3 线性代数

3.1 高斯消元

求解线性方程组，a[i][0] ~ a[i][n-1] 为系数，a[i][n] 为等号右边的常数，最后结果保存在a[i][n]中。

const int N = 110;
const double eps = 1e-8;
LL n;
double a[N][N];
LL gauss(){
    LL c, r;
    for (c = 0, r = 0; c < n; c ++ ){
        LL t = r;
        for (int i = r; i < n; i ++ )    //找到绝对值最大的行 
            if (fabs(a[i][c]) > fabs(a[t][c]))
                t = i;
        if (fabs(a[t][c]) < eps) continue;
        for (int j = c; j < n + 1; j ++ ) swap(a[t][j], a[r][j]);    //将绝对值最大的一行换到最顶端
        for (int j = n; j >= c; j -- ) a[r][j] /= a[r][c];    //将当前行首位变成 1
        for (int i = r + 1; i < n; i ++ )    //将下面列消成 0 
            if (fabs(a[i][c]) > eps)
                for (int j = n; j >= c; j -- )
                    a[i][j] -= a[r][j] * a[i][c];
        r ++ ;
    }
    if (r < n){
        for (int i = r; i < n; i ++ )
            if (fabs(a[i][n]) > eps)
                return 2;
        return 1;
    }
    for (int i = n - 1; i >= 0; i -- )
        for (int j = i + 1; j < n; j ++ )
            a[i][n] -= a[i][j] * a[j][n];
    return 0;
}
int main(){
    cin >> n;
    for (int i = 0; i < n; i ++ )
        for (int j = 0; j < n + 1; j ++ )
            cin >> a[i][j];
    LL t = gauss();
    if (t == 0){
        for (int i = 0; i < n; i ++ ){
            if (fabs(a[i][n]) < eps) a[i][n] = abs(a[i][n]);
            printf("%.2lf\n", a[i][n]);
        }
    }
    else if (t == 1) cout << "Infinite group solutions\n";
    else cout << "No solution\n";
    return 0;
}

3.2 线性基

constexpr int L = 60;
ll basis[L];
int cnt = 0;
bool insert(ll x) {
    for (int k = L - 1; k >= 0; k--) {
        if (((x >> k) & 1LL) == 0) continue;
        if (basis[k] == 0) {
            basis[k] = x;
            cnt++;
            return true;
        }
        x ^= basis[k];
    }
    return false;
}