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Polynomial Interpolation
(src/algebra/interpolation.hpp)
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- Last update: 2022-05-07 02:21:47+09:00
- Include:
#include "src/algebra/interpolation.hpp"
Depends on
- Complex Number
(src/algebra/complex.hpp)
- Fast Fourier Transform
(src/algebra/fft.hpp)
- Polynomial
(src/algebra/polynomial.hpp)
- Operation Traits
(src/algebra/system/operation.hpp)
- Extended Euclidean Algorithm
(src/number_theory/ext_gcd.hpp)
- Primitive Root
(src/number_theory/primitive_root.hpp)
- SFINAE
(src/utils/sfinae.hpp)
Verified with
Code
#pragma once
/**
* @file interpolation.hpp
* @brief Polynomial Interpolation
*/
#include <vector>
#include "polynomial.hpp"
#include "system/operation.hpp"
namespace workspace {
/**
* @brief Lagrange interpolation on the points 0, 1, 2, ...
* @param __x Point to evaluate at.
* @param __first
* @param __last
*/
template <class _Tp, class _InputIter,
typename = std::_RequireInputIter<_InputIter>,
typename = require_arithmetic<
typename std::iterator_traits<_InputIter>::value_type>>
auto interpolate(_Tp __x, _InputIter __first, _InputIter __last) noexcept {
using value_type = typename std::iterator_traits<_InputIter>::value_type;
std::vector<value_type> __f(__first, __last);
decltype(__f.size()) __k = 0;
value_type __d = 1;
while (__k != __f.size()) __k += 1, __d *= value_type{__k};
__d = value_type{1} / __d;
value_type __c = __k & 1 ? -__d : __d;
auto __i = __f.begin();
while (__i != __f.end()) {
__c *= -value_type{__k}, __k -= 1;
*__i++ *= __c;
__c *= __x, __x -= 1;
}
__c = __d, __k = __f.size();
value_type __y = 0;
while (__i != __f.begin()) {
__c *= value_type{__k}, __k -= 1;
__y += *--__i * __c;
__x += 1, __c *= __x;
}
return __y;
}
/**
* @brief Polynomial interpolation in O(n log(n)^2) time.
* @param __first
* @param __last
*/
template <class _InputIter, typename = std::_RequireInputIter<_InputIter>>
auto interpolate(_InputIter __first, _InputIter __last) noexcept {
size_t __n = std::distance(__first, __last);
auto [__1, __2] = typename std::iterator_traits<_InputIter>::value_type{};
using poly = polynomial<decltype(__1)>;
if (!__n) return poly{};
struct node {
poly __all, __lack;
};
auto __tree = new node[__n << 1];
auto __iter = __first;
for (size_t __i = 0; __i != __n; ++__i) {
auto&& [__a, __b] = *__iter++;
__tree[__i + __n].__all = {-__a, 1}, __tree[__i + __n].__lack = {1};
}
for (size_t __i = __n; --__i;)
__tree[__i].__all = __tree[__i << 1].__all * __tree[__i << 1 | 1].__all,
__tree[__i].__lack =
__tree[__i << 1].__all * std::move(__tree[__i << 1 | 1].__lack) +
__tree[__i << 1 | 1].__all * std::move(__tree[__i << 1].__lack);
for (size_t __i = 2; __i != __n << 1; __i += 2)
__tree[__i].__lack = __tree[__i >> 1].__lack % __tree[__i].__all,
__tree[__i | 1].__lack =
std::move(__tree[__i >> 1].__lack %= __tree[__i | 1].__all);
for (size_t __i = 0; __i != __n; ++__i) {
auto&& [__a, __b] = *__first++;
__tree[__i + __n].__lack[0] =
std::move(__b) / std::move(__tree[__i + __n].__lack[0]);
}
for (size_t __i = __n; --__i;)
__tree[__i].__lack = std::move(__tree[__i << 1].__all) *
std::move(__tree[__i << 1 | 1].__lack) +
std::move(__tree[__i << 1 | 1].__all) *
std::move(__tree[__i << 1].__lack);
auto __result = std::move(__tree[1].__lack);
delete[] __tree;
return __result;
}
} // namespace workspace#line 2 "src/algebra/interpolation.hpp"
/**
* @file interpolation.hpp
* @brief Polynomial Interpolation
*/
#include <vector>
#line 2 "src/algebra/polynomial.hpp"
/**
* @file polynomial.hpp
* @brief Polynomial
*/
#include <algorithm>
#include <cassert>
#line 11 "src/algebra/polynomial.hpp"
#line 2 "src/algebra/fft.hpp"
/**
* @file fft.hpp
* @brief Fast Fourier Transform
*/
#line 9 "src/algebra/fft.hpp"
#line 2 "src/algebra/complex.hpp"
/**
* @file complex.hpp
* @brief Complex Number
*/
namespace workspace {
// Complex number.
