ASSISTANT
我将分别给出串行Jacobi迭代、串行Gauss-Seidel迭代、并行Jacobi迭代以及并行Gauss-Seidel迭代的C++代码和其运行时间度量。
```cpp
#include <iostream>
#include <vector>
#include <cmath>
#include <omp.h>
using namespace std;
const double EPS = 1e-6;
// 串行Jacobi迭代
vector<double> jacobiSerial(const vector<vector<double>> &A, const vector<double> &b, const vector<double> &x0){
int n = A.size();
vector<double> x(x0);
vector<double> x_new(n);
double diff;
do{
diff = 0;
for (int i = 0; i < n; i++){
double sum = 0;
for (int j = 0; j < n; j++){
if (i != j)
sum += A[i][j] * x[j];
}
x_new[i] = (b[i] - sum) / A[i][i];
diff += abs(x_new[i] - x[i]);
}
x.swap(x_new);
} while (diff > EPS);
return x;
}
// 串行Gauss-Seidel迭代
vector<double> gaussSeidelSerial(const vector<vector<double>> &A, const vector<double> &b, const vector<double> &x0){
int n = A.size();
vector<double> x(x0);
double diff;
do{
diff = 0;
for (int i = 0; i < n; i++){
double sum = 0;
for (int j = 0; j < n; j++){
if (i != j)
sum += A[i][j] * x[j];
}
double x_new = (b[i] - sum) / A[i][i];
diff += abs(x_new - x[i]);
x[i] = x_new;
}
} while (diff > EPS);
return x;
}
// 并行Jacobi迭代
vector<double> jacobiParallel(const vector<vector<double>> &A, const vector<double> &b, const vector<double> &x0){
int n = A.size();
vector<double> x(x0);
vector<double> x_new(n);
double diff;
do {
diff = 0;
#pragma omp parallel for reduction(+:diff)
for (int i = 0; i < n; i++) {
double sum = 0;
for (int j = 0; j < n; j++) {
if (i != j)
sum += A[i][j] * x[j];
}
x_new[i] = (b[i] - sum) / A[i][i];
diff += abs(x_new[i] - x[i]);
}
x.swap(x_new);
} while (diff > EPS);
return x;
}
// 并行Gauss-Seidel迭代
vector<double> gaussSeidelParallel(const vector<vector<double>> &A, const vector<double> &b, const vector<double> &x0) {
int n = A.size();
vector<double> x(x0);
double diff;
do {
diff = 0;
#pragma omp parallel for reduction(+:diff)
for (int i = 0; i < n; i++) {
double sum = 0;
for (int j = 0; j < n; j++) {
if (i != j)
sum += A[i][j] * x[j];
}
double x_new = (b[i] - sum) / A[i][i];
diff += abs(x_new - x[i]);
x[i] = x_new;
}
} while (diff > EPS);
return x;
}
int main() {
vector<vector<double>> A = {
{4, -1, 0, 0},
{-1, 4, -1, 0},
{0, -1, 4, -1},
{0, 0, -1, 4}
};
vector<double> b = {15, 10, 10, 15};
vector<double> x0 = {0, 0, 0, 0};
double start, end;
start = omp_get_wtime();
vector<double> x_jacobi_serial = jacobiSerial(A, b, x0);
end = omp_get_wtime();
cout << "串行Jacobi迭代的解为:";
for (double x : x_jacobi_serial)
cout << x << " ";
cout << endl << "运行时间:" << end - start << endl;
start = omp_get_wtime();
vector<double> x_gauss_seidel_serial = gaussSeidelSerial(A, b, x0);
end = omp_get_wtime();
cout << "串行Gauss-Seidel迭代的解为:";
for (double x : x_gauss_seidel_serial)
cout << x << " ";
cout << endl << "运行时间:" << end - start << endl;
start = omp_get_wtime();
vector<double> x_jacobi_parallel = jacobiParallel(A, b, x0);
end = omp_get_wtime();
cout << "并行Jacobi迭代的解为:";
for (double x : x_jacobi_parallel)
cout << x << " ";
cout << endl << "运行时间:" << end - start << endl;
start = omp_get_wtime();
vector<double> x_gauss_seidel_parallel = gaussSeidelParallel(A, b, x0);
end = omp_get_wtime();
cout << "并行Gauss-Seidel迭代的解为:";
for (double x : x_gauss_seidel_parallel)
cout << x << " ";
cout << endl << "运行时间:" << end - start << endl;
return 0;
}
```
请注意,为了运行此代码,您还需要启用OpenMP支持。 在GCC编译器中,您需要使用`-fopenmp`标志进行编译。
为了评估程序的运行时间,请检查输入矩阵A和向量b的大小,因为它们会影响代码的运行时间。 另外,运行时间可能会因硬件和编译器设置而异。 请将编译器选项设置为优化的版本,并在本地计算机上运行这些代码,以获得对运行时间的准确度量。