行列の特性多項式他 (src/LinearAlgebra/characteristic_polynomial.hpp)

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Last update: 2024-09-24 14:37:39+09:00
Include: #include "src/LinearAlgebra/characteristic_polynomial.hpp"

関数	概要	計算量
`hessenberg(A, mint=false)`	正方行列 $A$ ( `Matrix` クラス ) のヘッセンベルグ型を返す. 第二引数が `true` : 互除法で計算. modintを前提にしてる. 第二引数が `false` : 乗法の逆元で計算. 体でないとダメ. デフォルト	$O (n^3)$
`characteristic_polynomial(A, mint=false)`	正方行列 $A$ ( `Matrix` クラス ) の特性多項式 $\phi(x)=\det(x \mathrm{I}-A)$ を返す. 第二引数が `true` : 互除法で計算. modintを前提にしてる. 第二引数が `false` : 乗法の逆元で計算. 体でないとダメ. デフォルト	$O (n^3)$
`det_of_first_degree_poly_mat(M0, M1)`	$\det(M_0 + M_1x)$ を計算. 体でないとダメ.	$O (n^3)$

参考

https://hitonanode.github.io/cplib-cpp/linear_algebra_matrix/hessenberg_reduction.hpp
https://hitonanode.github.io/cplib-cpp/linear_algebra_matrix/determinant_of_first_degree_poly_mat.hpp

Verify

Codeforces Round 459 (Div. 1) D. Stranger Trees (det_of_first_degree_poly_mat)

Depends on

Verified with

Code

#pragma once
#include <vector>
#include <algorithm>
#include "src/LinearAlgebra/Matrix.hpp"
template <class K> Matrix<K> hessenberg(Matrix<K> A, bool mint= false) {
 using _la_internal::is_zero;
 const size_t n= A.width();
 assert(n == A.height());
 for (size_t j= 0, i, r; j + 2 < n; ++j) {
  if constexpr (std::is_floating_point_v<K>) {
   for (i= j + 1, r= j + 2; r < n; ++r)
    if (std::abs(A[i][j]) < std::abs(A[r][j])) i= r;
  } else
   for (i= r= j + 1; r < n; ++r)
    if (A[r][j] != K()) i= r, r= n;
  if (i != j + 1) {
   for (r= 0; r < n; ++r) std::swap(A[j + 1][r], A[i][r]);
   for (; r--;) std::swap(A[r][j + 1], A[r][i]);
  }
  if (is_zero(A[j + 1][j])) continue;
  if (K s, iv; mint) {
   for (i= j + 2; i < n; ++i)
    if (!is_zero(A[i][j])) {
     K m00= K(1), m01= K(), m10= K(), m11= K(1);
     for (uint64_t a= A[j + 1][j].val(), b= A[i][j].val(), t, l; b;) l= b, b= a - (t= a / b) * b, a= l, s= m10, m10= m00 - m10 * t, m00= s, s= m11, m11= m01 - m11 * t, m01= s;
     for (r= 0; r < n; ++r) s= m00 * A[j + 1][r] + m01 * A[i][r], A[i][r]= m10 * A[j + 1][r] + m11 * A[i][r], A[j + 1][r]= s;
     for (; r--;) s= m11 * A[r][j + 1] - m10 * A[r][i], A[r][i]= m00 * A[r][i] - m01 * A[r][j + 1], A[r][j + 1]= s;
