Hessenberg reduction of matrix
(linear_algebra_matrix/hessenberg_reduction.hpp)

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• Last update: 2023-05-21 18:11:51+09:00
• Include: #include "linear_algebra_matrix/hessenberg_reduction.hpp"
• Link: http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf

体上の $n$ 次正方行列の Upper Hessenberg reduction を $O(n^3)$ で行う．

やっていること

(Upper) Hessenberg reduction とは，行列に相似変換を施すことでその対角成分より2つ以上左下側の成分を全てゼロにするというもので，このような変換は特に Householder 変換の組合せによって可能である．相似変換で特性多項式は不変なため，本ライブラリでは特性多項式の導出などに応用されている．

使用方法

matrix<T> に対して upper Hessenberg reduction を行う関数は以下のように使用する．

T が逆元がとれるデータ構造の場合

matrix<T> mat(N, N);
hessenberg_reduction(mat);

T が逆元がとれないがユークリッドの互除法が可能なデータ構造の場合（例：合成数 modint）

matrix<T> mat(N, N);
ring_hessenberg_reduction(mat);

問題例

• Library Checker: Determinant of Matrix (arbitrary mod): 任意 mod のケースでも特性多項式が $O(n^3)$ で計算可能である．

Depends on

• linear_algebra_matrix/matrix.hpp

Required by

• Determinant of $M_0 + M_1 x$ （各要素が高々 $1$ 次の行列の行列式） (linear_algebra_matrix/determinant_of_first_degree_poly_mat.hpp)

Verified with

• linear_algebra_matrix/test/characteristic_poly.test.cpp
• linear_algebra_matrix/test/determinant_arbitrary_mod.test.cpp
• linear_algebra_matrix/test/determinant_of_first_degree_poly_mat.yuki1907.test.cpp

Code

#pragma once
#include "matrix.hpp"
#include <cassert>
#include <utility>

// Upper Hessenberg reduction of square matrices
// Complexity: O(n^3)
// Reference:
// http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf
template <class Tp> void hessenberg_reduction(matrix<Tp> &M) {
    assert(M.height() == M.width());
    const int N = M.height();
    for (int r = 0; r < N - 2; r++) {
        int piv = matrix<Tp>::choose_pivot(M, r + 1, r);
        if (piv < 0) continue;
        for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
        for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);

        const auto rinv = Tp(1) / M[r + 1][r];
        for (int i = r + 2; i < N; i++) {
            const auto n = M[i][r] * rinv;
            for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * n;
            for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * n;
        }
    }
}

template <class Ring> void ring_hessenberg_reduction(matrix<Ring> &M) {
    assert(M.height() == M.width());
    const int N = M.height();
    for (int r = 0; r < N - 2; r++) {
        int piv = matrix<Ring>::choose_pivot(M, r + 1, r);
        if (piv < 0) continue;
        for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
        for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);

        for (int i = r + 2; i < N; i++) {
            if (M[i][r] == Ring()) continue;
            Ring a = M[r + 1][r], b = M[i][r], m00 = 1, m01 = 0, m10 = 0, m11 = 1;
            while (a != Ring() and b != Ring()) {
                if (a.val() > b.val()) {
                    auto d = a.val() / b.val();
                    a -= b * d, m00 -= m10 * d, m01 -= m11 * d;
                } else {
                    auto d = b.val() / a.val();
                    b -= a * d, m10 -= m00 * d, m11 -= m01 * d;
                }
            }
            if (a == Ring()) std::swap(a, b), std::swap(m00, m10), std::swap(m01, m11);

            for (int j = 0; j < N; j++) {
                Ring anew = M[r + 1][j] * m00 + M[i][j] * m01;
                Ring bnew = M[r + 1][j] * m10 + M[i][j] * m11;
                M[r + 1][j] = anew;
                M[i][j] = bnew;
            }
            assert(M[i][r] == 0);

            for (int j = 0; j < N; j++) {
                Ring anew = M[j][r + 1] * m11 - M[j][i] * m10;
                Ring bnew = -M[j][r + 1] * m01 + M[j][i] * m00;
                M[j][r + 1] = anew;
                M[j][i] = bnew;
            }
        }
    }
}

