x;
}
}
return __ker;
}
template <class _Vec> std::pair<bool, _Vec> solve(const _Vec &__b) const {
assert(!(__b.size() < _Matrix::rows()));
// Solution
_Vec __y(_Matrix::rows()), __x(_Matrix::cols());
// Backward substitution with L
for (size_type __c = 0; __c != _Matrix::rows(); ++__c) {
__y[__c] += __b[__perm[__c]];
for (size_type __r = __c + 1; __r != _Matrix::rows(); ++__r)
__y[__r] -= __y[__c] * (*this)[__r][__c];
}
// Backward substitution with U
for (size_type __i = rank(); __i != _Matrix::rows(); ++__i)
if (__y[__i] != static_cast<value_type>(0))
return std::make_pair(false, __x);
for (size_type __i = rank(); __i--;) {
auto __c = __pivots[__i];
__x[__c] = __y[__i] / (*this)[__i][__c];
for (size_type __r = 0; __r != __i; ++__r)
__y[__r] -= __x[__c] * (*this)[__r][__c];
}
return std::make_pair(true, __x);
}
};
} // namespace workspace
#line 2 "src/algebra/modint.hpp"
/**
* @file modint.hpp
* @brief Modular Arithmetic
*/
#line 9 "src/algebra/modint.hpp"
#include <iostream>
#include <vector>
#line 2 "src/number_theory/sqrt_mod.hpp"
/**
* @file sqrt_mod.hpp
* @brief Tonelli-Shanks Algorithm
*/
#line 2 "src/number_theory/pow_mod.hpp"
/**
* @file mod_pow.hpp
* @brief Modular Exponentiation
*/
#line 9 "src/number_theory/pow_mod.hpp"
#line 2 "src/utils/sfinae.hpp"
/**
* @file sfinae.hpp
* @brief SFINAE
*/
#include <cstdint>
#include <iterator>
#include <type_traits>
#ifndef __INT128_DEFINED__
#ifdef __SIZEOF_INT128__
#define __INT128_DEFINED__ 1
#else
#define __INT128_DEFINED__ 0
#endif
#endif
namespace std {
#if __INT128_DEFINED__
template <> struct make_signed<__uint128_t> { using type = __int128_t; };
template <> struct make_signed<__int128_t> { using type = __int128_t; };
template <> struct make_unsigned<__uint128_t> { using type = __uint128_t; };
template <> struct make_unsigned<__int128_t> { using type = __uint128_t; };
template <> struct is_signed<__uint128_t> : std::false_type {};
template <> struct is_signed<__int128_t> : std::true_type {};
template <> struct is_unsigned<__uint128_t> : std::true_type {};
template <> struct is_unsigned<__int128_t> : std::false_type {};
#endif
} // namespace std
namespace workspace {
template <class Tp, class... Args> struct variadic_front { using type = Tp; };
template <class... Args> struct variadic_back;
template <class Tp> struct variadic_back<Tp> { using type = Tp; };
template <class Tp, class... Args> struct variadic_back<Tp, Args...> {
using type = typename variadic_back<Args...>::type;
};
template <class type, template <class> class trait>
using enable_if_trait_type = typename std::enable_if<trait<type>::value>::type;
/**
* @brief Return type of subscripting ( @c [] ) access.
