行列式 (determinant)
(include/emthrm/math/matrix/determinant.hpp)

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• Last update: 2023-12-25 04:31:42+09:00
• Include: #include "emthrm/math/matrix/determinant.hpp"

時間計算量

$O(N^3)$

仕様

名前	戻り値
template <typename T, typename U> U det(const Matrix<T>& a, const U eps);	$\lvert A \rvert$

参考文献

• https://github.com/beet-aizu/library/blob/ebcad58b8480962339b18e316b826e56752b90c8/matrix/matrix.cpp

TODO

• 固有多項式 (characteristic polynomial)
  • https://ja.wikipedia.org/wiki/%E5%9B%BA%E6%9C%89%E5%A4%9A%E9%A0%85%E5%BC%8F
  • https://github.com/yosupo06/library-checker-problems/issues/665
  • https://judge.yosupo.jp/problem/characteristic_polynomial
  • 問題例 “DETERMINATION”
  • 問題例 “Inversion of Tree”
• 任意の法の下での行列式
  • https://judge.yosupo.jp/problem/matrix_det_arbitrary_mod
  • https://github.com/yosupo06/library-checker-problems/issues/750
• 巡回行列 (circulant matrix) の行列式
  • https://ja.wikipedia.org/wiki/%E5%B7%A1%E5%9B%9E%E8%A1%8C%E5%88%97
  • https://mathlandscape.com/circulant-matrix/
  • 問題例「第59回TechFUL Coding Battle『Techちゃんと魔法の宝石』」
    • https://kotatsugame.hatenablog.com/entry/2023/06/26/235357

Submissons

https://judge.yosupo.jp/submission/2714

Depends on

• 行列 (matrix) (include/emthrm/math/matrix/matrix.hpp)

Required by

• 行列木定理 (Kirchhoff's matrix tree theorem) (include/emthrm/graph/matrix_tree_theorem.hpp)

Verified with

• グラフ/行列木定理 (test/graph/matrix_tree_theorem.test.cpp)
• 数学/行列/行列式 (test/math/matrix/determinant.test.cpp)

Code

#ifndef EMTHRM_MATH_MATRIX_DETERMINANT_HPP_
#define EMTHRM_MATH_MATRIX_DETERMINANT_HPP_

#include <algorithm>
#include <utility>

#include "emthrm/math/matrix/matrix.hpp"

namespace emthrm {

template <typename T, typename U>
U det(const Matrix<T>& a, const U eps) {
  const int n = a.nrow();
  Matrix<U> b(n, n);
  for (int i = 0; i < n; ++i) {
    std::copy(a[i].begin(), a[i].end(), b[i].begin());
  }
  U res = 1;
  for (int j = 0; j < n; ++j) {
    int pivot = -1;
    U mx = eps;
    for (int i = j; i < n; ++i) {
      const U abs = (b[i][j] < 0 ? -b[i][j] : b[i][j]);
      if (abs > mx) {
        pivot = i;
        mx = abs;
      }
    }
    if (pivot == -1) return 0;
    if (pivot != j) {
      std::swap(b[j], b[pivot]);
      res = -res;
    }
    res *= b[j][j];
    for (int k = j + 1; k < n; ++k) {
      b[j][k] /= b[j][j];
    }
    for (int i = j + 1; i < n; ++i) {
      for (int k = j + 1; k < n; ++k) {
        b[i][k] -= b[i][j] * b[j][k];
      }
    }
  }
  return res;
}

}  // namespace emthrm

#endif  // EMTHRM_MATH_MATRIX_DETERMINANT_HPP_

#line 1 "include/emthrm/math/matrix/determinant.hpp"



#include <algorithm>
#include <utility>

#line 1 "include/emthrm/math/matrix/matrix.hpp"



#include <vector>

namespace emthrm {

template <typename T>
struct Matrix {
  explicit Matrix(const int m, const int n, const T def = 0)
      : data(m, std::vector<T>(n, def)) {}

  int nrow() const { return data.size(); }
  int ncol() const { return data.empty() ? 0 : data.front().size(); }

  Matrix pow(long long exponent) const {
    const int n = nrow();
    Matrix<T> res(n, n, 0), tmp = *this;
    for (int i = 0; i < n; ++i) {
      res[i][i] = 1;
    }
    for (; exponent > 0; exponent >>= 1) {
      if (exponent & 1) res *= tmp;
      tmp *= tmp;
    }
    return res;
  }

  inline const std::vector<T>& operator[](const int i) const { return data[i]; }
  inline std::vector<T>& operator[](const int i) { return data[i]; }

  Matrix& operator=(const Matrix& x) = default;

  Matrix& operator+=(const Matrix& x) {
    const int m = nrow(), n = ncol();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        data[i][j] += x[i][j];
      }
    }
    return *this;
  }

  Matrix& operator-=(const Matrix& x) {
    const int m = nrow(), n = ncol();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        data[i][j] -= x[i][j];
      }
    }
    return *this;
  }

  Matrix& operator*=(const Matrix& x) {
    const int m = nrow(), l = ncol(), n = x.ncol();
    std::vector<std::vector<T>> res(m, std::vector<T>(n, 0));
    for (int i = 0; i < m; ++i) {
      for (int k = 0; k < l; ++k) {
        for (int j = 0; j < n; ++j) {
          res[i][j] += data[i][k] * x[k][j];
        }
      }
    }
    data.swap(res);
    return *this;
  }

  Matrix operator+(const Matrix& x) const { return Matrix(*this) += x; }
  Matrix operator-(const Matrix& x) const { return Matrix(*this) -= x; }
  Matrix operator*(const Matrix& x) const { return Matrix(*this) *= x; }

 private:
  std::vector<std::vector<T>> data;
};

}  // namespace emthrm


#line 8 "include/emthrm/math/matrix/determinant.hpp"

namespace emthrm {

template <typename T, typename U>
U det(const Matrix<T>& a, const U eps) {
  const int n = a.nrow();
  Matrix<U> b(n, n);
  for (int i = 0; i < n; ++i) {
    std::copy(a[i].begin(), a[i].end(), b[i].begin());
  }
  U res = 1;
  for (int j = 0; j < n; ++j) {
    int pivot = -1;
    U mx = eps;
    for (int i = j; i < n; ++i) {
      const U abs = (b[i][j] < 0 ? -b[i][j] : b[i][j]);
      if (abs > mx) {
        pivot = i;
        mx = abs;
      }
    }
    if (pivot == -1) return 0;
    if (pivot != j) {
      std::swap(b[j], b[pivot]);
      res = -res;
    }
    res *= b[j][j];
    for (int k = j + 1; k < n; ++k) {
      b[j][k] /= b[j][j];
    }
    for (int i = j + 1; i < n; ++i) {
      for (int k = j + 1; k < n; ++k) {
        b[i][k] -= b[i][j] * b[j][k];
      }
    }
  }
  return res;
}

}  // namespace emthrm

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