#line 2 "linear_algebra_matrix/matrix.hpp"
#include <algorithm>
#include <cassert>
#include <cmath>
#include <iterator>
#include <type_traits>
#include <utility>
#include <vector>

namespace matrix_ {
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 matrix_

template <typename T> struct matrix {
    int H, W;
    std::vector<T> elem;
    typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; }
    inline T &at(int i, int j) { return elem[i * W + j]; }
    inline T get(int i, int j) const { return elem[i * W + j]; }
    int height() const { return H; }
    int width() const { return W; }
    std::vector<std::vector<T>> vecvec() const {
        std::vector<std::vector<T>> ret(H);
        for (int i = 0; i < H; i++) {
            std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i]));
        }
        return ret;
    }
    operator std::vector<std::vector<T>>() const { return vecvec(); }
    matrix() = default;
    matrix(int H, int W) : H(H), W(W), elem(H * W) {}
    matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) {
        for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem));
    }

    template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2::id();
    }
    template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2(1);
    }

    static matrix Identity(int N) {
        matrix ret(N, N);
        for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>();
        return ret;
    }

    matrix operator-() const {
        matrix ret(H, W);
        for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i];
        return ret;
    }
    matrix operator*(const T &v) const {
        matrix ret = *this;
        for (auto &x : ret.elem) x *= v;
        return ret;
    }
    matrix operator/(const T &v) const {
        matrix ret = *this;
        const T vinv = _T_id<T>() / v;
        for (auto &x : ret.elem) x *= vinv;
        return ret;
    }
    matrix operator+(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i];
        return ret;
    }
    matrix operator-(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i];
        return ret;
    }
    matrix operator*(const matrix &r) const {
        matrix ret(H, r.W);
        for (int i = 0; i < H; i++) {
            for (int k = 0; k < W; k++) {
                for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j);
            }
        }
        return ret;
    }
    matrix &operator*=(const T &v) { return *this = *this * v; }
    matrix &operator/=(const T &v) { return *this = *this / v; }
    matrix &operator+=(const matrix &r) { return *this = *this + r; }
    matrix &operator-=(const matrix &r) { return *this = *this - r; }
    matrix &operator*=(const matrix &r) { return *this = *this * r; }
    bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; }
    bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; }
    bool operator<(const matrix &r) const { return elem < r.elem; }
    matrix pow(int64_t n) const {
        matrix ret = Identity(H);
        bool ret_is_id = true;
        if (n == 0) return ret;
        for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
            if (!ret_is_id) ret *= ret;
            if ((n >> i) & 1) ret *= (*this), ret_is_id = false;
        }
        return ret;
    }
    std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const {
        matrix x = *this;
        while (n) {
            if (n & 1) vec = x * vec;
            x *= x;
            n >>= 1;
        }
        return vec;
    };
    matrix transpose() const {
        matrix ret(W, H);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j);
        }
        return ret;
    }
    // Gauss-Jordan elimination
    // - Require inverse for every non-zero element
    // - Complexity: O(H^2 W)
    template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        int piv = -1;
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c))))
                piv = j;
        }
        return piv;
    }
    template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) != T2()) return j;
        }
        return -1;
    }
    matrix gauss_jordan() const {
        int c = 0;
        matrix mtr(*this);
        std::vector<int> ws;
        ws.reserve(W);
        for (int h = 0; h < H; h++) {
            if (c == W) break;
            int piv = choose_pivot(mtr, h, c);
            if (piv == -1) {
                c++;
                h--;
                continue;
            }
            if (h != piv) {
                for (int w = 0; w < W; w++) {
                    std::swap(mtr[piv][w], mtr[h][w]);
                    mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant
                }
            }
            ws.clear();
            for (int w = c; w < W; w++) {
                if (mtr.at(h, w) != T()) ws.emplace_back(w);
            }
            const T hcinv = _T_id<T>() / mtr.at(h, c);
            for (int hh = 0; hh < H; hh++)
                if (hh != h) {
                    const T coeff = mtr.at(hh, c) * hcinv;
                    for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff;
                    mtr.at(hh, c) = T();
                }
            c++;
        }
        return mtr;
    }
    int rank_of_gauss_jordan() const {
        for (int i = H * W - 1; i >= 0; i--) {
            if (elem[i] != 0) return i / W + 1;
        }
        return 0;
    }
    int rank() const { return gauss_jordan().rank_of_gauss_jordan(); }