template <class _Tp> class complex {
_Tp re, im;
friend constexpr complex conj(const complex &x) noexcept {
return {x.re, -x.im};
}
friend constexpr _Tp abs(const complex &x) noexcept {
return hypot(x.re, x.im);
}
friend constexpr _Tp arg(const complex &x) noexcept {
return atan2(x.re, x.im);
}
template <class _Is>
friend constexpr _Is &operator>>(_Is &__is, complex &x) noexcept {
return __is >> x.re >> x.im;
}
template <class _Os>
friend constexpr _Os &operator<<(_Os &__os, const complex &x) noexcept {
return __os << x.re << ' ' << x.im;
}
public:
constexpr complex() noexcept : re{}, im{} {}
constexpr complex(_Tp _re) noexcept : re{_re}, im{} {}
constexpr complex(_Tp _re, _Tp _im) noexcept : re{_re}, im{_im} {}
constexpr _Tp real() const noexcept { return re; }
constexpr void real(_Tp _re) noexcept { re = _re; }
constexpr _Tp imag() const noexcept { return im; }
constexpr void imag(_Tp _im) noexcept { im = _im; }
constexpr complex operator+() const noexcept { return *this; }
constexpr complex operator-() const noexcept { return {-re, -im}; }
constexpr complex &operator+=(const complex &x) noexcept {
return re += x.re, im += x.im, *this;
}
constexpr complex &operator-=(const complex &x) noexcept {
return re -= x.re, im -= x.im, *this;
}
constexpr complex &operator*=(const complex &x) noexcept {
_Tp _re{re * x.re - im * x.im};
return im = im * x.re + x.im * re, re = _re, *this;
}
constexpr complex &operator*=(_Tp x) noexcept {
return re *= x, im *= x, *this;
}
constexpr complex &operator/=(const complex &x) noexcept {
return (*this *= conj(x)) /= re * re + im * im;
}
constexpr complex &operator/=(_Tp x) noexcept {
return re /= x, im /= x, *this;
}
constexpr complex operator+(const complex &x) const noexcept {
return {re + x.re, im + x.im};
}
constexpr complex operator-(const complex &x) const noexcept {
return {re - x.re, im - x.im};
}
constexpr complex operator*(const complex &x) const noexcept {
return complex(*this) *= x;
}
constexpr complex operator*(_Tp x) const noexcept { return {re * x, im * x}; }
constexpr complex operator/(const complex &x) const noexcept {
return complex(*this) /= x;
}
constexpr complex operator/(_Tp x) const noexcept { return {re / x, im / x}; }
};
} // namespace workspace
#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
#include <iterator>
#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 2 "src/number_theory/ext_gcd.hpp"
/**
* @file ext_gcd.hpp
* @brief Extended Euclidean Algorithm
*/
#include <tuple>
#line 2 "src/utils/sfinae.hpp"
/**
* @file sfinae.hpp
* @brief SFINAE
*/
#include <cstdint>
#line 10 "src/utils/sfinae.hpp"
#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/ext_gcd.hpp"
namespace workspace {
/**
* @param __a Integer
* @param __b Integer
* @return Pair of integers (x, y) s.t. ax + by = g = gcd(a, b) and (b = 0 or 0
* <= x < |b/g|) and (a = 0 or -|a/g| < y <= 0). Return (0, 0) if (a, b) = (0,
* 0).
*/
template <typename _T1, typename _T2>
constexpr auto ext_gcd(_T1 __a, _T2 __b) noexcept {
static_assert(is_integral_ext<_T1>::value);
static_assert(is_integral_ext<_T2>::value);
using value_type = typename std::make_signed<
typename std::common_type<_T1, _T2>::type>::type;
using result_type = std::pair<value_type, value_type>;
value_type a{__a}, b{__b}, p{1}, q{}, r{}, s{1};
while (b != value_type(0)) {
auto t = a / b;
r ^= p ^= r ^= p -= t * r;
s ^= q ^= s ^= q -= t * s;
b ^= a ^= b ^= a -= t * b;
}
if (a < 0) p = -p, q = -q, a = -a;
if (p < 0) {
__a /= a, __b /= a;
if (__b > 0)
p += __b, q -= __a;
else
p -= __b, q += __a;
}
return result_type{p, q};
}
/**
* @param __a Integer
* @param __b Integer
* @param __c Integer
* @return Pair of integers (x, y) s.t. ax + by = c and (b = 0 or 0 <= x <
* |b/g|). Return (0, 0) if there is no solution.
*/
template <typename _T1, typename _T2, typename _T3>
constexpr auto ext_gcd(_T1 __a, _T2 __b, _T3 __c) noexcept {
static_assert(is_integral_ext<_T1>::value);
static_assert(is_integral_ext<_T2>::value);
static_assert(is_integral_ext<_T3>::value);
using value_type = typename std::make_signed<
typename std::common_type<_T1, _T2, _T3>::type>::type;
using result_type = std::pair<value_type, value_type>;
value_type a{__a}, b{__b}, p{1}, q{}, r{}, s{1};
while (b != value_type(0)) {
auto t = a / b;
r ^= p ^= r ^= p -= t * r;
s ^= q ^= s ^= q -= t * s;
b ^= a ^= b ^= a -= t * b;
}
if (__c % a) return result_type{};
__a /= a, __b /= a, __c /= a;
p *= __c, q *= __c;
if (__b != value_type(0)) {
auto t = p / __b;
p -= __b * t;
q += __a * t;
if (p < 0) {
if (__b > 0)
p += __b, q -= __a;
else
p -= __b, q += __a;
}
}
return result_type{p, q};
}
} // namespace workspace
#line 2 "src/number_theory/primitive_root.hpp"
/**
* @file primitive_root.hpp
* @brief Primitive Root
* @date 2020-12-28
*/
#line 10 "src/number_theory/primitive_root.hpp"
namespace workspace {
/**
* @brief Compile time primitive root.
*
* @tparam __mod Positive integer
* @return Minimum positive one if it exists. Otherwise 0.
*/
template <class Tp>
constexpr typename std::enable_if<(is_integral_ext<Tp>::value), Tp>::type
primitive_root(const Tp __mod) noexcept {
assert(__mod > 0);
using int_type = typename multiplicable_uint<Tp>::type;
int_type __r = __mod, __p[16] = {}, *__q = __p;
for (int_type __i = 2; __i <= __r / __i; ++__i) {
if (__r % __i) continue;
*__q++ = __i;
while (!(__r % __i)) __r /= __i;
}
if (__r != 1) *__q++ = __r;
int_type __tot = __mod;
for (__q = __p; *__q; *__q++ = 0) (__tot /= *__q) *= *__q - 1;
__r = __tot, __q = __p + 1, __p[0] = 1;
for (int_type __i = 2; __i <= __r / __i; ++__i) {
if (__r % __i) continue;
*__q++ = __i;
while (!(__r % __i)) __r /= __i;
}
if (__r != 1) *__q++ = __r;
for (Tp __r = 1; __r != __mod; ++__r) {
auto __cnt = 0;
for (__q = __p; *__q; ++__q) {
int_type __w = 1;
for (int_type __e = __tot / *__q, __x = __r; __e;
__e >>= 1, (__x *= __x) %= __mod)
if (__e & 1) (__w *= __x) %= __mod;
if (__w == 1 && ++__cnt > 1) break;
}
if (__cnt == 1) return __r;
}
return 0;
};
} // namespace workspace
#line 15 "src/algebra/fft.hpp"
namespace workspace {
namespace _fft_impl {
template <class _Tp, bool = std::is_floating_point<_Tp>::value, class = void>
struct to_float {
using type = double;
};
template <class _Tp> struct to_float<_Tp, true> { using type = _Tp; };
// template <class _Tp>
// struct to_float<_Tp, false, std::enable_if_t<sizeof(_Tp) <= sizeof(float)>> {
// using type = float;
// };
template <class _Tp>
struct to_float<_Tp, false, std::enable_if_t<(sizeof(_Tp) > sizeof(double))>> {
using type = long double;
};
// Assume ntt-friendly mod.