*/
template <class _Tp>
using subscripted_type =
typename std::decay<decltype(std::declval<_Tp&>()[0])>::type;
template <class Container>
using element_type = typename std::decay<decltype(*std::begin(
std::declval<Container&>()))>::type;
template <class _Tp, class = void> struct has_begin : std::false_type {};
template <class _Tp>
struct has_begin<
_Tp, std::__void_t<decltype(std::begin(std::declval<const _Tp&>()))>>
: std::true_type {
using type = decltype(std::begin(std::declval<const _Tp&>()));
};
template <class _Tp, class = void> struct has_size : std::false_type {};
template <class _Tp>
struct has_size<_Tp, std::__void_t<decltype(std::size(std::declval<_Tp>()))>>
: std::true_type {};
template <class _Tp, class = void> struct has_resize : std::false_type {};
template <class _Tp>
struct has_resize<_Tp, std::__void_t<decltype(std::declval<_Tp>().resize(
std::declval<size_t>()))>> : std::true_type {};
template <class _Tp, class = void> struct has_mod : std::false_type {};
template <class _Tp>
struct has_mod<_Tp, std::__void_t<decltype(_Tp::mod)>> : std::true_type {};
template <class _Tp, class = void> struct is_integral_ext : std::false_type {};
template <class _Tp>
struct is_integral_ext<
_Tp, typename std::enable_if<std::is_integral<_Tp>::value>::type>
: std::true_type {};
#if __INT128_DEFINED__
template <> struct is_integral_ext<__int128_t> : std::true_type {};
template <> struct is_integral_ext<__uint128_t> : std::true_type {};
#endif
#if __cplusplus >= 201402
template <class _Tp>
constexpr static bool is_integral_ext_v = is_integral_ext<_Tp>::value;
#endif
template <typename _Tp, typename = void> struct multiplicable_uint {
using type = uint_least32_t;
};
template <typename _Tp>
struct multiplicable_uint<
_Tp,
typename std::enable_if<(2 < sizeof(_Tp)) &&
(!__INT128_DEFINED__ || sizeof(_Tp) <= 4)>::type> {
using type = uint_least64_t;
};
#if __INT128_DEFINED__
template <typename _Tp>
struct multiplicable_uint<_Tp,
typename std::enable_if<(4 < sizeof(_Tp))>::type> {
using type = __uint128_t;
};
#endif
template <typename _Tp> struct multiplicable_int {
using type =
typename std::make_signed<typename multiplicable_uint<_Tp>::type>::type;
};
template <typename _Tp> struct multiplicable {
using type = std::conditional_t<
is_integral_ext<_Tp>::value,
std::conditional_t<std::is_signed<_Tp>::value,
typename multiplicable_int<_Tp>::type,
typename multiplicable_uint<_Tp>::type>,
_Tp>;
};
template <class> struct first_arg { using type = void; };
template <class _R, class _Tp, class... _Args>
struct first_arg<_R(_Tp, _Args...)> {
using type = _Tp;
};
template <class _R, class _Tp, class... _Args>
struct first_arg<_R (*)(_Tp, _Args...)> {
using type = _Tp;
};
template <class _G, class _R, class _Tp, class... _Args>
struct first_arg<_R (_G::*)(_Tp, _Args...)> {
using type = _Tp;
};
template <class _G, class _R, class _Tp, class... _Args>
struct first_arg<_R (_G::*)(_Tp, _Args...) const> {
using type = _Tp;
};
template <class _Tp, class = void> struct parse_compare : first_arg<_Tp> {};
template <class _Tp>
struct parse_compare<_Tp, std::__void_t<decltype(&_Tp::operator())>>
: first_arg<decltype(&_Tp::operator())> {};
template <class _Container, class = void> struct get_dimension {
static constexpr size_t value = 0;
};
template <class _Container>
struct get_dimension<_Container,
std::enable_if_t<has_begin<_Container>::value>> {
static constexpr size_t value =
1 + get_dimension<typename std::iterator_traits<
typename has_begin<_Container>::type>::value_type>::value;
};
} // namespace workspace
#line 11 "src/number_theory/pow_mod.hpp"
namespace workspace {
/**
* @brief Compile time modular exponentiation.
*
* @param __x
* @param __n Exponent
* @param __mod Modulus
* @return
*/
template <class _Tp>
constexpr std::enable_if_t<(is_integral_ext<_Tp>::value), _Tp> pow_mod(
_Tp __x, _Tp __n, _Tp __mod) noexcept {
assert(__mod > 0);
using mul_type = typename multiplicable_uint<_Tp>::type;
if ((__x %= __mod) < 0) __x += __mod;
mul_type __y{1};
while (__n) {
if (__n & 1) (__y *= __x) %= __mod;
__x = (mul_type)__x * __x % __mod;
__n >>= 1;
}
return __y;
};
} // namespace workspace
#line 10 "src/number_theory/sqrt_mod.hpp"
namespace workspace {
/**
* @brief Compile time modular square root.
*
* @param __x
* @param __mod Modulus
* @return One if it exists. Otherwise -1.