    }
  } else {
   for (iv= K(1) / A[j + 1][j], i= j + 2; i < n; ++i)
    if (!is_zero(A[i][j])) {
     for (s= A[i][r= j] * iv; r < n; ++r) A[i][r]-= s * A[j + 1][r];
     for (; r--;) A[r][j + 1]+= s * A[r][i];
    }
  }
 }
 return A;
}
template <class K> std::vector<K> characteristic_polynomial(const Matrix<K> &A, bool mint= false) {
 size_t n= A.width(), i= 0, k, j;
 assert(n == A.height());
 auto b= hessenberg(A, mint);
 std::vector<K> fss((n + 1) * (n + 2) / 2);
 K *pr= fss.data(), *nx= pr, prod, tmp, s;
 for (fss[0]= 1; i < n; ++i, pr= nx) {
  for (prod= 1, tmp= -b[i][i], nx= pr + i + 1, std::copy_n(pr, i + 1, nx + 1), k= 0; k <= i; ++k) nx[k]+= tmp * pr[k];
  for (j= i; j--;)
   for (pr-= j + 1, s= (prod*= b[j + 1][j]) * -b[j][i], k= 0; k <= j; ++k) nx[k]+= s * pr[k];
 }
 return std::vector<K>(fss.begin() + n * (n + 1) / 2, fss.end());
}
template <class K> std::vector<K> det_of_first_degree_poly_mat(Matrix<K> M0, Matrix<K> M1) {
 const size_t n= M0.height();
 assert(n == M1.height()), assert(n == M0.width()), assert(n == M1.width());
 size_t cnt= 0;
 K det= 1, v, iv;
 for (size_t p= 0, piv, r, i; p < n;) {
  if constexpr (std::is_floating_point_v<K>) {
   for (piv= p, r= p + 1; r < n; ++r)
    if (std::abs(M1[piv][p]) < std::abs(M1[r][p])) piv= r;
  } else
   for (piv= p; piv < n; ++piv)
    if (M1[piv][p] != K()) break;
  if (piv == n || _la_internal::is_zero(M1[piv][p])) {
   if (++cnt > n) return std::vector<K>(n + 1);
   for (r= p; r--;)
    for (v= M1[r][p], M1[r][p]= K(), i= n; i--;) M0[i][p]-= v * M0[i][r];
   for (i= n; i--;) std::swap(M0[i][p], M1[i][p]);
   continue;
  }
  if (piv != p) {
   for (det*= -1, i= 0; i < n; ++i) std::swap(M0[p][i], M0[piv][i]);
   for (; i--;) std::swap(M1[p][i], M1[piv][i]);
  }
  for (det*= v= M1[p][p], iv= K(1) / v, i= 0; i < n; ++i) M0[p][i]*= iv;
  for (; i--;) M1[p][i]*= iv;
  for (r= n; r--;)
   if (r != p) {
    for (v= M1[r][p], i= 0; i < n; ++i) M0[r][i]-= v * M0[p][i];
    for (; i--;) M1[r][i]-= v * M1[p][i];
   }
  ++p;
 }
 auto poly= characteristic_polynomial(M0 * -1);
 poly.erase(poly.begin(), poly.begin() + cnt);
 for (auto &x: poly) x*= det;
 return poly.resize(n + 1), poly;
}

