C++ Library for Competitive Programming
#include "emthrm/math/matrix/gauss_jordan.hpp"
行基本変形を用いて行列を行階段形に変形するアルゴリズムである。
$O(M^2 N)$
名前 | 戻り値 | 要件 | 備考 |
---|---|---|---|
template <bool IS_EXTENDED = false, typename T> int gauss_jordan(Matrix<T>* a, const T eps = 1e-8);
|
行列 $A$ のランク | 要素の型は実数型または ModInt である。 |
IS_EXTENDED は $A$ が拡大係数行列かを表す。$A$ は行階段形に変形される。 |
https://judge.yosupo.jp/submission/179224
#ifndef EMTHRM_MATH_MATRIX_GAUSS_JORDAN_HPP_
#define EMTHRM_MATH_MATRIX_GAUSS_JORDAN_HPP_
#include <utility>
#include "emthrm/math/matrix/matrix.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
#endif // EMTHRM_MATH_MATRIX_GAUSS_JORDAN_HPP_
#line 1 "include/emthrm/math/matrix/gauss_jordan.hpp"
#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 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