template <class _Tp> struct field {
using type = std::conditional_t<has_mod<_Tp>::value, _Tp,
complex<typename to_float<_Tp>::type>>;
};
template <class _Tp> struct field<complex<_Tp>> : field<_Tp> {};
// Modular
template <class _Tp, int _Nm = 29, bool = has_mod<_Tp>::value> struct coef {
_Tp s[_Nm], is[_Nm], ip2[_Nm];
_CXX14_CONSTEXPR coef() : s{}, is{}, ip2{1, (1 + _Tp::mod) / 2} {
if (_Tp::mod < 2) return;
int cnt2 = std::min(__builtin_ctz(_Tp::mod - 1), _Nm + 1);
_Tp e = 1;
_Tp w = primitive_root(_Tp::mod);
for (auto p = (_Tp::mod - 1) >> cnt2; p; p >>= 1, w *= w)
if (p & 1) e *= w;
_Tp ie = ext_gcd(decltype(_Tp::mod)(e), _Tp::mod).first;
_Tp es[_Nm]{}, ies[_Nm]{};
for (int i = cnt2; i >= 2; i--) {
es[i - 2] = e, e *= e;
ies[i - 2] = ie, ie *= ie;
}
e = ie = 1;
for (int i = 0; i < cnt2 - 1; i++) {
s[i] = es[i] * e, e *= ies[i];
is[i] = ies[i] * ie, ie *= es[i];
}
for (int i = 1; i < _Nm - 1; ++i) ip2[i + 1] = ip2[i] * ip2[1];
}
};
// Complex
template <class _Tp, int _Nm> struct coef<_Tp, _Nm, false> {
_Tp s[_Nm], is[_Nm], ip2[_Nm];
static_assert(_Nm < 30);
_CXX14_CONSTEXPR static _Tp es[29] = {
{0, 1},
{0.70710678118654752438189403651, 0.70710678118654752443610414514},
{0.92387953251128675610142140795, 0.38268343236508977172325753068},
{0.98078528040323044911909938781, 0.19509032201612826785692544201},
{0.99518472667219688623102546998, 0.09801714032956060199569840382},
{0.99879545620517239270077028412, 0.04906767432741801425693899119},
{0.99969881869620422009748220149, 0.02454122852291228803212346128},
{0.99992470183914454093764001552, 0.01227153828571992607945510345},
{0.99998117528260114264494415325, 0.00613588464915447535972750246},
{0.99999529380957617150137498041, 0.00306795676296597627029751672},
{0.99999882345170190993313003025, 0.00153398018628476561237225788},
{0.99999970586288221914474799723, 0.00076699031874270452695124765},
{0.99999992646571785113833452651, 0.00038349518757139558906815188},
{0.99999998161642929381167504976, 0.00019174759731070330743679009},
{0.99999999540410731290905263501, 0.00009587379909597734587360460},
{0.99999999885102682753608427379, 0.00004793689960306688454884772},
{0.99999999971275670682981095982, 0.00002396844980841821872882467},
{0.99999999992818917670745273995, 0.00001198422490506970642183282},
{0.99999999998204729416331065783, 0.00000599211245264242784278378},
{0.99999999999551182356793271877, 0.00000299605622633466075058210},
{0.99999999999887795586487812538, 0.00000149802811316901122883643},
{0.99999999999971948897977205850, 0.00000074901405658471572113723},
{0.99999999999992987223139048746, 0.00000037450702829238412391495},
{0.99999999999998246807140014902, 0.00000018725351414619534486931},
{0.99999999999999561700429751010, 0.00000009362675707309808280024},
{0.99999999999999890425107437752, 0.00000004681337853654909269501},
{0.99999999999999972607632112153, 0.00000002340668926827455275977},
{0.99999999999999993153263280754, 0.00000001170334463413727718121},
{0.99999999999999998286960567472, 0.00000000585167231706863869077}};
_CXX14_CONSTEXPR coef() : s{}, is{}, ip2{1, .5} {
_Tp ies[_Nm];
for (int i = 0; i < _Nm; ++i) ies[i] = _Tp(1) / es[i];
_Tp e = 1, ie = 1;
for (int i = 0; i < _Nm; i++) {
s[i] = es[i] * e, e *= ies[i];
is[i] = ies[i] * ie, ie *= es[i];
}
for (int i = 1; i < _Nm - 1; ++i) ip2[i + 1] = ip2[i] * ip2[1];
}
};
} // namespace _fft_impl
template <bool _Inverse = false, class _Iterator>
void fft(_Iterator __first, _Iterator __last) noexcept {
using value_type = typename std::iterator_traits<_Iterator>::value_type;
using difference_type =
typename std::iterator_traits<_Iterator>::difference_type;
_CXX14_CONSTEXPR _fft_impl::coef<value_type> c;
auto __h = __builtin_ctz(std::distance(__first, __last));
if _CXX17_CONSTEXPR (_Inverse) {
for (difference_type __p = 1; __p >> __h ^ 1; __p <<= 1) {
value_type __iw = 1;
auto __l = __first;
for (auto __i = 1 << __h; __l != __last;
__iw *= c.is[__builtin_ctz(--__i)]) {
auto __r = std::next(__l, __p);
for (auto __mid = __r; __l != __mid; ++__l, ++__r) {
auto __tmp = (*__l - *__r) * __iw;
*__l += *__r;
*__r = __tmp;
}
__l = __r;
}
}
while (__first != __last) *--__last *= c.ip2[__h];
}
else {
for (difference_type __p = 1 << __h; __p >>= 1;) {
value_type __w = -1;
auto __l = __first;
for (auto __i = 1 << __h; __l != __last;
__w *= c.s[__builtin_ctz(--__i)]) {
auto __r = std::next(__l, __p);
for (auto __mid = __r; __l != __mid; ++__l, ++__r) {
auto __tmp = *__l;
*__l -= *__r *= __w;
*__r += __tmp;
}
__l = __r;
}
}
}
}
template <class _Iterator>
void fft(_Iterator __first, std::size_t __n) noexcept {
fft(__first, std::next(__first, __n));
}
template <class _Iterator>
void ifft(_Iterator __first, _Iterator __last) noexcept {
fft<true>(__first, __last);
}
template <class _Iterator>
void ifft(_Iterator __first, std::size_t __n) noexcept {
ifft(__first, std::next(__first, __n));
}
template <size_t _Nm, size_t _Dm, class _Container, class _Index>
decltype(auto) access(_Container &__a, const _Index &__i) {
if _CXX17_CONSTEXPR (_Nm != _Dm)
return access<_Nm + 1, _Dm>(__a[__i[_Nm]], __i);
else
return __a;
}
template <bool _Inverse, size_t _Dm, class _Container, class _Tp, class _Index>
void dive(_Container &__a, const _Tp &__t, _Index &__i) {
if _CXX17_CONSTEXPR (has_size<_Tp>::value) {
for (__i.emplace_back(0); __i.back() != std::size(__t); ++__i.back())
dive<_Inverse, _Dm + 1>(__a, __t[__i.back()], __i);
__i.pop_back();
}
else {
static std::vector<_Tp> __work;
// Resize to a power of 2.