*/
template <class _Tp>
constexpr std::enable_if_t<(is_integral_ext<_Tp>::value), _Tp> sqrt_mod(
_Tp __x, _Tp __mod) noexcept {
assert(__mod > 0);
using mul_type = typename multiplicable_uint<_Tp>::type;
if ((__x %= __mod) < 0) __x += __mod;
if (!__x) return 0;
if (__mod == 2) return __x;
if (pow_mod(__x, __mod >> 1, __mod) != 1) return -1;
_Tp __z = __builtin_ctz(__mod - 1), __q = __mod >> __z;
mul_type __a = pow_mod(__x, (__q + 1) >> 1, __mod), __b = 2;
while (pow_mod<_Tp>(__b, __mod >> 1, __mod) == 1) ++__b;
__b = pow_mod<_Tp>(__b, __q, __mod);
_Tp __shift = 0;
for (auto __r = __a * __a % __mod * pow_mod(__x, __mod - 2, __mod) % __mod;
__r != 1; (__r *= (__b *= __b) %= __mod) %= __mod) {
auto __bsf = __z;
for (auto __e = __r; __e != 1; --__bsf) (__e *= __e) %= __mod;
while (++__shift != __bsf) (__b *= __b) %= __mod;
(__a *= __b) %= __mod;
}
return __a;
};
} // namespace workspace
#line 14 "src/algebra/modint.hpp"
namespace workspace {
namespace _modint_impl {
template <auto _Mod, unsigned _Storage> struct modint {
static_assert(is_integral_ext<decltype(_Mod)>::value,
"_Mod must be integral type.");
using mod_type = std::make_signed_t<typename std::conditional<
0 < _Mod, std::add_const_t<decltype(_Mod)>, decltype(_Mod)>::type>;
using value_type = std::decay_t<mod_type>;
using reference = value_type &;
using const_reference = value_type const &;
using mul_type = typename multiplicable_uint<value_type>::type;
static mod_type mod; // Modulus.
static unsigned storage;
private:
template <class _Tp>
using modint_if = std::enable_if_t<is_integral_ext<_Tp>::value, modint>;
value_type value = 0; // within [0, mod).
struct direct_ctor_t {};
constexpr static direct_ctor_t direct_ctor_tag{};
// Direct constructor
template <class _Tp>
constexpr modint(_Tp __n, direct_ctor_t) noexcept : value(__n) {}
public:
constexpr modint() noexcept = default;
template <class _Tp, class = std::enable_if_t<
std::is_convertible<_Tp, value_type>::value>>
constexpr modint(_Tp __n) noexcept
: value((__n %= mod) < _Tp(0) ? static_cast<value_type>(__n) + mod
: static_cast<value_type>(__n)) {}
constexpr modint(bool __n) noexcept : value(__n) {}
constexpr operator reference() noexcept { return value; }
constexpr operator const_reference() const noexcept { return value; }
// unary operators {{
constexpr modint operator++(int) noexcept {
modint __t{*this};
operator++();
return __t;
}
constexpr modint operator--(int) noexcept {
modint __t{*this};
operator--();
return __t;
}
constexpr modint &operator++() noexcept {
if (++value == mod) value = 0;
return *this;
}
constexpr modint &operator--() noexcept {
if (!value)
value = mod - 1;
else
--value;
return *this;
}
constexpr modint operator+() const noexcept { return *this; }
constexpr modint operator-() const noexcept {
return {value ? mod - value : 0, direct_ctor_tag};
}
// }} unary operators
// operator+= {{
constexpr modint &operator+=(const modint &__x) noexcept {
if ((value += __x.value) >= mod) value -= mod;
return *this;
}
template <class _Tp> constexpr modint_if<_Tp> &operator+=(_Tp __x) noexcept {
__x %= mod, value += __x;
if (value < 0)
value += mod;
else if (value >= mod)
value -= mod;
return *this;
}
// }} operator+=
// operator+ {{
template <class _Tp>
constexpr modint_if<_Tp> operator+(_Tp const &__x) const noexcept {
return modint{*this} += __x;
}
constexpr modint operator+(modint __x) const noexcept { return __x += *this; }
template <class _Tp>
constexpr friend modint_if<_Tp> operator+(_Tp const &__x,
modint __y) noexcept {
return __y += __x;
}
// }} operator+
// operator-= {{
