cplib-cpp

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:heavy_check_mark: linear_algebra_matrix/test/hessenberg_system.stress.test.cpp

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Code

#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 2 "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";
}
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