size_t __len = 1 << (31 - __builtin_clz(std::size(__a)));
if (__work.size() < __len) __work.resize(__len);
for (size_t __k = 0; __k != __len; ++__k)
__work[__k] = std::move(access<0, _Dm>(__a[__k], __i));
fft<_Inverse>(__work.data(), __work.data() + __len);
for (size_t __k = 0; __k != __len; ++__k)
access<0, _Dm>(__a[__k], __i) = std::move(__work[__k]);
}
}
template <bool _Inverse, class _Container> void fft(_Container &__a) {
if _CXX17_CONSTEXPR (has_size<_Container>::value) {
if _CXX17_CONSTEXPR (has_resize<_Container>::value)
// Resize to a power of 2.
__a.resize(1 << (32 - __builtin_clz(__a.size() - 1)));
std::vector<size_t> __i;
dive<_Inverse, 0>(__a, __a[0], __i);
for (size_t __k = 0; __k != std::size(__a); ++__k) fft<_Inverse>(__a[__k]);
}
}
template <class _Container> auto conv_resize(_Container &__a, _Container &__b) {
std::array<size_t, get_dimension<_Container>::value> __s;
rec(__a, __s);
rec(__b, __s);
return __s;
}
template <size_t _Nm, class _Container, class _Size>
void rec(const _Container &__a, _Size &__s) {
if _CXX17_CONSTEXPR (_Nm != __s.size()) {
__s[_Nm] = std::max(__s[_Nm], std::size(__a));
for (auto &__x : __a) rec<_Nm + 1>(__x, __s);
}
}
} // namespace workspace
#line 15 "src/algebra/polynomial.hpp"
namespace workspace {
/**
* @brief Polynomial.
*
* @tparam _Tp Ring structure
* @tparam _Conv_threshold Threshold for convolution method
*/
template <class _Tp, std::size_t _Conv_threshold = 64>
class polynomial : public std::vector<_Tp> {
using vec = std::vector<_Tp>;
using poly = polynomial;
template <class _Os> friend _Os& operator<<(_Os& __os, const poly& __x) {
bool __head = true;
for (const auto& __a : __x) {
if (!__head) __os << ' ';
__head = false;
__os << __a;
}
return __os;
}
public:
using typename vec::size_type;
using typename vec::value_type;
using vec::size;
using vec::vec;
using field = typename _fft_impl::field<_Tp>::type;
protected:
constexpr static _fft_impl::coef<field> __coef{};
static std::vector<field> __work1, __work2;
void _erase_leading_zeros() noexcept {
auto __i = vec::_M_impl._M_finish;
while (__i != vec::_M_impl._M_start && *(__i - 1) == _Tp(0)) --__i;
vec::_M_erase_at_end(__i);
}
template <class _Iter> void _dft(_Iter __first, _Iter __last) const noexcept {
fft<false>(__first, __last);
}
template <class _Iter>
void _idft(_Iter __first, _Iter __last) const noexcept {
fft<true>(__first, __last);
}
void _conv_naive(const poly& __x) noexcept {
auto __n = vec::_M_impl._M_finish - vec::_M_impl._M_start;
if (__n == 0) return;
if (__x._M_impl._M_start == __x._M_impl._M_finish) {
vec::_M_erase_at_end(vec::_M_impl._M_start); // clear
return;
}
vec::_M_default_append(__x._M_impl._M_finish - __x._M_impl._M_start - 1);
for (auto __h = vec::_M_impl._M_start + __n, __i = vec::_M_impl._M_finish;
__i != vec::_M_impl._M_start;) {
auto __k = __x._M_impl._M_start;
if (__i != __h) {
__k += __i - __h;
--__i;
} else {
--__i, --__h;
*__i *= *__k++;
}
for (auto __j = __h;
__j != vec::_M_impl._M_start && __k != __x._M_impl._M_finish;)
*__i += *--__j * *__k++;
}
}
template <class _Poly> void _conv_dft(_Poly&& __x) noexcept {
size_type __n = vec::_M_impl._M_finish - vec::_M_impl._M_start,
__m = __x._M_impl._M_finish - __x._M_impl._M_start,
__len = 1 << (32 - __builtin_clz(__n + __m - 1));
if (__work1.size() < __len) __work1.resize(__len);
if (__work2.size() < __len) __work2.resize(__len);
vec::_M_default_append(__m - 1);
if _CXX17_CONSTEXPR (std::is_same<_Tp, field>::value) {
std::fill(std::move(vec::_M_impl._M_start, vec::_M_impl._M_finish,
__work1.data()),
__work1.data() + __len, _Tp(0));
std::fill(std::move(__x._M_impl._M_start, __x._M_impl._M_finish,
__work2.data()),
__work2.data() + __len, _Tp(0));
fft(__work1.data(), __len);
fft(__work2.data(), __len);
for (size_type __i = 0; __i < __len; ++__i)
__work1[__i] *= std::move(__work2[__i]);
ifft(__work1.data(), __len);
std::move(__work1.data(), __work1.data() + __n + __m - 1,
vec::_M_impl._M_start);
}
else {
std::fill_n(__work1.data(), __len, _Tp(0));
std::fill_n(__work2.data(), __len, _Tp(0));
for (size_type __i = 0; __i < __n; ++__i)
__work1[__i].real(vec::_M_impl._M_start[__i]);
for (size_type __i = 0; __i < __m; ++__i)