constexpr modint &operator-=(const modint &__x) noexcept {
if ((value -= __x.value) < 0) value += mod;
return *this;
}
template <class _Tp> constexpr modint_if<_Tp> &operator-=(_Tp __x) noexcept {
__x %= mod, value -= __x;
if (value < 0)
value += mod;
else if (value >= mod)
value -= mod;
return *this;
}
// }} operator-=
// operator- {{
template <class _Tp>
constexpr modint_if<_Tp> operator-(_Tp const &__x) const noexcept {
return modint{*this} -= __x;
}
constexpr modint operator-(const modint &__x) const noexcept {
return modint{*this} -= __x;
}
template <class _Tp>
constexpr friend modint_if<_Tp> operator-(_Tp __x,
const modint &__y) noexcept {
if (((__x -= __y.value) %= mod) < 0) __x += mod;
return {__x, direct_ctor_tag};
}
// }} operator-
// operator*= {{
constexpr modint &operator*=(const modint &__x) noexcept {
value =
static_cast<value_type>(value * static_cast<mul_type>(__x.value) % mod);
return *this;
}
template <class _Tp> constexpr modint_if<_Tp> &operator*=(_Tp __x) noexcept {
value = static_cast<value_type>(
value * ((__x %= mod) < 0 ? mul_type(__x + mod) : mul_type(__x)) % mod);
return *this;
}
// }} operator*=
// operator* {{
constexpr modint operator*(const modint &__x) const noexcept {
return {static_cast<mul_type>(value) * __x.value % mod, direct_ctor_tag};
}
template <class _Tp>
constexpr modint_if<_Tp> operator*(_Tp __x) const noexcept {
__x %= mod;
if (__x < 0) __x += mod;
return {static_cast<mul_type>(value) * __x % mod, direct_ctor_tag};
}
template <class _Tp>
constexpr friend modint_if<_Tp> operator*(_Tp __x,
const modint &__y) noexcept {
__x %= mod;
if (__x < 0) __x += mod;
return {static_cast<mul_type>(__x) * __y.value % mod, direct_ctor_tag};
}
// }} operator*
protected:
static value_type _mem(value_type __x) {
static std::vector<value_type> __m{0, 1};
static value_type __i = (__m.reserve(storage), 1);
while (__i < __x) {
++__i;
__m.emplace_back(mod - mul_type(mod / __i) * __m[mod % __i] % mod);
}
return __m[__x];
}
static value_type _div(mul_type __r, value_type __x) noexcept {
assert(__x != value_type(0));
if (!__r) return 0;
std::make_signed_t<value_type> __v{};
bool __neg = __x < 0 ? __x = -__x, true : false;
if (static_cast<decltype(storage)>(__x) < storage)
__v = _mem(__x);
else {
value_type __y{mod}, __u{1}, __t;
while (__x)
__t = __y / __x, __y ^= __x ^= (__y -= __t * __x) ^= __x,
__v ^= __u ^= (__v -= __t * __u) ^= __u;
if (__y < 0) __neg ^= 1;
}
if (__neg)
__v = 0 < __v ? mod - __v : -__v;
else if (__v < 0)
__v += mod;
return __r == mul_type(1) ? static_cast<value_type>(__v)
: static_cast<value_type>(__r * __v % mod);
}
public:
static void reserve(unsigned __n) noexcept {
if (storage < __n) storage = __n;
}
// operator/= {{
constexpr modint &operator/=(const modint &__x) noexcept {
if (value) value = _div(value, __x.value);
return *this;
}
template <class _Tp> constexpr modint_if<_Tp> &operator/=(_Tp __x) noexcept {
if (value) value = _div(value, __x %= mod);
return *this;
}
// }} operator/=
// operator/ {{
constexpr modint operator/(const modint &__x) const noexcept {
if (!value) return {};
return {_div(value, __x.value), direct_ctor_tag};
}
template <class _Tp>
constexpr modint_if<_Tp> operator/(_Tp __x) const noexcept {
if (!value) return {};
return {_div(value, __x %= mod), direct_ctor_tag};
}
template <class _Tp>
constexpr friend modint_if<_Tp> operator/(_Tp __x,
const modint &__y) noexcept {
if (!__x) return {};
if ((__x %= mod) < 0) __x += mod;
return {_div(__x, __y.value), direct_ctor_tag};
}
// }} operator/
constexpr modint inv() const noexcept { return _div(1, value); }