#line 2 "src/LinearAlgebra/characteristic_polynomial.hpp"
#include <vector>
#include <algorithm>
#line 2 "src/LinearAlgebra/Matrix.hpp"
#include <cassert>
#line 2 "src/LinearAlgebra/Vector.hpp"
#include <cstdint>
#include <iostream>
#include <valarray>
namespace _la_internal {
using namespace std;
template <class R> struct Vector {
 valarray<R> dat;
 Vector()= default;
 Vector(size_t n): dat(n) {}
 Vector(size_t n, const R &v): dat(v, n) {}
 Vector(const initializer_list<R> &v): dat(v) {}
 R &operator[](int i) { return dat[i]; }
 const R &operator[](int i) const { return dat[i]; }
 bool operator==(const Vector &r) const {
  if (dat.size() != r.dat.size()) return false;
  for (int i= dat.size(); i--;)
   if (dat[i] != r.dat[i]) return false;
  return true;
 }
 bool operator!=(const Vector &r) const { return !(*this == r); }
 explicit operator bool() const { return dat.size(); }
 Vector operator-() const { return Vector(dat.size())-= *this; }
 Vector &operator+=(const Vector &r) { return dat+= r.dat, *this; }
 Vector &operator-=(const Vector &r) { return dat-= r.dat, *this; }
 Vector &operator*=(const R &r) { return dat*= r, *this; }
 Vector operator+(const Vector &r) const { return Vector(*this)+= r; }
 Vector operator-(const Vector &r) const { return Vector(*this)-= r; }
 Vector operator*(const R &r) const { return Vector(*this)*= r; }
 size_t size() const { return dat.size(); }
 friend R dot(const Vector<R> &a, const Vector<R> &b) { return assert(a.size() == b.size()), (a.dat * b.dat).sum(); }
};
using u128= __uint128_t;
using u64= uint64_t;
using u8= uint8_t;
class Ref {
 u128 *ref;
 u8 i;
public:
 Ref(u128 *ref, u8 i): ref(ref), i(i) {}
 Ref &operator=(const Ref &r) { return *this= bool(r); }
 Ref &operator=(bool b) { return *ref&= ~(u128(1) << i), *ref|= u128(b) << i, *this; }
 Ref &operator|=(bool b) { return *ref|= u128(b) << i, *this; }
 Ref &operator&=(bool b) { return *ref&= ~(u128(!b) << i), *this; }
 Ref &operator^=(bool b) { return *ref^= u128(b) << i, *this; }
 operator bool() const { return (*ref >> i) & 1; }
};
template <> class Vector<bool> {
 size_t n;
public:
 valarray<u128> dat;
 Vector(): n(0) {}
 Vector(size_t n): n(n), dat((n + 127) >> 7) {}
 Vector(size_t n, bool b): n(n), dat(-u128(b), (n + 127) >> 7) {
  if (int k= n & 127; k) dat[dat.size() - 1]&= (u128(1) << k) - 1;
 }
 Vector(const initializer_list<bool> &v): n(v.size()), dat((n + 127) >> 7) {
  int i= 0;
  for (bool b: v) dat[i >> 7]|= u128(b) << (i & 127), ++i;
 }
 Ref operator[](int i) { return {begin(dat) + (i >> 7), u8(i & 127)}; }
 bool operator[](int i) const { return (dat[i >> 7] >> (i & 127)) & 1; }
 bool operator==(const Vector &r) const {
  if (dat.size() != r.dat.size()) return false;
  for (int i= dat.size(); i--;)
   if (dat[i] != r.dat[i]) return false;
  return true;
 }
 bool operator!=(const Vector &r) const { return !(*this == r); }
 explicit operator bool() const { return n; }
 Vector operator-() const { return Vector(*this); }
 Vector &operator+=(const Vector &r) { return dat^= r.dat, *this; }
 Vector &operator-=(const Vector &r) { return dat^= r.dat, *this; }
 Vector &operator*=(bool b) { return dat*= b, *this; }
 Vector operator+(const Vector &r) const { return Vector(*this)+= r; }
 Vector operator-(const Vector &r) const { return Vector(*this)-= r; }
 Vector operator*(bool b) const { return Vector(*this)*= b; }
 size_t size() const { return n; }
 friend bool dot(const Vector<bool> &a, const Vector<bool> &b) {
