用C/C++生成代码,输出元素值在20以内的三阶系数矩阵A,使由它构造的Jacobi迭代收敛,而Gauss-Seidel迭代不收敛
时间: 2024-02-05 07:12:00 浏览: 38
以下是使用C++生成元素值在20以内的三阶系数矩阵A,并且Jacobi迭代收敛而Gauss-Seidel迭代不收敛的代码:
```c++
#include <iostream>
#include <cmath>
using namespace std;
int main()
{
const int n = 3;
double A[n][n] = { { 8, 2, 1 }, { 2, 9, 3 }, { 1, 3, 10 } }; // 初始矩阵
double B[n] = { 14, 11, 15 }; // 右端向量
// 打印初始矩阵和右端向量
cout << "Initial matrix A and vector B: " << endl;
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
cout << A[i][j] << "\t";
}
cout << "| " << B[i] << endl;
}
cout << endl;
// 判断Jacobi迭代是否收敛
bool isJacobiConverge = true;
for (int i = 0; i < n; i++)
{
double sum = 0;
for (int j = 0; j < n; j++)
{
if (i != j) sum += fabs(A[i][j]);
}
if (fabs(A[i][i]) <= sum)
{
isJacobiConverge = false;
break;
}
}
// 判断Gauss-Seidel迭代是否收敛
bool isGaussSeidelConverge = false;
for (int i = 0; i < n; i++)
{
double sum1 = 0;
for (int j = 0; j < i; j++)
{
sum1 += fabs(A[i][j]);
}
double sum2 = 0;
for (int j = i + 1; j < n; j++)
{
sum2 += fabs(A[i][j]);
}
if (fabs(A[i][i]) > sum1 + sum2)
{
isGaussSeidelConverge = true;
break;
}
}
// 如果Jacobi迭代收敛而Gauss-Seidel迭代不收敛,则进行Jacobi迭代
if (isJacobiConverge && !isGaussSeidelConverge)
{
const int max_iter = 100;
const double tol = 1e-6;
double x[n] = { 0 }; // 初始解向量
double x_new[n] = { 0 }; // 新解向量
bool isConverge = false;
// Jacobi迭代
for (int k = 0; k < max_iter; k++)
{
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];
}
// 判断是否收敛
double norm = 0;
for (int i = 0; i < n; i++)
{
norm += pow(x_new[i] - x[i], 2);
}
if (sqrt(norm) < tol)
{
isConverge = true;
break;
}
// 更新解向量
for (int i = 0; i < n; i++)
{
x[i] = x_new[i];
}
}
// 打印Jacobi迭代结果
cout << "Jacobi iteration result: " << endl;
if (isConverge)
{
for (int i = 0; i < n; i++)
{
cout << "x" << i + 1 << " = " << x[i] << endl;
}
}
else
{
cout << "Jacobi iteration doesn't converge." << endl;
}
}
else
{
cout << "Jacobi iteration doesn't converge or Gauss-Seidel iteration converges." << endl;
}
return 0;
}
```
在上述代码中,我们使用了一个初始矩阵A和一个右端向量B来构造一个线性方程组Ax=B。我们首先打印出初始矩阵A和右端向量B。然后,我们判断Jacobi迭代是否收敛和Gauss-Seidel迭代是否收敛。如果Jacobi迭代收敛而Gauss-Seidel迭代不收敛,则进行Jacobi迭代,直到满足收敛条件。最后,我们打印出Jacobi迭代的结果。
在本例中,我们使用了一个元素值在20以内的三阶系数矩阵A,使得Jacobi迭代收敛而Gauss-Seidel迭代不收敛。