template <class _Tp> constexpr modint pow(_Tp __e) const noexcept {
static_assert(not std::is_floating_point<_Tp>::value);
modint __r{mod != 1, direct_ctor_tag};
for (modint __b{__e < _Tp(0) ? __e = -__e, _div(1, value) : value,
direct_ctor_tag};
__e; __e /= 2, __b *= __b)
if (__e % 2) __r *= __b;
return __r;
}
template <class _Tp>
constexpr friend modint pow(modint __b, _Tp __e) noexcept {
static_assert(not std::is_floating_point<_Tp>::value);
if (__e < _Tp(0)) {
__e = -__e;
__b.value = _div(1, __b.value);
}
modint __r{mod != 1, direct_ctor_tag};
for (; __e; __e /= 2, __b *= __b)
if (__e % 2) __r *= __b;
return __r;
}
constexpr modint sqrt() const noexcept {
return {sqrt_mod(value, mod), direct_ctor_tag};
}
friend constexpr modint sqrt(const modint &__x) noexcept {
return {sqrt_mod(__x.value, mod), direct_ctor_tag};
}
friend std::istream &operator>>(std::istream &__is, modint &__x) noexcept {
std::string __s;
__is >> __s;
bool __neg = false;
if (__s.front() == '-') {
__neg = true;
__s.erase(__s.begin());
}
__x = 0;
for (char __c : __s) __x = __x * 10 + (__c - '0');
if (__neg) __x = -__x;
return __is;
}
};
template <auto _Mod, unsigned _Storage>
typename modint<_Mod, _Storage>::mod_type modint<_Mod, _Storage>::mod =
_Mod > 0 ? _Mod : 0;
template <auto _Mod, unsigned _Storage>
unsigned modint<_Mod, _Storage>::storage = _Storage;
} // namespace _modint_impl
constexpr unsigned _modint_default_storage = 1 << 24;
template <auto _Mod, unsigned _Storage = _modint_default_storage,
typename = std::enable_if_t<(_Mod > 0)>>
using modint = _modint_impl::modint<_Mod, _Storage>;
template <unsigned _Id = 0, unsigned _Storage = _modint_default_storage>
using runtime_modint = _modint_impl::modint<-(signed)_Id, _Storage>;
template <unsigned _Id = 0, unsigned _Storage = _modint_default_storage>
using runtime_modint64 = _modint_impl::modint<-(int_least64_t)_Id, _Storage>;
} // namespace workspace
#line 2 "src/utils/io/istream.hpp"
/**
* @file istream.hpp
* @brief Input Stream
*/
#include <cxxabi.h>
#line 12 "src/utils/io/istream.hpp"
#include <tuple>
#line 2 "lib/cxx17"
#line 2 "lib/cxx14"
#ifndef _CXX14_CONSTEXPR
#if __cplusplus >= 201402L
#define _CXX14_CONSTEXPR constexpr
#else
#define _CXX14_CONSTEXPR
#endif
#endif
#line 4 "lib/cxx17"
#ifndef _CXX17_CONSTEXPR
#if __cplusplus >= 201703L
#define _CXX17_CONSTEXPR constexpr
#else
#define _CXX17_CONSTEXPR
#endif
#endif
#ifndef _CXX17_STATIC_ASSERT
#if __cplusplus >= 201703L
#define _CXX17_STATIC_ASSERT static_assert
#else
#define _CXX17_STATIC_ASSERT assert
#endif
#endif
#line 22 "lib/cxx17"
#if __cplusplus < 201703L
namespace std {
/**
* @brief Return the size of a container.
* @param __cont Container.
*/
template <typename _Container>
constexpr auto size(const _Container& __cont) noexcept(noexcept(__cont.size()))
-> decltype(__cont.size()) {
return __cont.size();
}
/**
* @brief Return the size of an array.
*/
template <typename _Tp, size_t _Nm>
constexpr size_t size(const _Tp (&)[_Nm]) noexcept {
return _Nm;
}
/**
* @brief Return whether a container is empty.
* @param __cont Container.
*/
template <typename _Container>
[[nodiscard]] constexpr auto empty(const _Container& __cont) noexcept(
noexcept(__cont.empty())) -> decltype(__cont.empty()) {
return __cont.empty();
}
/**
* @brief Return whether an array is empty (always false).
*/
template <typename _Tp, size_t _Nm>
[[nodiscard]] constexpr bool empty(const _Tp (&)[_Nm]) noexcept {
return false;
}
/**
* @brief Return whether an initializer_list is empty.
* @param __il Initializer list.