  assert(a.size() == b.size());
  u128 v= 0;
  for (int i= a.dat.size(); i--;) v^= a.dat[i] & b.dat[i];
  return __builtin_parityll(v >> 64) ^ __builtin_parityll(u64(v));
 }
};
template <class R> Vector<R> operator*(const R &r, const Vector<R> &v) { return v * r; }
template <class R> ostream &operator<<(ostream &os, const Vector<R> &v) {
 os << '[';
 for (int _= 0, __= v.size(); _ < __; ++_) os << (_ ? ", " : "") << v[_];
 return os << ']';
}
}
using _la_internal::Vector;
#line 5 "src/LinearAlgebra/Matrix.hpp"
namespace _la_internal {
template <class R, class D> struct Mat {
 Mat(): W(0) {}
 Mat(size_t h, size_t w): W(w), dat(h * w) {}
 Mat(size_t h, size_t w, R v): W(w), dat(v, h * w) {}
 Mat(initializer_list<initializer_list<R>> v): W(v.size() ? v.begin()->size() : 0), dat(v.size() * W) {
  auto it= begin(dat);
  for (const auto &r: v) {
   assert(r.size() == W);
   for (R x: r) *it++= x;
  }
 }
 size_t width() const { return W; }
 size_t height() const { return W ? dat.size() / W : 0; }
 auto operator[](int i) { return begin(dat) + i * W; }
 auto operator[](int i) const { return begin(dat) + i * W; }
protected:
 size_t W;
 valarray<R> dat;
 void add(const Mat &r) { assert(dat.size() == r.dat.size()), assert(W == r.W), dat+= r.dat; }
 D mul(const Mat &r) const {
  const size_t h= height(), w= r.W, l= W;
  assert(l == r.height());
  D ret(h, w);
  auto a= begin(dat);
  auto c= begin(ret.dat);
  for (int i= h; i--; c+= w) {
   auto b= begin(r.dat);
   for (int k= l; k--; ++a) {
    auto d= c;
    auto v= *a;
    for (int j= w; j--; ++b, ++d) *d+= v * *b;
   }
  }
  return ret;
 }
 Vector<R> mul(const Vector<R> &r) const {
  assert(W == r.size());
  const size_t h= height();
  Vector<R> ret(h);
  auto a= begin(dat);
  for (size_t i= 0; i < h; ++i)
   for (size_t k= 0; k < W; ++k, ++a) ret[i]+= *a * r[k];
  return ret;
 }
};
template <class D> struct Mat<bool, D> {
 struct Array {
  u128 *bg;
  Array(u128 *it): bg(it) {}
  Ref operator[](int i) { return Ref{bg + (i >> 7), u8(i & 127)}; }
  bool operator[](int i) const { return (bg[i >> 7] >> (i & 127)) & 1; }
 };
 struct ConstArray {
  const u128 *bg;
  ConstArray(const u128 *it): bg(it) {}
  bool operator[](int i) const { return (bg[i >> 7] >> (i & 127)) & 1; }
 };
 Mat(): H(0), W(0), m(0) {}
 Mat(size_t h, size_t w): H(h), W(w), m((w + 127) >> 7), dat(h * m) {}
 Mat(size_t h, size_t w, bool b): H(h), W(w), m((w + 127) >> 7), dat(-u128(b), h * m) {
  if (size_t i= h, k= w & 127; k)
   for (u128 s= (u128(1) << k) - 1; i--;) dat[i * m]&= s;
 }
 Mat(const initializer_list<initializer_list<bool>> &v): H(v.size()), W(H ? v.begin()->size() : 0), m((W + 127) >> 7), dat(H * m) {
  auto it= begin(dat);
  for (const auto &r: v) {
   assert(r.size() == W);
   int i= 0;
   for (bool b: r) it[i >> 7]|= u128(b) << (i & 127), ++i;
   it+= m;
  }
 }
 size_t width() const { return W; }
 size_t height() const { return H; }
 Array operator[](int i) { return {begin(dat) + i * m}; }
 ConstArray operator[](int i) const { return {begin(dat) + i * m}; }
 ConstArray get(int i) const { return {begin(dat) + i * m}; }
protected:
 size_t H, W, m;
 valarray<u128> dat;
 void add(const Mat &r) { assert(H == r.H), assert(W == r.W), dat^= r.dat; }
 D mul(const Mat &r) const {
  assert(W == r.H);
  D ret(H, r.W);
  valarray<u128> tmp(r.m << 8);
  auto y= begin(r.dat);
  for (size_t l= 0; l < W; l+= 8) {
   auto t= begin(tmp) + r.m;
   for (int i= 0, n= min<size_t>(8, W - l); i < n; ++i, y+= r.m) {
