# C++ Program to Perform LU Decomposition of any Matrix

The LU decomposition of a matrix produces a matrix as a product of its lower triangular matrix and upper triangular matrix. The LU in LU Decomposition of a matrix stands for Lower Upper.

An example of LU Decomposition of a matrix is given below −

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

A program that performs LU Decomposition of a matrix is given below −

## Example

#include<iostream>
using namespace std;
void LUdecomposition(float a, float l, float u, 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, l, u;
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;
}

## Output

The output of the above program is as follows

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

In the above program, the function LU decomposition finds the L and U decompositions of the given matrices. This is done by using nested for loops that calculate the L and U decompositions and store them in l[][] and u[][] matrix from the matrix a[][].

The code snippet that demonstrates this is given as follows −

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]);
}
}
}
}

In the main() function, the size of the matrix and its elements are obtained from the user. This is given as follows −

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];

Then the LU decomposition function is called and the L and U decomposition are displayed.This is given below −

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;
}