cp-library

C++ Library for Competitive Programming

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:heavy_check_mark: 連立一次方程式 (linear equation)
(include/emthrm/math/matrix/linear_equation.hpp)

時間計算量

$O(M^2 N)$

仕様

名前 戻り値
template <typename T, typename U = double>
std::vector<U> linear_equation(const Matrix<T>& a, const std::vector<T>& b, const U eps = 1e-8); 		$A \boldsymbol{x} = \boldsymbol{b}$ を満たす $\boldsymbol{x}$。ただし解なしのときは空配列を返す。

TODO

Submissons

https://onlinejudge.u-aizu.ac.jp/solutions/problem/2171/review/5899058/emthrm/C++17

Depends on

Verified with

Code

#ifndef EMTHRM_MATH_MATRIX_LINEAR_EQUATION_HPP_
#define EMTHRM_MATH_MATRIX_LINEAR_EQUATION_HPP_

#include <algorithm>
#include <cmath>
#include <vector>

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

namespace emthrm {

template <typename T, typename U = double>
std::vector<U> linear_equation(const Matrix<T>& a, const std::vector<T>& b,
                               const U eps = 1e-8) {
  const int m = a.nrow(), n = a.ncol();
  Matrix<U> c(m, n + 1);
  for (int i = 0; i < m; ++i) {
    std::copy(a[i].begin(), a[i].end(), c[i].begin());
    c[i].back() = b[i];
  }
  const int rank = gauss_jordan<true>(&c, eps);
  for (int row = rank; row < m; ++row) {
    if ((c[row].back() < 0 ? -c[row].back() : c[row].back()) > eps) {
      return std::vector<U>{};
    }
  }
  std::vector<U> res(n, 0);
  for (int i = 0, j = 0; i < rank; ++i) {
    while ((c[i][j] < 0 ? -c[i][j] : c[i][j]) <= eps) ++j;
    res[j++] = c[i].back();
  }
  return res;
}

}  // namespace emthrm

#endif  // EMTHRM_MATH_MATRIX_LINEAR_EQUATION_HPP_
#line 1 "include/emthrm/math/matrix/linear_equation.hpp"



#include <algorithm>
#include <cmath>
#include <vector>

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



#include <utility>

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



#line 5 "include/emthrm/math/matrix/matrix.hpp"

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 7 "include/emthrm/math/matrix/gauss_jordan.hpp"

namespace emthrm {

template <bool IS_EXTENDED = false, typename T>
int gauss_jordan(Matrix<T>* a, const T eps = 1e-8) {
  const int m = a->nrow(), n = a->ncol();
  int rank = 0;
  for (int col = 0; col < (IS_EXTENDED ? n - 1 : n); ++col) {
    int pivot = -1;
    T mx = eps;
    for (int row = rank; row < m; ++row) {
      const T abs = ((*a)[row][col] < 0 ? -(*a)[row][col] : (*a)[row][col]);
      if (abs > mx) {
        pivot = row;
        mx = abs;
      }
    }
    if (pivot == -1) continue;
    std::swap((*a)[rank], (*a)[pivot]);
    T tmp = (*a)[rank][col];
    for (int col2 = 0; col2 < n; ++col2) {
      (*a)[rank][col2] /= tmp;
    }
    for (int row = 0; row < m; ++row) {
      if (row != rank &&
          ((*a)[row][col] < 0 ? -(*a)[row][col] : (*a)[row][col]) > eps) {
        tmp = (*a)[row][col];
        for (int col2 = 0; col2 < n; ++col2) {
          (*a)[row][col2] -= (*a)[rank][col2] * tmp;
        }
      }
    }
    ++rank;
  }
  return rank;
}

}  // namespace emthrm


#line 10 "include/emthrm/math/matrix/linear_equation.hpp"

namespace emthrm {

template <typename T, typename U = double>
std::vector<U> linear_equation(const Matrix<T>& a, const std::vector<T>& b,
                               const U eps = 1e-8) {
  const int m = a.nrow(), n = a.ncol();
  Matrix<U> c(m, n + 1);
  for (int i = 0; i < m; ++i) {
    std::copy(a[i].begin(), a[i].end(), c[i].begin());
    c[i].back() = b[i];
  }
  const int rank = gauss_jordan<true>(&c, eps);
  for (int row = rank; row < m; ++row) {
    if ((c[row].back() < 0 ? -c[row].back() : c[row].back()) > eps) {
      return std::vector<U>{};
    }
  }
  std::vector<U> res(n, 0);
  for (int i = 0, j = 0; i < rank; ++i) {
    while ((c[i][j] < 0 ? -c[i][j] : c[i][j]) <= eps) ++j;
    res[j++] = c[i].back();
  }
  return res;
}

}  // namespace emthrm
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