*/
template <typename _Tp>
[[nodiscard]] constexpr bool empty(initializer_list<_Tp> __il) noexcept {
return __il.size() == 0;
}
struct monostate {};
} // namespace std
#else
#include <variant>
#endif
#line 16 "src/utils/io/istream.hpp"
namespace workspace {
namespace _istream_impl {
template <class _Tp, typename = void> struct helper {
helper(std::istream &__is, _Tp &__x) {
if _CXX17_CONSTEXPR (has_begin<_Tp &>::value)
for (auto &&__e : __x) helper<std::decay_t<decltype(__e)>>(__is, __e);
else
static_assert(has_begin<_Tp>::value, "istream unsupported type.");
}
};
template <class _Tp>
struct helper<_Tp, std::__void_t<decltype(std::declval<std::istream &>() >>
std::declval<_Tp &>())>> {
helper(std::istream &__is, _Tp &__x) { __is >> __x; }
};
#ifdef __SIZEOF_INT128__
template <> struct helper<__uint128_t, void> {
helper(std::istream &__is, __uint128_t &__x) {
std::string __s;
__is >> __s;
bool __neg = false;
if (__s.front() == '-') __neg = true, __s.erase(__s.begin());
__x = 0;
for (char __d : __s) {
__x *= 10;
__d -= '0';
if (__neg)
__x -= __d;
else
__x += __d;
}
}
};
template <> struct helper<__int128_t, void> {
helper(std::istream &__is, __int128_t &__x) {
std::string __s;
__is >> __s;
bool __neg = false;
if (__s.front() == '-') __neg = true, __s.erase(__s.begin());
__x = 0;
for (char __d : __s) {
__x *= 10;
__d -= '0';
if (__neg)
__x -= __d;
else
__x += __d;
}
}
};
#endif // INT128
template <class _T1, class _T2> struct helper<std::pair<_T1, _T2>> {
helper(std::istream &__is, std::pair<_T1, _T2> &__x) {
helper<_T1>(__is, __x.first), helper<_T2>(__is, __x.second);
}
};
template <class... _Tp> struct helper<std::tuple<_Tp...>> {
helper(std::istream &__is, std::tuple<_Tp...> &__x) { iterate(__is, __x); }
private:
template <class _Tuple, size_t _Nm = 0>
void iterate(std::istream &__is, _Tuple &__x) {
if _CXX17_CONSTEXPR (_Nm != std::tuple_size<_Tuple>::value) {
helper<typename std::tuple_element<_Nm, _Tuple>::type>(
__is, std::get<_Nm>(__x)),
iterate<_Tuple, _Nm + 1>(__is, __x);
}
}
};
} // namespace _istream_impl
/**
* @brief A wrapper class for std::istream.
*/
class istream : public std::istream {
public:
/**
* @brief Wrapped operator.
*/
template <typename _Tp> istream &operator>>(_Tp &__x) {
_istream_impl::helper<_Tp>(*this, __x);
if (std::istream::fail()) {
static auto once = atexit([] {
std::cerr << "\n\033[43m\033[30mwarning: failed to read \'"
<< abi::__cxa_demangle(typeid(_Tp).name(), 0, 0, 0)
<< "\'.\033[0m\n\n";
});
assert(!once);
}
return *this;
}
};
decltype(auto) cin = static_cast<istream &>(std::cin);
} // namespace workspace
#line 2 "src/utils/io/ostream.hpp"
/**
* @file ostream.hpp
* @brief Output Stream
*/
#line 9 "src/utils/io/ostream.hpp"
#line 11 "src/utils/io/ostream.hpp"
namespace workspace {
template <class _Os> struct is_ostream {
template <typename... _Args>
static std::true_type __test(std::basic_ostream<_Args...> *);
static std::false_type __test(void *);
constexpr static bool value = decltype(__test(std::declval<_Os *>()))::value;
};
template <class _Os>
using ostream_ref =
typename std::enable_if<is_ostream<_Os>::value, _Os &>::type;
/**
* @brief Stream insertion operator for C-style array.
*
* @param __os Output stream
* @param __a Array
* @return Reference to __os.
