理论部分：
简言之
- jacobi迭代法就是把线性方程组化为关于第i行xi的等式，然后利用等式循环计算xi
- Gauss-Seidel迭代法就是在jacobi迭代的基础上，在计算第i+1行的xi+1时，带入了刚刚计算出的xi,xi-1…，以减少迭代次数
- 迭代法不是万能的，收敛时才能用
C++代码
雅可比迭代法：
#include<bits/stdc++.h>
#define MAXSIZE 100
using namespace std;
int main() {
double A[MAXSIZE][MAXSIZE], x[MAXSIZE], b[MAXSIZE],re[MAXSIZE];
int n;
double e;
cout << " 请输入原方程的阶数：";
cin >> n;
cout << "请输入原方程的增广矩阵：";
for (int i = 0; i < n; ++i) {
for (int j = 0; j < n ; ++j) {
cin >> A[i][j];
}
cin >> b[i];
}
cout << "请输入初始迭代向量值";
for (int i = 0; i < n; ++i) {
cin >> x[i];
}
cout << "请输入误差上限";
cin >> e;
int count = 0;
while (true) {
cout << "第" << ++count << "次迭代：";
int flag = 0;
for (int i = 0; i < n; ++i) {
re[i] = 0;
for (int j = 0; j < n; ++j) {
if (i != j) {
re[i] += - A[i][j] * x[j];
}
}
re[i] = (re[i] + b[i]) / A[i][i];
if (fabs(x[i] - re[i]) < e)
++flag;
cout << re[i]<<" ";
}
cout << endl;
if (flag == n) break;
for (int i = 0; i < n; ++i) {
x[i] = re[i];
}
}
for (int i = 0; i < n; ++i) {
cout << re[i] << " ";
}
}
高斯赛德尔迭代法：
#include<bits/stdc++.h>
#define MAXSIZE 100
using namespace std;
int main() {
double A[MAXSIZE][MAXSIZE], x[MAXSIZE], b[MAXSIZE];
int n;
double e;
cout << " 请输入原方程的阶数：";
cin >> n;
cout << "请输入原方程的增广矩阵：";
for (int i = 0; i < n; ++i) {
for (int j = 0; j < n ; ++j) {
cin >> A[i][j];
}
cin >> b[i];
}
cout << "请输入初始迭代向量值";
for (int i = 0; i < n; ++i) {
cin >> x[i];
}
cout << "请输入误差上限";
cin >> e;
int count = 0;
while (true) {
cout << "第" << ++count << "次迭代：";
int flag = 0;
for (int i = 0; i < n; ++i) {
double tmp = x[i];
x[i] = 0;
for (int j = 0; j < n; ++j) {
if (i != j) {
x[i] += - A[i][j] * x[j];
}
}
x[i] = (x[i] + b[i]) / A[i][i];
if (fabs(x[i] - tmp) < e)
++flag;
cout << x[i]<<" ";
}
cout << endl;
if (flag == n) break;
}
for (int i = 0; i < n; ++i) {
cout << x[i] << " ";
}
}