__work1[__i].imag(__x._M_impl._M_start[__i]);
fft(__work1.data(), __len);
__work2[0].imag(__work1[0].real() * __work1[0].imag());
for (size_type __b = 1; __b != __len; __b <<= 1)
for (size_type __i = __b, __j = __b << 1; __j-- != __b; ++__i)
__work2[__i] = (__work1[__i] + conj(__work1[__j])) *
(__work1[__i] - conj(__work1[__j])) / 4;
ifft(__work2.data(), __len);
for (size_type __i = 0; __i < __n + __m - 1; ++__i)
if _CXX17_CONSTEXPR (std::is_floating_point<_Tp>::value)
vec::_M_impl._M_start[__i] = __work2[__i].imag();
else
vec::_M_impl._M_start[__i] = roundl(__work2[__i].imag());
}
}
size_type _divmod_naive(const poly& __x) {
auto __xfin = __x._M_impl._M_finish;
auto __xlen = __x.size();
while (__xfin != __x._M_impl._M_start && *(__xfin - 1) == _Tp(0))
--__xfin, --__xlen;
assert(__xlen != 0);
_erase_leading_zeros();
auto __p = vec::_M_impl._M_finish;
while (size_type(__p - vec::_M_impl._M_start) >= __xlen) {
--__p;
auto __src = __xfin;
auto __dst = __p;
*__dst /= *--__src;
while (__src != __x._M_impl._M_start) *--__dst -= *--__src * *__p;
}
return std::min<size_type>(__xlen - 1, __p - vec::_M_impl._M_start);
}
void _div_naive(const poly& __x) { operator>>=(_divmod_naive(__x)); }
void _div_doubling(poly&& __x) noexcept {
_erase_leading_zeros();
__x._erase_leading_zeros();
auto __n = vec::_M_impl._M_finish - vec::_M_impl._M_start;
auto __m = __x._M_impl._M_finish - __x._M_impl._M_start;
if (__n < __m)
vec::clear();
else {
assert(__m != 0);
std::reverse(__x._M_impl._M_start, __x._M_impl._M_finish);
__x = __x.inv(__n - __m + 1);
std::reverse(vec::_M_impl._M_start, vec::_M_impl._M_finish);
vec::_M_erase_at_end(vec::_M_impl._M_finish - (__m - 1));
operator*=(__x).resize(__n - __m + 1);
std::reverse(vec::_M_impl._M_start, vec::_M_impl._M_finish);
}
}
public:
/**
* @return Degree of %polynomial. Return -1 if it equals zero.
*/
size_type deg() const noexcept { return size() - 1; }
/**
* @param __i Not exceeding the degree.
* @return Coefficient of x^i.
*/
typename vec::reference operator[](size_type __i) noexcept {
assert(__i < size());
return *(vec::_M_impl._M_start + __i);
}
/**
* @param __i Not exceeding the degree.
* @return Coefficient of x^i.
*/
typename vec::const_reference operator[](size_type __i) const noexcept {
assert(__i < size());
return *(vec::_M_impl._M_start + __i);
}
/**
* @brief Evaluate at given point.
*/
_Tp eval(const _Tp& __a) const noexcept {
_Tp __v(0), __p(1);
for (auto __i = vec::_M_impl._M_start; __i != vec::_M_impl._M_finish;
++__i, __p *= __a)
__v += *__i * __p;
return __v;
}
/**
* @brief In-place multipoint evaluation.
*/
template <class _Iter, typename = std::_RequireInputIter<_Iter>>
_Iter eval(_Iter __first, _Iter __last) const noexcept {
return eval(__first, __last, __first);
}
/**
* @brief Multipoint evaluation.
*/
template <class _InputIter, class _OutputIter,
typename = std::_RequireInputIter<_InputIter>>
_OutputIter eval(_InputIter __first, _InputIter __last,
_OutputIter __result) const noexcept {
size_type __n = std::distance(__first, __last);
if (!__n) return __result;
auto __tree = new poly[__n << 1];
for (auto __p = __tree + __n; __first != __last; ++__p, ++__first)
*__p = {-*__first, 1};
for (size_type __i = __n; --__i;)
__tree[__i] = __tree[__i << 1] * __tree[__i << 1 | 1];
__tree[1] = operator%(std::move(__tree[1]));
for (size_type __i = 2; __i != __n << 1; __i += 2)
__tree[__i] = __tree[__i >> 1] % std::move(__tree[__i]),
__tree[__i | 1] =
std::move(__tree[__i >> 1] %= std::move(__tree[__i | 1]));
for (size_type __i = 0; __i != __n; ++__i)
*__result++ = std::move(*__tree[__n + __i]._M_impl._M_start);
delete[] __tree;
return __result;
}
/**
* @brief Multiply by x^i.
*/
poly& operator<<=(size_type __i) noexcept {
vec::insert(vec::begin(), __i, _Tp(0));
return *this;
}
/**
* @brief Divide by x^i.
*/
poly& operator>>=(size_type __i) noexcept {
vec::_M_erase_at_end(
std::move(vec::_M_impl._M_start + std::min(__i, size()),
vec::_M_impl._M_finish, vec::_M_impl._M_start));
return *this;
}
/**
* @brief Multiply by x^i.
*/
poly operator<<(size_type __i) const noexcept {
return poly(*this).operator<<=(__i);
}
/**
* @brief Divide by x^i.
