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#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY #include "../hessenberg_system.hpp" #include "../../modint.hpp" #include "../../random/xorshift.hpp" #include <cassert> #include <iostream> #include <vector> using namespace std; using mint = ModInt<1000000007>; void test_lower_hessenberg_random() { for (int t = 0; t < 10000; ++t) { int N = t / 100; vector<vector<mint>> A(N, vector<mint>(N)); vector<mint> x(N); const double p = rand_double(); for (int i = 0; i < N; ++i) { x[i] = rand_int() % mint::mod(); for (int j = 0; j < i; ++j) A[i][j] = rand_int() % mint::mod(); A[i][i] = 1 + rand_int() % (mint::mod() - 1); if (i + 1 < N and rand_double() < p) A[i][i + 1] = rand_int() % mint::mod(); } vector<mint> b(N); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) b[i] += A[i][j] * x[j]; } assert(x == solve_lower_hessenberg<mint>(A, b)); } } void test_upper_hessenberg_random() { for (int t = 0; t < 10000; ++t) { int N = t / 100; vector<vector<mint>> A(N, vector<mint>(N)); vector<mint> x(N); const double p = rand_double(); for (int i = 0; i < N; ++i) { x[i] = rand_int() % mint::mod(); for (int j = i + 1; j < N; ++j) A[i][j] = rand_int() % mint::mod(); A[i][i] = 1 + rand_int() % (mint::mod() - 1); if (i and rand_double() < p) A[i][i - 1] = rand_int() % mint::mod(); } vector<mint> b(N); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) b[i] += A[i][j] * x[j]; } assert(x == solve_upper_hessenberg<mint>(A, b)); } } void test_lower_hessenberg_hand() { vector<vector<mint>> A{{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}; vector<mint> b{2, 3, 4}, x{4, 2, 3}; assert(solve_lower_hessenberg<mint>(A, b) == x); } int main() { test_lower_hessenberg_random(); test_lower_hessenberg_hand(); test_upper_hessenberg_random(); cout << "Hello World\n"; }
#line 1 "linear_algebra_matrix/test/hessenberg_system.stress.test.cpp" #define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ITP1_1_A" // DUMMY #line 1 "number/dual_number.hpp" #include <type_traits> namespace dual_number_ { struct has_id_method_impl { template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type()); template <class T_> static auto check(...) -> std::false_type; }; template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {}; } // namespace dual_number_ // Dual number (二重数) // Verified: https://atcoder.jp/contests/abc235/tasks/abc235_f template <class T> struct DualNumber { T a, b; // a + bx template <typename T2, typename std::enable_if<dual_number_::has_id<T2>::value>::type * = nullptr> static T2 _T_id() { return T2::id(); } template <typename T2, typename std::enable_if<!dual_number_::has_id<T2>::value>::type * = nullptr> static T2 _T_id() { return T2(1); } DualNumber(T x = T(), T y = T()) : a(x), b(y) {} static DualNumber id() { return DualNumber(_T_id<T>(), T()); } explicit operator bool() const { return a != T() or b != T(); } DualNumber operator+(const DualNumber &x) const { return DualNumber(a + x.a, b + x.b); } DualNumber operator-(const DualNumber &x) const { return DualNumber(a - x.a, b - x.b); } DualNumber operator*(const DualNumber &x) const { return DualNumber(a * x.a, b * x.a + a * x.b); } DualNumber operator/(const DualNumber &x) const { T cinv = _T_id<T>() / x.a; return DualNumber(a * cinv, (b * x.a - a * x.b) * cinv * cinv); } DualNumber operator-() const { return DualNumber(-a, -b); } DualNumber &operator+=(const DualNumber &x) { return *this = *this + x; } DualNumber &operator-=(const DualNumber &x) { return *this = *this - x; } DualNumber &operator*=(const DualNumber &x) { return *this = *this * x; } DualNumber &operator/=(const DualNumber &x) { return *this = *this / x; } bool operator==(const DualNumber &x) const { return a == x.a and b == x.b; } bool operator!=(const DualNumber &x) const { return !(*this == x); } bool operator<(const DualNumber &x) const { return (a != x.a ? a < x.a : b < x.b); } template <class OStream> friend OStream &operator<<(OStream &os, const DualNumber &x) { return os << '{' << x.a << ',' << x.b << '}'; } T eval(const T &x) const { return a + b * x; } T root() const { return (-a) / b; } // Solve a + bx = 0 (b \neq 0 is assumed) }; #line 3 "linear_algebra_matrix/hessenberg_system.hpp" #include <algorithm> #include <cassert> #include <vector> // Solve Ax = b, where A is n x n (square), lower Hessenberg, and non-singular. // Complexity: O(n^2) // Verified: https://atcoder.jp/contests/abc249/tasks/abc249_h template <class T> std::vector<T> solve_lower_hessenberg(const std::vector<std::vector<T>> &A, const std::vector<T> &b) { const int N = A.size(); if (!N) return {}; assert(int(A[0].size()) == N and int(b.size()) == N); using dual = DualNumber<T>; std::vector<dual> sol(N); for (int h = 0; h < N;) { sol[h] = dual(0, 1); for (int i = h;; ++i) { dual y = b[i]; for (int j = 0; j <= i; ++j) y -= sol[j] * A[i][j]; T a = i + 1 < N ? A[i][i + 1] : T(); if (a == T()) { T x0 = y.root(); while (h <= i) sol[h] = sol[h].eval(x0), ++h; break; } else { sol[i + 1] = y / a; } } } std::vector<T> ret(N); for (int i = 0; i < N; ++i) ret[i] = sol[i].a; return ret; } // Solve Ax = b, where A is n x n (square), upper Hessenberg, and non-singular // Complexity: O(n^2) template <class T> std::vector<T> solve_upper_hessenberg(std::vector<std::vector<T>> A, std::vector<T> b) { std::reverse(A.begin(), A.end()); for (auto &v : A) std::reverse(v.begin(), v.end()); std::reverse(b.begin(), b.end()); auto ret = solve_lower_hessenberg(A, b); std::reverse(ret.begin(), ret.end()); return ret; } #line 3 "modint.hpp" #include <iostream> #include <set> #line 6 "modint.hpp" template <int md> struct ModInt { using lint = long long; constexpr static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val_; int val() const noexcept { return val_; } constexpr ModInt() : val_(0) {} constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; } constexpr ModInt(lint v) { _setval(v % md + md); } constexpr explicit operator bool() const { return val_ != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val_ + x.val_); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val_ - x.val_ + md); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.val_ % md); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.inv().val() % md); } constexpr ModInt operator-() const { return ModInt()._setval(md - val_); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; } constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; } constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; } constexpr bool operator<(const ModInt &x) const { return val_ < x.val_; } // To use std::map<ModInt, T> friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val_; } constexpr ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static constexpr int cache_limit = std::min(md, 1 << 21); static std::vector<ModInt> facs, facinvs, invs; constexpr static void _precalculation(int N) { const int l0 = facs.size(); if (N > md) N = md; if (N <= l0) return; facs.resize(N), facinvs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i; facinvs[N - 1] = facs.back().pow(md - 2); for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1); for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1]; } constexpr ModInt inv() const { if (this->val_ < cache_limit) { if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0}; while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val_]; } else { return this->pow(md - 2); } } constexpr ModInt fac() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val_]; } constexpr ModInt facinv() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val_]; } constexpr ModInt doublefac() const { lint k = (this->val_ + 1) / 2; return (this->val_ & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } constexpr ModInt nCr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv() * ModInt(r).facinv(); } constexpr ModInt nPr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv(); } static ModInt binom(int n, int r) { static long long bruteforce_times = 0; if (r < 0 or n < r) return ModInt(0); if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r); r = std::min(r, n - r); ModInt ret = ModInt(r).facinv(); for (int i = 0; i < r; ++i) ret *= n - i; bruteforce_times += r; return ret; } // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!) // Complexity: O(sum(ks)) template <class Vec> static ModInt multinomial(const Vec &ks) { ModInt ret{1}; int sum = 0; for (int k : ks) { assert(k >= 0); ret *= ModInt(k).facinv(), sum += k; } return ret * ModInt(sum).fac(); } // Catalan number, C_n = binom(2n, n) / (n + 1) // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ... // https://oeis.org/A000108 // Complexity: O(n) static ModInt catalan(int n) { if (n < 0) return ModInt(0); return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).facinv(); } ModInt sqrt() const { if (val_ == 0) return 0; if (md == 2) return val_; if (pow((md - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val_, md - x.val_)); } }; template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0}; using ModInt998244353 = ModInt<998244353>; // using mint = ModInt<998244353>; // using mint = ModInt<1000000007>; #line 2 "random/xorshift.hpp" #include <cstdint> // CUT begin uint32_t rand_int() // XorShift random integer generator { static uint32_t x = 123456789, y = 362436069, z = 521288629, w = 88675123; uint32_t t = x ^ (x << 11); x = y; y = z; z = w; return w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); } double rand_double() { return (double)rand_int() / UINT32_MAX; } #line 8 "linear_algebra_matrix/test/hessenberg_system.stress.test.cpp" using namespace std; using mint = ModInt<1000000007>; void test_lower_hessenberg_random() { for (int t = 0; t < 10000; ++t) { int N = t / 100; vector<vector<mint>> A(N, vector<mint>(N)); vector<mint> x(N); const double p = rand_double(); for (int i = 0; i < N; ++i) { x[i] = rand_int() % mint::mod(); for (int j = 0; j < i; ++j) A[i][j] = rand_int() % mint::mod(); A[i][i] = 1 + rand_int() % (mint::mod() - 1); if (i + 1 < N and rand_double() < p) A[i][i + 1] = rand_int() % mint::mod(); } vector<mint> b(N); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) b[i] += A[i][j] * x[j]; } assert(x == solve_lower_hessenberg<mint>(A, b)); } } void test_upper_hessenberg_random() { for (int t = 0; t < 10000; ++t) { int N = t / 100; vector<vector<mint>> A(N, vector<mint>(N)); vector<mint> x(N); const double p = rand_double(); for (int i = 0; i < N; ++i) { x[i] = rand_int() % mint::mod(); for (int j = i + 1; j < N; ++j) A[i][j] = rand_int() % mint::mod(); A[i][i] = 1 + rand_int() % (mint::mod() - 1); if (i and rand_double() < p) A[i][i - 1] = rand_int() % mint::mod(); } vector<mint> b(N); for (int i = 0; i < N; ++i) { for (int j = 0; j < N; ++j) b[i] += A[i][j] * x[j]; } assert(x == solve_upper_hessenberg<mint>(A, b)); } } void test_lower_hessenberg_hand() { vector<vector<mint>> A{{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}; vector<mint> b{2, 3, 4}, x{4, 2, 3}; assert(solve_lower_hessenberg<mint>(A, b) == x); } int main() { test_lower_hessenberg_random(); test_lower_hessenberg_hand(); test_upper_hessenberg_random(); cout << "Hello World\n"; }