LU列主元法解线性方程组

拼搏现实的明天。 2023-06-16 08:29 3阅读 0赞

##  ##

## 一、 L-U列主元法解线性方程组 ##

## 1.1 实验原理 ##

定理1：设![20191125202814610.png][]非奇异，若其顺序主子式![20191125202856250.png][]均不为零，则存在唯一的单位下三角矩阵L和上三角矩阵U，使得A=LU.

![20191125202943304.png][]

设A = LU，其中L为一个单位下三角矩阵，U为一个上角矩阵，即

![20191125202957475.png][]

转化为Ly = b及Ux = y的两个三角形方程组，由三角形方程组很容易通过回代方法求解方程组的解。

## 1.2 算法说明 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70][]

![20191125203052581.png][]

## 1.3 流程图 ##

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 1][]

## 1.4 程序设计说明 ##

![20191125203131122.png][]

## 1.5 程序例题 ##

### 1.5.1 实验题目 ###

计算下列线性方程组的解

![20191125203155336.png][]

### 1.5.2 程序运行结果 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 2][]

### 1.5.3 手算验证过程 ###

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 3][]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 4][]

## 1.6 代码 ##

// Note:Your choice is C++ IDE
    #include<iostream>
    using namespace std;
    #include<math.h>
    
    #define N 3        //矩阵维数，可自定义
    
    double A[N][N] = {1，2，3，5，4，10，3，0.2，5};    //系数矩阵
    double B[N] = {1，0，2};        //右端项
    double Y[N];         //中间项
    double X[N];         //输出
    double S[N];         //选取列主元的比较器
    
    int i, j, k;         //计数器
    
    int main()
    {
        
        for (k = 0; k < N; k++)
        {
            //选列主元
            int index = k;
            for (i = k; i < N; i++)
            {
                double temp = 0;
                for (int m = 0; m < k; m++)
                {
                    temp = temp + A[i][m] * A[m][k];
                }
                S[i] = A[i][k] - temp;
                if (S[index] < S[i])
                {
                    index = i;
                }
            }
            //交换行
            double temp;
            for (i = k; i < N; i++)
            {
                temp = A[index][i];
                A[index][i] = A[k][i];
                A[k][i] = temp;
            }
            temp = B[index];
            B[index] = B[k];
            B[k] = temp;
            // 构造L、U矩阵
            for (j = k; j < N; j++)
            {
                double temp = 0;
                for (int m = 0; m < k; m++)
                {
    
                    temp = temp + A[k][m] * A[m][j];
                }
                A[k][j] = A[k][j] - temp;   //先构造U一行的向量
            }
            for (i = k + 1; i < N; i++)
            {
                double temp = 0;
                for (int m = 0; m < k; m++)
                {
                    temp = temp + A[i][m] * A[m][k];
                }
                A[i][k] = (A[i][k] - temp) / A[k][k];  //再构造L一列的向量
            }
        }
        //求解LY = B
        Y[0] = B[0];
        for (i = 1; i < N; i++)
        {
            double temp = 0;
            for (int j = 0; j < i; j++)
            {
                temp = temp + A[i][j] * Y[j];
            }
            Y[i] = B[i] - temp;
        }
        //求解UX = Y
        X[N - 1] = Y[N - 1] / A[N - 1][N - 1];
        for (i = N - 2; i >= 0; i--)
        {
    
            double  temp = 0;
            for (int j = i + 1; j < N; j++)
            {
                temp = temp + A[i][j] * X[j];
            }
            X[i] = (Y[i] - temp) / A[i][i];
        }
        //打印X
        cout << "此方程组的解为：" << endl;
        for (i = 0; i < N; i++)
        {
            cout << X[i] << " ";
        }
        cout << endl;
    
        return 0;
    }

[20191125202814610.png]: https://img-blog.csdnimg.cn/20191125202814610.png
[20191125202856250.png]: https://img-blog.csdnimg.cn/20191125202856250.png
[20191125202943304.png]: https://img-blog.csdnimg.cn/20191125202943304.png
[20191125202957475.png]: https://img-blog.csdnimg.cn/20191125202957475.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70]: https://img-blog.csdnimg.cn/20191125203032911.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz,size_16,color_FFFFFF,t_70
[20191125203052581.png]: https://img-blog.csdnimg.cn/20191125203052581.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 1]: https://img-blog.csdnimg.cn/20191125203116200.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz,size_16,color_FFFFFF,t_70
[20191125203131122.png]: https://img-blog.csdnimg.cn/20191125203131122.png
[20191125203155336.png]: https://img-blog.csdnimg.cn/20191125203155336.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 2]: https://img-blog.csdnimg.cn/20191125203208315.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz,size_16,color_FFFFFF,t_70
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 3]: https://img-blog.csdnimg.cn/20191125203244558.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz,size_16,color_FFFFFF,t_70
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz_size_16_color_FFFFFF_t_70 4]: https://img-blog.csdnimg.cn/20191125203302164.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3FxXzQ0MDU3NDQz,size_16,color_FFFFFF,t_70