*/
poly operator>>(size_type __i) const noexcept {
return poly(*this).operator>>=(__i);
}
poly operator+() const noexcept { return *this; }
poly operator-() const noexcept {
poly __x = *this;
for (auto __i = __x._M_impl._M_start; __i != __x._M_impl._M_finish; ++__i)
*__i = -*__i;
return __x;
}
poly& operator+=(const poly& __x) noexcept {
if (size() < __x.size()) vec::_M_default_append(__x.size() - size());
for (auto __i = vec::_M_impl._M_start, __j = __x._M_impl._M_start;
__j != __x._M_impl._M_finish; ++__i, ++__j)
*__i += *__j;
_erase_leading_zeros();
return *this;
}
poly& operator+=(const _Tp& __c) noexcept {
if (__c != static_cast<_Tp>(0)) {
if (vec::_M_impl._M_start == vec::_M_impl._M_finish)
vec::emplace_back(__c);
else
*vec::_M_impl._M_start += __c, _erase_leading_zeros();
}
return *this;
}
poly& operator-=(const poly& __x) noexcept {
if (size() < __x.size()) vec::_M_default_append(__x.size() - size());
for (auto __i = vec::_M_impl._M_start, __j = __x._M_impl._M_start;
__j != __x._M_impl._M_finish; ++__i, ++__j)
*__i -= *__j;
_erase_leading_zeros();
return *this;
}
poly& operator-=(const _Tp& __c) noexcept {
if (__c != static_cast<_Tp>(0)) {
if (vec::_M_impl._M_start == vec::_M_impl._M_finish)
vec::emplace_back(-__c);
else
*vec::_M_impl._M_start -= __c, _erase_leading_zeros();
}
return *this;
}
poly& operator*=(const poly& __x) noexcept {
if (this == std::addressof(__x)) // with itself
return operator*=(poly(__x));
std::min(size(), __x.size()) > _Conv_threshold ? _conv_dft(__x)
: _conv_naive(__x);
return *this;
}
poly& operator*=(poly&& __x) noexcept {
if (this == std::addressof(__x)) // with itself
return operator*=(poly(__x));
std::min(size(), __x.size()) > _Conv_threshold
? _conv_dft(std::move(__x))
: _conv_naive(std::move(__x));
return *this;
}
poly& operator*=(const _Tp& __c) noexcept {
if (__c == static_cast<_Tp>(0))
vec::_M_erase_at_end(vec::_M_impl._M_start);
else
for (auto __i = vec::_M_impl._M_start; __i != vec::_M_impl._M_finish;
++__i)
*__i *= __c;
return *this;
}
poly& operator/=(const _Tp& __c) noexcept {
assert(__c != static_cast<_Tp>(0));
for (auto __i = vec::_M_impl._M_start; __i != vec::_M_impl._M_finish; ++__i)
*__i /= __c;
return *this;
}
poly pow(size_type __e) const noexcept {
if (vec::empty()) return *this;
if (!__e) return {1};
if (size() == 1) {
_Tp __x = vec::front(), __y = 1;
for (auto __i = __e; __i; __i >>= 1, __x *= __x)
if (__i & 1) __y *= __x;
return {__y};
}
size_type __deg = (size() - 1) * __e;
assert(__deg > 0);
poly __p(1 << (32 - __builtin_clz(__deg)));
std::copy(vec::_M_impl._M_start, vec::_M_impl._M_finish,
__p._M_impl._M_start);
fft(__p._M_impl._M_start, __p._M_impl._M_finish);
for (auto&& __x : __p) {
_Tp __y = 1;
for (auto __i = __e; __i; __i >>= 1, __x *= __x)
if (__i & 1) __y *= __x;
__x = __y;
}
ifft(__p._M_impl._M_start, __p._M_impl._M_finish);
__p.resize(__deg + 1);
return __p;
}
poly rev() const noexcept { return rev(size()); }
poly rev(size_type __n) const noexcept {
poly __r(__n);
auto __src = vec::_M_impl._M_start;
auto __dst = __r._M_impl._M_finish;
for (size_type __i = std::min(__n, size()); __i; --__i) *--__dst = *__src++;
return __r;
}
poly inv() const noexcept { return inv(size()); }
/**
* @brief Multiplicative inverse modulo x^n.
*
* @param __n Degree of modulus
* @return
*/
poly inv(size_type __n) const noexcept {
if (!__n) return {};
assert(*vec::_M_impl._M_start != _Tp(0));
size_type __len = 1;
while (__len < __n) __len <<= 1;
poly __y(__len);
auto __xp = new _Tp[__len], __yp = __y._M_impl._M_start,
__zp = new _Tp[__len];
*__yp = _Tp(1) / *vec::_M_impl._M_start;
for (size_type __i = 1; __i != __len; __i <<= 1) {
std::fill(std::copy_n(__yp, __i, __zp), __zp + (__i << 1), _Tp(0));
_dft(__zp, __zp + (__i << 1));
std::fill(
std::copy_n(vec::_M_impl._M_start, std::min(__i << 1, size()), __xp),
__xp + (__i << 1), _Tp(0));
_dft(__xp, __xp + (__i << 1));
for (size_type __j = 0; __j != (__i << 1); ++__j) __xp[__j] *= -__zp[__j];
_idft(__xp, __xp + (__i << 1));
std::fill(std::move(__xp + __i, __xp + (__i << 1), __xp),
__xp + (__i << 1), _Tp(0));
_dft(__xp, __xp + (__i << 1));
for (size_type __j = 0; __j != (__i << 1); ++__j)
__xp[__j] *= static_cast<_Tp&&>(__zp[__j]);
_idft(__xp, __xp + (__i << 1));
std::move(__xp, __xp + __i, __yp + __i);
}
delete[] __xp;
delete[] __zp;
__y._M_erase_at_end(__yp + __n);
return __y;
}
poly& operator/=(const poly& __x) noexcept {
if (__x.size() > _Conv_threshold)
_div_doubling(poly(__x));
else
_div_naive(__x);
return *this;
}
poly& operator/=(poly&& __x) noexcept {
if (__x.size() > _Conv_threshold)
_div_doubling(std::move(__x));
else
_div_naive(__x);
return *this;
}
poly& operator%=(const poly& __x) noexcept {
if (__x.size() > _Conv_threshold)
return operator-=(__x.operator*(operator/(__x)));
vec::_M_erase_at_end(vec::_M_impl._M_start + _divmod_naive(__x));
return *this;
}
template <class _T> poly operator+(_T&& __x) const noexcept {
return poly(*this).operator+=(std::forward<_T>(__x));
}
template <class _T> poly operator-(_T&& __x) const noexcept {
return poly(*this).operator-=(std::forward<_T>(__x));
}
template <class _T> poly operator*(_T&& __x) const noexcept {
return poly(*this).operator*=(std::forward<_T>(__x));
}
template <class _T> poly operator/(_T&& __x) const noexcept {
return poly(*this).operator/=(std::forward<_T>(__x));
}
template <class _T> poly operator%(_T&& __x) const noexcept {
return poly(*this).operator%=(std::forward<_T>(__x));
}
std::pair<poly, poly> divmod(const poly& __x) const {
if (__x.size() > _Conv_threshold) return {operator/(__x), operator%(__x)};
poly __rem(*this);
auto __p = __rem._M_impl._M_start + __rem._divmod_naive(__x);
poly __quot(__p, __rem._M_impl._M_finish);
__rem._M_erase_at_end(__p);
return {__quot, __rem};
}
/**
* @brief Differentiate.