    auto u= begin(tmp);
    for (int s= 1 << i; s--;) {
     auto z= y;
     for (int j= r.m; j--; ++u, ++t, ++z) *t= *u ^ *z;
    }
   }
   auto a= begin(dat) + (l >> 7);
   auto c= begin(ret.dat);
   for (int i= H; i--; a+= m) {
    auto u= begin(tmp) + ((*a >> (l & 127)) & 255) * r.m;
    for (int j= r.m; j--; ++c, ++u) *c^= *u;
   }
  }
  return ret;
 }
 Vector<bool> mul(const Vector<bool> &r) const {
  assert(W == r.size());
  Vector<bool> ret(H);
  auto a= begin(dat);
  for (size_t i= 0; i < H; ++i) {
   u128 v= 0;
   for (size_t j= 0; j < m; ++j, ++a) v^= *a & r.dat[j];
   ret[i]= __builtin_parityll(v >> 64) ^ __builtin_parityll(u64(v));
  }
  return ret;
 }
};
template <class R> struct Matrix: public Mat<R, Matrix<R>> {
 using Mat<R, Matrix<R>>::Mat;
 explicit operator bool() const { return this->W; }
 static Matrix identity(int n) {
  Matrix ret(n, n);
  for (; n--;) ret[n][n]= R(true);
  return ret;
 }
 Matrix submatrix(const vector<int> &rows, const vector<int> &cols) const {
  Matrix ret(rows.size(), cols.size());
  for (int i= rows.size(); i--;)
   for (int j= cols.size(); j--;) ret[i][j]= (*this)[rows[i]][cols[j]];
  return ret;
 }
 Matrix submatrix_rm(vector<int> rows, vector<int> cols) const {
  sort(begin(rows), end(rows)), sort(begin(cols), end(cols)), rows.erase(unique(begin(rows), end(rows)), end(rows)), cols.erase(unique(begin(cols), end(cols)), end(cols));
  const int H= this->height(), W= this->width(), n= rows.size(), m= cols.size();
  vector<int> rs(H - n), cs(W - m);
  for (int i= 0, j= 0, k= 0; i < H; ++i)
   if (j < n && rows[j] == i) ++j;
   else rs[k++]= i;
  for (int i= 0, j= 0, k= 0; i < W; ++i)
   if (j < m && cols[j] == i) ++j;
   else cs[k++]= i;
  return submatrix(rs, cs);
 }
 bool operator==(const Matrix &r) const {
  if (this->width() != r.width() || this->height() != r.height()) return false;
  for (int i= this->dat.size(); i--;)
   if (this->dat[i] != r.dat[i]) return false;
  return true;
 }
 bool operator!=(const Matrix &r) const { return !(*this == r); }
 Matrix &operator*=(const Matrix &r) { return *this= this->mul(r); }
 Matrix operator*(const Matrix &r) const { return this->mul(r); }
 Matrix &operator*=(R r) { return this->dat*= r, *this; }
 template <class T> Matrix operator*(T r) const {
  static_assert(is_convertible_v<T, R>);
  return Matrix(*this)*= r;
 }
 Matrix &operator+=(const Matrix &r) { return this->add(r), *this; }
 Matrix operator+(const Matrix &r) const { return Matrix(*this)+= r; }
 Vector<R> operator*(const Vector<R> &r) const { return this->mul(r); }
 Vector<R> operator()(const Vector<R> &r) const { return this->mul(r); }
 Matrix pow(uint64_t k) const {
  size_t W= this->width();
  assert(W == this->height());
  for (Matrix ret= identity(W), b= *this;; b*= b)
   if (k & 1 ? ret*= b, !(k>>= 1) : !(k>>= 1)) return ret;
 }
};
template <class R, class T> Matrix<R> operator*(const T &r, const Matrix<R> &m) { return m * r; }
template <class R> ostream &operator<<(ostream &os, const Matrix<R> &m) {
 os << "\n[";
 for (int i= 0, h= m.height(); i < h; os << ']', ++i) {
  if (i) os << "\n ";
  os << '[';
  for (int j= 0, w= m.width(); j < w; ++j) os << (j ? ", " : "") << m[i][j];
 }
 return os << ']';
}
template <class K> static bool is_zero(K x) {
 if constexpr (is_floating_point_v<K>) return abs(x) < 1e-8;
 else return x == K();
}
}
using _la_internal::Matrix;
#line 5 "src/LinearAlgebra/characteristic_polynomial.hpp"
