Надо реализовать метод отражений Хаусхолдера для решения матриц разной размерности (от 2 до 14). Разобрался с примером для матрицы 3х3, но не понимаю как сделать алгоритм универсальным. Чем больше размерность, тем больше составляется матриц, которые перемножаются. Есть ли универсальный алгоритм, чтобы в одной функции описать его? Для матрицы 3х3 придумал только это, но даже для такой матрицы это ужасно не эффективно.
| C++ (Qt) |
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| int main()
{
SetConsoleOutputCP(1251); SetConsoleCP(1251);
int N;
cout << "Введите размерность системы: "; cin >> N;
double **arr = new double*[N];
for (int i = 0; i < N; i++)
arr[i] = new double [N];
double* f = new double[N];
srand(time(NULL));
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
arr[i][j] = rand() % 20 - 10;
}
}
for (int i = 0; i < N; i++)
f[i] = 1;
cout << "\nИсходная матрица: " << endl;
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
cout << arr[i][j] << " "<<"\t";
}
cout << endl;
}
cout << "\nВектор правой части: " << endl;
for (int i = 0; i < N; i++)
cout << f[i] << endl;
double* b = new double[N];
for (int i = 0; i < N; i++)
b[i] = arr[i][0];
double* c = new double[N];
c[0] = 1;
for (int i = 1; i < N; i++)
c[i] = 0;
double bm = 0, cm = 0, alpha = 0;
for (int i = 0; i < N; i++)
{
bm += b[i] * b[i];
cm += c[i] * c[i];
}
bm = sqrt(bm);
cm = sqrt(cm);
alpha = bm / cm;
double* bac = new double[N];
for (int i = 0; i < N; i++)
bac[i] = b[i] - alpha * c[i];
double bbac = 0;
for (int i = 0; i < N; i++)
bbac += b[i] * bac[i];
double* omega = new double[N];
for (int i = 0; i < N; i++)
omega[i] = bac[i] / sqrt(2 * bbac);
double** Omega = new double* [N];
for (int i = 0; i < N; i++)
Omega[i] = new double[N];
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
Omega[i][j] = omega[i] * omega[j];
}
}
double** E = new double* [N];
for (int i = 0; i < N; i++)
E[i] = new double[N];
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
if (i == j)
E[i][j] = 1;
else
E[i][j] = 0;
}
}
double** U1 = new double* [N];
for (int i = 0; i < N; i++)
U1[i] = new double[N];
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
U1[i][j] = E[i][j] - 2 * Omega[i][j];
}
}
double** U1A = new double* [N];
for (int i = 0; i < N; i++)
U1A[i] = new double[N];
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
double tmp = 0;
for (int k = 0; k < N; k++)
{
tmp += U1[i][k] * arr[k][j];
}
U1A[i][j] = tmp;
}
}
double* U1B = new double[N];
for (int i = 0; i < N; i++)
{
double temp = 0;
for (int j = 0; j < N; j++)
{
temp += U1[i][j]* f[j];
}
U1B[i] = temp;
}
double* b1 = new double[N];
for (int i = 1; i < N; i++)
b1[i] = U1A[i][1];
double* c1 = new double[N];
c1[1] = 1;
for (int i = 2; i < N; i++)
c1[i] = 0;
double bm1 = 0, cm1 = 0, alpha1 = 0;
for (int i = 1; i < N; i++)
{
bm1 += b1[i] * b1[i];
cm1 += c1[i] * c1[i];
}
bm1 = sqrt(bm1);
cm1 = sqrt(cm1);
alpha1 = bm1 / cm1;
double* bac1 = new double[N];
for (int i = 1; i < N; i++)
bac1[i] = b1[i] - alpha1 * c1[i];
double bbac1 = 0;
for (int i = 1; i < N; i++)
bbac1 += b1[i] * bac1[i];
double* omega1 = new double[N];
for (int i = 1; i < N; i++)
omega1[i] = bac1[i] / sqrt(2 * bbac1);
double** Omega1 = new double* [N];
for (int i = 1; i < N; i++)
Omega1[i] = new double[N];
for (int i = 1; i < N; i++)
{
for (int j = 1; j < N; j++)
{
Omega1[i][j] = omega1[i] * omega1[j];
}
}
double** EE = new double* [N];
for (int i = 1; i < N; i++)
EE[i] = new double[N];
for (int i = 1; i < N; i++)
{
for (int j = 1; j < N; j++)
{
EE[i][j] = E[i][j] - 2 * Omega1[i][j];
}
}
double** U2 = new double* [N];
for (int i = 0; i < N; i++)
U2[i] = new double[N];
U2[0][0] = 1; U2[0][1] = 0; U2[0][2] = 0;
U2[1][0] = 0; U2[2][0] = 0;
for (int i = 1; i < N; i++)
{
for (int j = 1; j < N; j++)
{
U2[i][j] = 0;
double tt = EE[i][j];
U2[i][j] = tt;
}
}
double** U2U1 = new double* [N];
for (int i = 0; i < N; i++)
U2U1[i] = new double[N];
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
double tmp = 0;
for (int k = 0; k < N; k++)
{
tmp += U2[i][k] * U1[k][j];
}
U2U1[i][j] = tmp;
}
}
double** U2U1A = new double* [N];
for (int i = 0; i < N; i++)
U2U1A[i] = new double[N];
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
{
double tmp = 0;
for (int k = 0; k < N; k++)
{
tmp += U2U1[i][k] * arr[k][j];
}
U2U1A[i][j] = tmp;
}
}
double* U2U1B = new double[N];
for (int i = 0; i < N; i++)
{
double temp = 0;
for (int j = 0; j < N; j++)
{
temp += U2U1[i][j] * f[j];
}
U2U1B[i] = temp;
}
double* x = new double[N];
x[N - 1] = U2U1B[N - 1] / U2U1A[N - 1][N - 1];
x[N - 2] = (U2U1B[N - 2] - U2U1A[N - 2][N - 1] * x[N - 1]) / U2U1A[N - 2][N - 2];
x[N-3]= (U2U1B[N - 3] - U2U1A[N - 3][N - 1] * x[N - 1] - U2U1A[N-3][N-2]*x[N-2]) / U2U1A[N - 3][N - 3];
cout << "\nx: " << endl;
for (int i = 0; i < N; i++)
{
cout << x[i] << endl;
}
for (int i = 0; i < N; i++)
delete[]arr[i];
delete[]arr;
delete[]f;
delete[]b;
delete[]c;
delete[]bac;
delete[]omega;
for (int i = 0; i < N; i++)
delete[]Omega[i];
delete[]Omega;
for (int i = 0; i < N; i++)
delete[]E[i];
delete[]E;
for (int i = 0; i < N; i++)
delete[]U1[i];
delete[]U1;
for (int i = 0; i < N; i++)
delete[]U1A[i];
delete[]U1A;
delete[]U1B;
delete[]b1;
delete[]c1;
delete[]bac1;
delete[]omega1;
for (int i = 1; i < N; i++)
delete[]Omega1[i];
delete[]Omega1;
for (int i = 1; i < N; i++)
delete[]EE[i];
delete[]EE;
for (int i = 0; i < N; i++)
delete[]U2[i];
delete[]U2;
for (int i = 0; i < N; i++)
delete[]U2U1[i];
delete[]U2U1;
for (int i = 0; i < N; i++)
delete[]U2U1A[i];
delete[]U2U1A;
delete[]U2U1B;
delete[]x;
system("pause");
return 0;
} |
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