# LQ Factorization

In this example we again compute the $$LQ$$ factorization and use it for solving a system of linear equations. However, in this example we do not setup matrix $$Q$$ explicitly.

## Example Code

#include <iostream>
#include <flens/flens.cxx>

using namespace std;
using namespace flens;

int
main()
{
GeMatrix<FullStorage<double> >     A(4,4), Q;
DenseVector<Array<double> >        b(4);
DenseVector<Array<double> >        tau;
//DenseVector<Array<double> >      work;

A =  2,   3,  -1,   0,
-6,  -5,   0,   2,
2,  -5,   6,  -6,
4,   6,   2,  -3;

b = 20,
-33,
-43,
49;

cout << "A = " << A << endl;
cout << "b = " << b << endl;

lapack::lqf(A, tau);
//lapack::lqf(A, tau, work);

const auto L = A.lower();
blas::sv(NoTrans, L, b);

Q = A;
lapack::orglq(Q, tau);
//lapack::orglq(Q, tau, work);

cout << "Q = " << Q << endl;

DenseVector<Array<double> >  x;
x = transpose(Q)*b;

cout << "x = " << x << endl;
}

Compute the factorization $$A = LQ$$. Note that the workspace gets created implicitly and temporarily. So you might not want to do this inside a loop.

lapack::lqf(A, tau);
//lapack::lqf(A, tau, work);

Solve $$L u = b$$. Vector $$b$$ gets overwritten with $$u$$.

const auto L = A.lower();
blas::sv(NoTrans, L, b);

Explicitly setup $$Q$$.

Q = A;
lapack::orglq(Q, tau);
//lapack::orglq(Q, tau, work);

Compute $$x = Q^T u$$.

DenseVector<Array<double> >  x;
x = transpose(Q)*b;

## Compile

$shell> cd flens/examples$shell> g++ -std=c++11 -Wall -I../.. -o lapack-orglq lapack-orglq.cc


## Run

$shell> cd flens/examples$shell> ./lapack-orglq
A =
2             3            -1             0
-6            -5             0             2
2            -5             6            -6
4             6             2            -3
b =
20            -33            -43             49
Q =
-0.534522     -0.801784      0.267261            -0
0.595961     -0.218519      0.536365      -0.55623
0.564904     -0.386513    -0.0297318      0.728428
-0.2           0.4           0.8           0.4
x =
1              9              9              9