template <class K> Matrix<K> hessenberg(Matrix<K> A, bool mint= false) {
 using _la_internal::is_zero;
 const size_t n= A.width();
 assert(n == A.height());
 for (size_t j= 0, i, r; j + 2 < n; ++j) {
  if constexpr (std::is_floating_point_v<K>) {
   for (i= j + 1, r= j + 2; r < n; ++r)
    if (std::abs(A[i][j]) < std::abs(A[r][j])) i= r;
  } else
   for (i= r= j + 1; r < n; ++r)
    if (A[r][j] != K()) i= r, r= n;
  if (i != j + 1) {
   for (r= 0; r < n; ++r) std::swap(A[j + 1][r], A[i][r]);
   for (; r--;) std::swap(A[r][j + 1], A[r][i]);
  }
  if (is_zero(A[j + 1][j])) continue;
  if (K s, iv; mint) {
   for (i= j + 2; i < n; ++i)
    if (!is_zero(A[i][j])) {
     K m00= K(1), m01= K(), m10= K(), m11= K(1);
     for (uint64_t a= A[j + 1][j].val(), b= A[i][j].val(), t, l; b;) l= b, b= a - (t= a / b) * b, a= l, s= m10, m10= m00 - m10 * t, m00= s, s= m11, m11= m01 - m11 * t, m01= s;
     for (r= 0; r < n; ++r) s= m00 * A[j + 1][r] + m01 * A[i][r], A[i][r]= m10 * A[j + 1][r] + m11 * A[i][r], A[j + 1][r]= s;
     for (; r--;) s= m11 * A[r][j + 1] - m10 * A[r][i], A[r][i]= m00 * A[r][i] - m01 * A[r][j + 1], A[r][j + 1]= s;
    }
  } else {
   for (iv= K(1) / A[j + 1][j], i= j + 2; i < n; ++i)
    if (!is_zero(A[i][j])) {
     for (s= A[i][r= j] * iv; r < n; ++r) A[i][r]-= s * A[j + 1][r];
     for (; r--;) A[r][j + 1]+= s * A[r][i];
    }
  }
 }
 return A;
}
template <class K> std::vector<K> characteristic_polynomial(const Matrix<K> &A, bool mint= false) {
 size_t n= A.width(), i= 0, k, j;
 assert(n == A.height());
 auto b= hessenberg(A, mint);
 std::vector<K> fss((n + 1) * (n + 2) / 2);
 K *pr= fss.data(), *nx= pr, prod, tmp, s;
 for (fss[0]= 1; i < n; ++i, pr= nx) {
  for (prod= 1, tmp= -b[i][i], nx= pr + i + 1, std::copy_n(pr, i + 1, nx + 1), k= 0; k <= i; ++k) nx[k]+= tmp * pr[k];
  for (j= i; j--;)
   for (pr-= j + 1, s= (prod*= b[j + 1][j]) * -b[j][i], k= 0; k <= j; ++k) nx[k]+= s * pr[k];
 }
 return std::vector<K>(fss.begin() + n * (n + 1) / 2, fss.end());
}
template <class K> std::vector<K> det_of_first_degree_poly_mat(Matrix<K> M0, Matrix<K> M1) {
 const size_t n= M0.height();
 assert(n == M1.height()), assert(n == M0.width()), assert(n == M1.width());
 size_t cnt= 0;
 K det= 1, v, iv;
 for (size_t p= 0, piv, r, i; p < n;) {
  if constexpr (std::is_floating_point_v<K>) {
   for (piv= p, r= p + 1; r < n; ++r)
    if (std::abs(M1[piv][p]) < std::abs(M1[r][p])) piv= r;
  } else
   for (piv= p; piv < n; ++piv)
    if (M1[piv][p] != K()) break;
  if (piv == n || _la_internal::is_zero(M1[piv][p])) {
   if (++cnt > n) return std::vector<K>(n + 1);
   for (r= p; r--;)
    for (v= M1[r][p], M1[r][p]= K(), i= n; i--;) M0[i][p]-= v * M0[i][r];
   for (i= n; i--;) std::swap(M0[i][p], M1[i][p]);
   continue;
  }
  if (piv != p) {
   for (det*= -1, i= 0; i < n; ++i) std::swap(M0[p][i], M0[piv][i]);
   for (; i--;) std::swap(M1[p][i], M1[piv][i]);
  }
  for (det*= v= M1[p][p], iv= K(1) / v, i= 0; i < n; ++i) M0[p][i]*= iv;
  for (; i--;) M1[p][i]*= iv;
  for (r= n; r--;)
   if (r != p) {
    for (v= M1[r][p], i= 0; i < n; ++i) M0[r][i]-= v * M0[p][i];
    for (; i--;) M1[r][i]-= v * M1[p][i];
   }
  ++p;
 }
 auto poly= characteristic_polynomial(M0 * -1);
 poly.erase(poly.begin(), poly.begin() + cnt);
 for (auto &x: poly) x*= det;
 return poly.resize(n + 1), poly;
}

行列の特性多項式 他 (src/LinearAlgebra/characteristic_polynomial.hpp)

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行列の特性多項式他 (src/LinearAlgebra/characteristic_polynomial.hpp)