    T determinant_of_upper_triangle() const {
        T ret = _T_id<T>();
        for (int i = 0; i < H; i++) ret *= get(i, i);
        return ret;
    }
    int inverse() {
        assert(H == W);
        std::vector<std::vector<T>> ret = Identity(H), tmp = *this;
        int rank = 0;
        for (int i = 0; i < H; i++) {
            int ti = i;
            while (ti < H and tmp[ti][i] == T()) ti++;
            if (ti == H) {
                continue;
            } else {
                rank++;
            }
            ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]);
            T inv = _T_id<T>() / tmp[i][i];
            for (int j = 0; j < W; j++) ret[i][j] *= inv;
            for (int j = i + 1; j < W; j++) tmp[i][j] *= inv;
            for (int h = 0; h < H; h++) {
                if (i == h) continue;
                const T c = -tmp[h][i];
                for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c;
                for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c;
            }
        }
        *this = ret;
        return rank;
    }
    friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) {
        assert(m.W == int(v.size()));
        std::vector<T> ret(m.H);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j];
        }
        return ret;
    }
    friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) {
        assert(int(v.size()) == m.H);
        std::vector<T> ret(m.W);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j);
        }
        return ret;
    }
    std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; }
    std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); }
    template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) {
        os << "[(" << x.H << " * " << x.W << " matrix)";
        os << "\n[column sums: ";
        for (int j = 0; j < x.W; j++) {
            T s = T();
            for (int i = 0; i < x.H; i++) s += x.get(i, j);
            os << s << ",";
        }
        os << "]";
        for (int i = 0; i < x.H; i++) {
            os << "\n[";
            for (int j = 0; j < x.W; j++) os << x.get(i, j) << ",";
            os << "]";
        }
        os << "]\n";
        return os;
    }
    template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) {
        for (auto &v : x.elem) is >> v;
        return is;
    }
};
#line 5 "linear_algebra_matrix/hessenberg_reduction.hpp"

// Upper Hessenberg reduction of square matrices
// Complexity: O(n^3)
// Reference:
// http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf
template <class Tp> void hessenberg_reduction(matrix<Tp> &M) {
    assert(M.height() == M.width());
    const int N = M.height();
    for (int r = 0; r < N - 2; r++) {
        int piv = matrix<Tp>::choose_pivot(M, r + 1, r);
        if (piv < 0) continue;
        for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
        for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);

        const auto rinv = Tp(1) / M[r + 1][r];
        for (int i = r + 2; i < N; i++) {
            const auto n = M[i][r] * rinv;
            for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * n;
            for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * n;
        }
    }
}

template <class Ring> void ring_hessenberg_reduction(matrix<Ring> &M) {
    assert(M.height() == M.width());
    const int N = M.height();
    for (int r = 0; r < N - 2; r++) {
        int piv = matrix<Ring>::choose_pivot(M, r + 1, r);
        if (piv < 0) continue;
        for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
        for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);

        for (int i = r + 2; i < N; i++) {
            if (M[i][r] == Ring()) continue;
            Ring a = M[r + 1][r], b = M[i][r], m00 = 1, m01 = 0, m10 = 0, m11 = 1;
            while (a != Ring() and b != Ring()) {
                if (a.val() > b.val()) {
                    auto d = a.val() / b.val();
                    a -= b * d, m00 -= m10 * d, m01 -= m11 * d;
                } else {
                    auto d = b.val() / a.val();
                    b -= a * d, m10 -= m00 * d, m11 -= m01 * d;
                }
            }
            if (a == Ring()) std::swap(a, b), std::swap(m00, m10), std::swap(m01, m11);

            for (int j = 0; j < N; j++) {
                Ring anew = M[r + 1][j] * m00 + M[i][j] * m01;
                Ring bnew = M[r + 1][j] * m10 + M[i][j] * m11;
                M[r + 1][j] = anew;
                M[i][j] = bnew;
            }
            assert(M[i][r] == 0);

            for (int j = 0; j < N; j++) {
                Ring anew = M[j][r + 1] * m11 - M[j][i] * m10;
                Ring bnew = -M[j][r + 1] * m01 + M[j][i] * m00;
                M[j][r + 1] = anew;
                M[j][i] = bnew;
            }
        }
    }
}

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