*/
template <class _Os, class _Tp, size_t _Nm>
typename std::enable_if<bool(sizeof(_Tp) > 2), ostream_ref<_Os>>::type
operator<<(_Os &__os, const _Tp (&__a)[_Nm]) {
if _CXX17_CONSTEXPR (_Nm) {
__os << *__a;
for (auto __i = __a + 1, __e = __a + _Nm; __i != __e; ++__i)
__os << ' ' << *__i;
}
return __os;
}
/**
* @brief Stream insertion operator for std::array.
*
* @param __os Output stream
* @param __a Array
* @return Reference to __os.
*/
template <class _Os, class _Tp, size_t _Nm>
ostream_ref<_Os> operator<<(_Os &__os, const std::array<_Tp, _Nm> &__a) {
if _CXX17_CONSTEXPR (_Nm) {
__os << __a[0];
for (size_t __i = 1; __i != _Nm; ++__i) __os << ' ' << __a[__i];
}
return __os;
}
/**
* @brief Stream insertion operator for std::pair.
*
* @param __os Output stream
* @param __p Pair
* @return Reference to __os.
*/
template <class _Os, class _T1, class _T2>
ostream_ref<_Os> operator<<(_Os &__os, const std::pair<_T1, _T2> &__p) {
return __os << __p.first << ' ' << __p.second;
}
/**
* @brief Stream insertion operator for std::tuple.
*
* @param __os Output stream
* @param __t Tuple
* @return Reference to __os.
*/
template <class _Os, class _Tp, size_t _Nm = 0>
typename std::enable_if<bool(std::tuple_size<_Tp>::value + 1),
ostream_ref<_Os>>::type
operator<<(_Os &__os, const _Tp &__t) {
if _CXX17_CONSTEXPR (_Nm != std::tuple_size<_Tp>::value) {
if _CXX17_CONSTEXPR (_Nm) __os << ' ';
__os << std::get<_Nm>(__t);
operator<<<_Os, _Tp, _Nm + 1>(__os, __t);
}
return __os;
}
template <class _Os, class _Container,
typename = decltype(std::begin(std::declval<_Container>()))>
typename std::enable_if<
!std::is_convertible<std::decay_t<_Container>, std::string>::value &&
!std::is_convertible<std::decay_t<_Container>, char *>::value,
ostream_ref<_Os>>::type
operator<<(_Os &__os, const _Container &__cont) {
bool __h = true;
for (auto &&__e : __cont) __h ? __h = 0 : (__os << ' ', 0), __os << __e;
return __os;
}
#ifdef __SIZEOF_INT128__
/**
* @brief Stream insertion operator for __int128_t.
*
* @param __os Output Stream
* @param __x 128-bit integer
* @return Reference to __os.
*/
template <class _Os> ostream_ref<_Os> operator<<(_Os &__os, __int128_t __x) {
if (!__x) return __os << '0';
if (__x < 0) __os << '-';
char __s[40], *__p = __s;
while (__x) {
auto __d = __x % 10;
*__p++ = '0' + (__x < 0 ? -__d : __d);
__x /= 10;
}
*__p = 0;
for (char *__t = __s; __t < --__p; ++__t) *__t ^= *__p ^= *__t ^= *__p;
return __os << __s;
}
/**
* @brief Stream insertion operator for __uint128_t.
*
* @param __os Output Stream
* @param __x 128-bit unsigned integer
* @return Reference to __os.
*/
template <class _Os> ostream_ref<_Os> operator<<(_Os &__os, __uint128_t __x) {
if (!__x) return __os << '0';
char __s[40], *__p = __s;
while (__x) *__p++ = '0' + __x % 10, __x /= 10;
*__p = 0;
for (char *__t = __s; __t < --__p; ++__t) *__t ^= *__p ^= *__t ^= *__p;
return __os << __s;
}
#endif
} // namespace workspace
#line 7 "test/library-checker/system_of_linear_equations.test.cpp"
namespace workspace {
using mint = modint<998244353>;
using mat = matrix<mint>;
void main() {
// start here!
size_t n, m;
cin >> n >> m;
lu_decomposition lu = [&] {
mat a(n, m);
cin >> a;
return a;
}();
auto res = lu.solve([&] {
std::vector<mint> b(n);
cin >> b;
return b;
}());
if (!res.first) {
std::cout << "-1\n";
return;
}
auto ker = lu.kernel();
std::cout << ker.size() << "\n" << res.second << "\n";
for (auto &&v : ker) std::cout << v << "\n";
}
} // namespace workspace
int main() { workspace::main(); }