*
* @return Derivative.
*/
poly deriv() const noexcept {
if (auto __s = vec::_M_impl._M_start, __f = vec::_M_impl._M_finish;
__s != __f) {
poly __der(++__s, __f);
__s = __der._M_impl._M_start, __f = __der._M_impl._M_finish;
for (_Tp __i(1); __s != __f; ++__s, __i += 1) *__s *= __i;
__der._erase_leading_zeros();
return __der;
}
return {};
}
/**
* @brief Differentiate at given point.
*
* @return Derivative coefficient.
*/
_Tp deriv(const _Tp& __a) const noexcept {
_Tp __der(0);
if (auto __s = vec::_M_impl._M_start, __f = vec::_M_impl._M_finish;
__s != __f)
for (_Tp __i(1), __p(1); ++__s != __f; __i += 1, __p *= __a)
__der += *__s * __i * __p;
return __der;
}
/**
* @brief Integrate.
*
* @return Integral indefinite at the degrees divisible by the characteristic
* of `_Tp`. Coefficients are set as 0 there.
*/
poly integ() const noexcept {
if (auto __s = vec::_M_impl._M_start, __f = vec::_M_impl._M_finish;
__s != __f) {
poly __int(__f - __s + 1);
__f = std::copy(__s, __f, __int._M_impl._M_start + 1);
__s = __int._M_impl._M_start + 1;
for (_Tp __i(1); __s != __f; ++__s, __i += 1)
__i == _Tp(0) ? assert(*__s == _Tp(0)) : void(*__s /= __i);
return __int;
}
return {};
}
/**
* @brief Integrate in given range.
*
* @return Definite integral over [0, __a].
*/
_Tp integ(const _Tp& __a) const noexcept {
_Tp __int(0);
auto __s = vec::_M_impl._M_start, __f = vec::_M_impl._M_finish;
for (_Tp __p(__a), __i(1); __s != __f; ++__s, __p *= __a, __i += 1)
__int += *__s / __i * __p;
return __int;
}
/**
* @brief Integrate in given range.
*
* @return Definite integral over [__a, __b].
*/
_Tp integ(const _Tp& __a, const _Tp& __b) const noexcept {
_Tp __int(0);
auto __s = vec::_M_impl._M_start, __f = vec::_M_impl._M_finish;
for (_Tp __pa(__a), __pb(__b), __i(1); __s != __f;
++__s, __pa *= __a, __pb *= __b, __i += 1)
__int += *__s / __i * (__pb - __pa);
return __int;
}
/**
* @brief
*
* @param __a
* @return f(x + a)
*/
poly shift(const _Tp& __a) const noexcept {
size_type __n = size();
poly __s(*this), __e(__n);
_Tp __cs(1), __ce(1);
for (size_type __i{0}; __i != __n;
__cs *= _Tp(++__i), __ce *= __a / _Tp(__i))
__s[__i] *= __cs, __e[__n - 1 - __i] = __ce;
__s *= std::move(__e);
__ce = 1;
for (size_type __i{0}; __i != __n; __ce /= _Tp(++__i))
__e[__i] = __s[__n - 1 + __i] * __ce;
return __e;
}
};
template <class _Tp, size_t _C>
std::vector<typename polynomial<_Tp, _C>::field> polynomial<_Tp, _C>::__work1;
template <class _Tp, size_t _C>
std::vector<typename polynomial<_Tp, _C>::field> polynomial<_Tp, _C>::__work2;
/**
* @brief Generating function of the sum of k-th powers of the first n
* non-negative integers. O(d \\log d) time in modulo x^d.
*
* @return \\sum_{k=0}^{d-1} x^k \\sum_{i=0}^{n-1} i^k.
*/
template <class _Tp> polynomial<_Tp> power_sum(_Tp __n, std::size_t __d) {
if (!__d) return {};
polynomial<_Tp> __f(__d), __e(__d);
__f[0] = __n;
for (std::size_t __i = 1; __i != __d; ++__i) __f[__i] = __f[__i - 1] * __n;
_Tp __c{1};
for (std::size_t __i = 0; __i != __d; ++__i)
__c /= __i + 1, __f[__i] *= __c, __e[__i] = __c;
(__f *= __e.inv(__d)).resize(__d);
__c = 1;
for (std::size_t __i = 0; __i != __d; __c *= ++__i) __f[__i] *= __c;
return __f;
}
} // namespace workspace
#line 2 "src/algebra/system/operation.hpp"
/**
* @file operation.hpp
* @brief Operation Traits
*/
#include <functional>
#line 10 "src/algebra/system/operation.hpp"
#line 12 "src/algebra/system/operation.hpp"
namespace workspace {
// Unary `+`
template <class _Tp>
using require_unary_plus = std::enable_if_t<
std::is_convertible<decltype(+std::declval<const _Tp &>()), _Tp>::value>;
template <class _Tp, class = void> struct has_unary_plus : std::false_type {};
template <class _Tp>
struct has_unary_plus<_Tp, require_unary_plus<_Tp>> : std::true_type {};
// Unary `-`
template <class _Tp>
using require_unary_minus = std::enable_if_t<
std::is_convertible<decltype(-std::declval<const _Tp &>()), _Tp>::value>;
template <class _Tp, class = void> struct has_unary_minus : std::false_type {};
template <class _Tp>
struct has_unary_minus<_Tp, require_unary_minus<_Tp>> : std::true_type {};
// Binary `+`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_plus =
std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() +
std::declval<const _Tp2 &>()),
_Tp1>::value>;
template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_plus : std::false_type {};
template <class _Tp1, class _Tp2>
struct has_binary_plus<_Tp1, _Tp2, require_binary_plus<_Tp1, _Tp2>>
: std::true_type {};
// Binary `-`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_minus =
std::__void_t<decltype(std::declval<const _Tp1 &>() -
std::declval<const _Tp2 &>())>;
template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_minus : std::false_type {};
template <class _Tp1, class _Tp2>
struct has_binary_minus<_Tp1, _Tp2, require_binary_minus<_Tp1, _Tp2>>
: std::true_type {};
