线性方程组 精确解 近似解 算法整理

此时我们只讨论当m==n时的情况，即n阶线性方程组，它的系数矩阵为n+1列为b。
note：
代码假设输入格式为n，代表n阶行列式，随后紧跟着n行，每行n+1个数，代表系数矩阵。

1. 求精确解的方法：
   1.1 高斯消元法:

//Gauss消元
#include using namespace std;double matrix[20][21], x[20];
int n;void print_matrix() {for(int i = 1; i <= n; i++) {for(int j = 1; j <= n+1; j++)cout << matrix[i][j] << " ";cout << endl;}cout << endl;
}int main() {//输入cin >> n;for(int i = 1; i <= n; i++)for(int j = 1; j <= n+1; j++)cin >> matrix[i][j];print_matrix();//消元过程for(int i = 1; i <= n-1; i++) {for(int j = i+1; j <= n; j++) {//第j行double l = matrix[j][i] / matrix[i][i];cout << "l[" << j << "][" << i << "] = " << l << endl;for(int k = i; k <= n+1; k++)//j行的每一个元素matrix[j][k] = matrix[j][k] - l*matrix[i][k];print_matrix();}}//回代过程for(int i = n; i >= 1; i--) {double sum = 0;for(int j = i+1; j <= n; j++)//求累加和sum += matrix[i][j] * x[j];x[i] = (matrix[i][n+1] - sum) / matrix[i][i];}for(int i = 1; i <= n; i++)cout << "x[" << i << "] = " << x[i] << endl;return 0;
}

1.2高斯主元素消元法:
可以有很好的数值稳定性，避免较大的舍入误差
1.2.1 完全主元素消元法:
note: 选最大主元素的时候不能选择常数列

#include 
#include using namespace std;double matrix[20][21], x[20];
int pos[20];//记录当前列对应的原列序号
int n;void print_matrix() {for(int i = 1; i <= n; i++) {for(int j = 1; j <= n+1; j++)cout << matrix[i][j] << " ";cout << endl;}cout << endl;
}void swap_row(int row1, int row2) {for(int i = 1; i <= n+1; i++)swap(matrix[row1][i], matrix[row2][i]);
}
void swap_col(int col1, int col2) {for(int i = 1; i <= n; i++)swap(matrix[i][col1], matrix[i][col2]);
}void find_max(int k) {int r = k, c = k;double maxs = abs(matrix[k][k]);for(int i = k; i <= n; i++) {for(int j = k; j <= n; j++) {if(abs(matrix[i][j]) > maxs) {r = i;c = j;maxs = abs(matrix[i][j]);}}}pos[k] = c;  pos[c] = k;swap_row(k, r);  swap_col(k, c);
}int main() {freopen("/Users/really/Documents/code/Gaussdata.txt","r",stdin);//输入cin >> n;for(int i = 1; i <= n; i++)for(int j = 1; j <= n+1; j++)cin >> matrix[i][j];print_matrix();//消元过程for(int i = 1; i <= n-1; i++) {find_max(i);//使主元素选取相较更大的值for(int j = i+1; j <= n; j++) {//第j行double l = matrix[j][i] / matrix[i][i];cout << "l[" << j << "][" << i << "] = " << l << endl;for(int k = i; k <= n+1; k++)//j行的每一个元素matrix[j][k] = matrix[j][k] - l*matrix[i][k];print_matrix();}}//回代过程for(int i = n; i >= 1; i--) {double sum = 0;for(int j = i+1; j <= n; j++)//求累加和sum += matrix[i][j] * x[j];x[i] = (matrix[i][n+1] - sum) / matrix[i][i];}for(int i = 1; i <= n; i++) {//将映射的解回归原位置if(pos[i] > i)//不用!=避免重复交换swap(x[i], x[pos[i]]);}for(int i = 1; i <= n; i++)cout << "x[" << i << "] = " << x[i] << endl;return 0;
}

1.2.2 列主元素消元法:
相比完全主元素效率更高，但是稳定性不如完全主元素，要求系数行列式不为0.

#include 
#include using namespace std;double matrix[20][21], x[20];
int n;void print_matrix() {for(int i = 1; i <= n; i++) {for(int j = 1; j <= n+1; j++)cout << matrix[i][j] << " ";cout << endl;}cout << endl;
}void swap_row(int row1, int row2) {for(int i = 1; i <= n+1; i++)swap(matrix[row1][i], matrix[row2][i]);
}void find_max(int k) {int r = k;double maxs = abs(matrix[k][k]);for(int i = k; i <= n; i++) {if(abs(matrix[i][k]) > maxs) {r = i;maxs = abs(matrix[i][k]);}}swap_row(k, r);
}int main() {freopen("/Users/really/Documents/code/Gaussdata.txt","r",stdin);//输入cin >> n;for(int i = 1; i <= n; i++)for(int j = 1; j <= n+1; j++)cin >> matrix[i][j];print_matrix();//消元过程for(int i = 1; i <= n-1; i++) {find_max(i);//使主元素选取相较更大的值for(int j = i+1; j <= n; j++) {//第j行double l = matrix[j][i] / matrix[i][i];cout << "l[" << j << "][" << i << "] = " << l << endl;for(int k = i; k <= n+1; k++)//j行的每一个元素matrix[j][k] = matrix[j][k] - l*matrix[i][k];print_matrix();}}//回代过程for(int i = n; i >= 1; i--) {double sum = 0;for(int j = i+1; j <= n; j++)//求累加和sum += matrix[i][j] * x[j];x[i] = (matrix[i][n+1] - sum) / matrix[i][i];}for(int i = 1; i <= n; i++)cout << "x[" << i << "] = " << x[i] << endl;return 0;
}

1.3 矩阵分解法:

#include using namespace std;double matrix[20][21], x[20];
int n;void print_matrix() {for(int i = 1; i <= n; i++) {for(int j = 1; j <= n+1; j++)cout << matrix[i][j] << " ";cout << endl;}cout << endl;
}int main() {//输入cin >> n;for(int i = 1; i <= n; i++)for(int j = 1; j <= n+1; j++)cin >> matrix[i][j];print_matrix();//LU分解for(int i = 1; i <= n-1; i++) {for(int j = i+1; j <= n; j++) {//第j行double l = matrix[j][i] / matrix[i][i];cout << "l[" << j << "][" << i << "] = " << l << endl;matrix[j][i] = l;for(int k = i+1; k <= n; k++)//j行的每一个元素matrix[j][k] = matrix[j][k] - l*matrix[i][k];print_matrix();}}//回代y的值并保存在x当中for(int i = 1; i <= n; i++) {double sum = 0;for(int j = 1; j < i; j++)sum += matrix[i][j] * x[j];x[i] = (matrix[i][n+1] - sum);}//更新常数列的值为y的值for(int i = 1; i <= n; i++)matrix[i][n+1] = x[i];//回代x的值for(int i = n; i >= 1; i--) {double sum = 0;for(int j = i+1; j <= n; j++)//求累加和sum += matrix[i][j] * x[j];x[i] = (matrix[i][n+1] - sum) / matrix[i][i];}for(int i = 1; i <= n; i++)cout << "x[" << i << "] = " << x[i] << endl;return 0;
}

1. 求近似解的方法：
   2.1 迭代法

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