利用赛德尔迭代法求解如下线性代数方程组的近似解系数矩阵a[3][3]={{10,-1,-2},{-1,10,-2},{-1,-2,5}},b[3]={7.2,8.3,4.2},写出迭代次数k和解的C语言和c++代码(误差控制限为5e-5,迭代初始值取[0,0,0])

迭代公式为：$x^{(k+1)}=D^{-1}(b-(L+U)x^{(k)})$，其中D为系数矩阵的对角线部分，L为系数矩阵的下三角部分，U为系数矩阵的上三角部分。

C语言代码如下：

#include <stdio.h>
#include <math.h>

#define N 3
#define MAX_ITER 1000
#define TOLERANCE 5e-5

void print_solution(double x[N]) {
    printf("Solution: ");
    for (int i = 0; i < N; i++) {
        printf("%lf ", x[i]);
    }
    printf("\n");
}

int main() {
    double a[N][N] = {{10, -1, -2}, {-1, 10, -2}, {-1, -2, 5}};
    double b[N] = {7.2, 8.3, 4.2};
    double x[N] = {0, 0, 0};
    double x_new[N];
    double D[N], L[N][N], U[N][N];

    // 分解系数矩阵
    for (int i = 0; i < N; i++) {
        D[i] = a[i][i];
        for (int j = 0; j < N; j++) {
            if (i < j) {
                U[i][j] = a[i][j];
            } else if (i > j) {
                L[i][j] = a[i][j];
            }
        }
    }

    // 判断对角线元素是否为0，如果是则无解
    for (int i = 0; i < N; i++) {
        if (D[i] == 0) {
            printf("No solution\n");
            return 0;
        }
    }

    // 迭代求解
    int k;
    for (k = 0; k < MAX_ITER; k++) {
        for (int i = 0; i < N; i++) {
            x_new[i] = b[i];
            for (int j = 0; j < N; j++) {
                x_new[i] -= (L[i][j] + U[i][j]) * x[j];
            }
            x_new[i] /= D[i];
        }
        // 判断误差是否满足要求
        double error = 0;
        for (int i = 0; i < N; i++) {
            error += pow(x_new[i] - x[i], 2);
        }
        if (sqrt(error) < TOLERANCE) {
            break;
        }
        // 更新迭代值
        for (int i = 0; i < N; i++) {
            x[i] = x_new[i];
        }
    }

    if (k == MAX_ITER) {
        printf("Iteration failed\n");
    } else {
        printf("Iteration succeeded in %d steps\n", k);
        print_solution(x_new);
    }

    return 0;
}

C++代码如下：

#include <iostream>
#include <cmath>

#define N 3
#define MAX_ITER 1000
#define TOLERANCE 5e-5

using namespace std;

void print_solution(double x[N]) {
    cout << "Solution: ";
    for (int i = 0; i < N; i++) {
        cout << x[i] << " ";
    }
    cout << endl;
}

int main() {
    double a[N][N] = {{10, -1, -2}, {-1, 10, -2}, {-1, -2, 5}};
    double b[N] = {7.2, 8.3, 4.2};
    double x[N] = {0, 0, 0};
    double x_new[N];
    double D[N], L[N][N], U[N][N];

    // 分解系数矩阵
    for (int i = 0; i < N; i++) {
        D[i] = a[i][i];
        for (int j = 0; j < N; j++) {
            if (i < j) {
                U[i][j] = a[i][j];
            } else if (i > j) {
                L[i][j] = a[i][j];
            }
        }
    }

    // 判断对角线元素是否为0，如果是则无解
    for (int i = 0; i < N; i++) {
        if (D[i] == 0) {
            cout << "No solution" << endl;
            return 0;
        }
    }

    // 迭代求解
    int k;
    for (k = 0; k < MAX_ITER; k++) {
        for (int i = 0; i < N; i++) {
            x_new[i] = b[i];
            for (int j = 0; j < N; j++) {
                x_new[i] -= (L[i][j] + U[i][j]) * x[j];
            }
            x_new[i] /= D[i];
        }
        // 判断误差是否满足要求
        double error = 0;
        for (int i = 0; i < N; i++) {
            error += pow(x_new[i] - x[i], 2);
        }
        if (sqrt(error) < TOLERANCE) {
            break;
        }
        // 更新迭代值
        for (int i = 0; i < N; i++) {
            x[i] = x_new[i];
        }
    }

    if (k == MAX_ITER) {
        cout << "Iteration failed" << endl;
    } else {
        cout << "Iteration succeeded in " << k << " steps" << endl;
        print_solution(x_new);
    }

    return 0;
}