// Binary `*`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_multiplies =
std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() *
std::declval<const _Tp2 &>()),
_Tp1>::value>;
template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_multiplies : std::false_type {};
template <class _Tp1, class _Tp2>
struct has_binary_multiplies<_Tp1, _Tp2, require_binary_multiplies<_Tp1, _Tp2>>
: std::true_type {};
// Binary `/`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_divides =
std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() /
std::declval<const _Tp2 &>()),
_Tp1>::value>;
template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_divides : std::false_type {};
template <class _Tp1, class _Tp2>
struct has_binary_divides<_Tp1, _Tp2, require_binary_divides<_Tp1, _Tp2>>
: std::true_type {};
// Binary `%`
template <class _Tp1, class _Tp2 = _Tp1>
using require_binary_modulus =
std::enable_if_t<std::is_convertible<decltype(std::declval<const _Tp1 &>() %
std::declval<const _Tp2 &>()),
_Tp1>::value>;
template <class _Tp1, class _Tp2 = _Tp1, class = void>
struct has_binary_modulus : std::false_type {};
template <class _Tp1, class _Tp2>
struct has_binary_modulus<_Tp1, _Tp2, require_binary_modulus<_Tp1, _Tp2>>
: std::true_type {};
template <class _Tp1, class _Tp2 = _Tp1, class = void, class = void,
class = void, class = void>
struct has_arithmetic : std::false_type {};
template <class _Tp1, class _Tp2>
struct has_arithmetic<_Tp1, _Tp2, require_binary_plus<_Tp1, _Tp2>,
require_binary_minus<_Tp1, _Tp2>,
require_binary_multiplies<_Tp1, _Tp2>,
require_binary_divides<_Tp1, _Tp2>> : std::true_type {};
template <class _Tp1, class _Tp2 = _Tp1>
using require_arithmetic = std::enable_if_t<has_arithmetic<_Tp1, _Tp2>::value>;
// Binary `<`
template <class _Tp, class = void> struct is_comparable : std::false_type {};
template <class _Tp>
struct is_comparable<_Tp, std::__void_t<decltype(std::declval<const _Tp &>() <
std::declval<const _Tp &>())>>
: std::true_type {};
template <class _Tp, bool _Default = false> struct try_less : std::less<_Tp> {
constexpr bool operator()(const _Tp &__x, const _Tp &__y) noexcept {
if _CXX17_CONSTEXPR (is_comparable<_Tp>::value)
return std::less<_Tp>::operator()(__x, __y);
else
return _Default;
}
};
} // namespace workspace
#line 12 "src/algebra/interpolation.hpp"
namespace workspace {
/**
* @brief Lagrange interpolation on the points 0, 1, 2, ...
* @param __x Point to evaluate at.
* @param __first
* @param __last
*/
template <class _Tp, class _InputIter,
typename = std::_RequireInputIter<_InputIter>,
typename = require_arithmetic<
typename std::iterator_traits<_InputIter>::value_type>>
auto interpolate(_Tp __x, _InputIter __first, _InputIter __last) noexcept {
using value_type = typename std::iterator_traits<_InputIter>::value_type;
std::vector<value_type> __f(__first, __last);
decltype(__f.size()) __k = 0;
value_type __d = 1;
while (__k != __f.size()) __k += 1, __d *= value_type{__k};
__d = value_type{1} / __d;
value_type __c = __k & 1 ? -__d : __d;
auto __i = __f.begin();
while (__i != __f.end()) {
__c *= -value_type{__k}, __k -= 1;
*__i++ *= __c;
__c *= __x, __x -= 1;
}
__c = __d, __k = __f.size();
value_type __y = 0;
while (__i != __f.begin()) {
__c *= value_type{__k}, __k -= 1;
__y += *--__i * __c;
__x += 1, __c *= __x;
}
return __y;
}
/**
* @brief Polynomial interpolation in O(n log(n)^2) time.
* @param __first
* @param __last
*/
template <class _InputIter, typename = std::_RequireInputIter<_InputIter>>
auto interpolate(_InputIter __first, _InputIter __last) noexcept {
size_t __n = std::distance(__first, __last);
auto [__1, __2] = typename std::iterator_traits<_InputIter>::value_type{};
using poly = polynomial<decltype(__1)>;
if (!__n) return poly{};
struct node {
poly __all, __lack;
};
auto __tree = new node[__n << 1];
auto __iter = __first;
for (size_t __i = 0; __i != __n; ++__i) {
auto&& [__a, __b] = *__iter++;
__tree[__i + __n].__all = {-__a, 1}, __tree[__i + __n].__lack = {1};
}
for (size_t __i = __n; --__i;)
__tree[__i].__all = __tree[__i << 1].__all * __tree[__i << 1 | 1].__all,
__tree[__i].__lack =
__tree[__i << 1].__all * std::move(__tree[__i << 1 | 1].__lack) +
__tree[__i << 1 | 1].__all * std::move(__tree[__i << 1].__lack);
for (size_t __i = 2; __i != __n << 1; __i += 2)
__tree[__i].__lack = __tree[__i >> 1].__lack % __tree[__i].__all,
__tree[__i | 1].__lack =
std::move(__tree[__i >> 1].__lack %= __tree[__i | 1].__all);
for (size_t __i = 0; __i != __n; ++__i) {
auto&& [__a, __b] = *__first++;
__tree[__i + __n].__lack[0] =
std::move(__b) / std::move(__tree[__i + __n].__lack[0]);
}
for (size_t __i = __n; --__i;)
__tree[__i].__lack = std::move(__tree[__i << 1].__all) *
std::move(__tree[__i << 1 | 1].__lack) +
std::move(__tree[__i << 1 | 1].__all) *
std::move(__tree[__i << 1].__lack);
auto __result = std::move(__tree[1].__lack);
delete[] __tree;
return __result;
}
} // namespace workspace