C ++程序执行任何矩阵的LU分解
矩阵的LU分解产生一个矩阵，作为其下三角矩阵和上三角矩阵的乘积。矩阵的LU分解中的LU表示LowerUpper。
下面是矩阵的LU分解的示例-
Given matrix is: 1 1 0 2 1 3 3 1 1 The L matrix is: 1 0 0 2 -1 0 3 -2 -5 The U matrix is: 1 1 0 0 1 -3 0 0 1
下面给出了执行矩阵的LU分解的程序-
示例
#include<iostream>
using namespace std;
void LUdecomposition(float a[10][10], float l[10][10], float u[10][10], int n) {
int i = 0, j = 0, k = 0;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
if (j < i)
l[j][i] = 0;
else {
l[j][i] = a[j][i];
for (k = 0; k < i; k++) {
l[j][i] = l[j][i] - l[j][k] * u[k][i];
}
}
}
for (j = 0; j < n; j++) {
if (j < i)
u[i][j] = 0;
else if (j == i)
u[i][j] = 1;
else {
u[i][j] = a[i][j] / l[i][i];
for (k = 0; k < i; k++) {
u[i][j] = u[i][j] - ((l[i][k] * u[k][j]) / l[i][i]);
}
}
}
}
}
int main() {
float a[10][10], l[10][10], u[10][10];
int n = 0, i = 0, j = 0;
cout << "Enter size of square matrix : "<<endl;
cin >> n;
cout<<"Enter matrix values: "<endl;
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
cin >> a[i][j];
LUdecomposition(a, l, u, n);
cout << "L Decomposition is as follows..."<<endl;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
cout<<l[i][j]<<" ";
}
cout << endl;
}
cout << "U Decomposition is as follows..."<<endl;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
cout<<u[i][j]<<" ";
}
cout << endl;
}
return 0;
}输出结果
上面程序的输出如下
Enter size of square matrix : 3 Enter matrix values: 1 1 0 2 1 3 3 1 1 L Decomposition is as follows... 1 0 0 2 -1 0 3 -2 -5 U Decomposition is as follows... 1 1 0 0 1 -3 0 0 1
在上面的程序中，函数LU分解找到给定矩阵的L和U分解。这是通过使用嵌套的for循环来完成的，该循环计算L和U分解并将它们存储在矩阵a[][]中的l[][]和u[][]矩阵中。
证明这一点的代码片段如下-
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
if (j < i)
l[j][i] = 0;
else {
l[j][i] = a[j][i];
for (k = 0; k < i; k++) {
l[j][i] = l[j][i] - l[j][k] * u[k][i];
}
}
}
for (j = 0; j < n; j++) {
if (j < i)
u[i][j] = 0;
else if (j == i)
u[i][j] = 1;
else {
u[i][j] = a[i][j] / l[i][i];
for (k = 0; k < i; k++) {
u[i][j] = u[i][j] - ((l[i][k] * u[k][j]) / l[i][i]);
}
}
}
}在该main()函数中，矩阵的大小及其元素是从用户那里获得的。这给出如下-
cout << "Enter size of square matrix : "<<endl; cin >> n; cout<<"Enter matrix values: "<endl; for (i = 0; i < n; i++) for (j = 0; j < n; j++) cin >> a[i][j];
然后调用LU分解函数并显示L和U分解，如下所示-
LUdecomposition(a, l, u, n);
cout << "L Decomposition is as follows..."<<endl;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
cout<<l[i][j]<<" ";
}
cout << endl;
}
cout << "U Decomposition is as follows..."<<endl;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
cout<u[i][j]<<" ";
}
cout << endl;
}