# QR Factorization

In this example we again compute the $$QR$$ 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::qrf(A, tau);
//lapack::qrf(A, tau, work);

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

cout << "Q = " << Q << endl;

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

const auto R = A.upper();
blas::sv(NoTrans, R, _b);

cout << "x = " << _b << endl;
}

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

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

Explicitly setup $$Q$$.

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

Compute $$\tilde{b} = Q^T b$$.

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

Solve $$R x = \tilde{b}$$. Vector $$b$$ gets overwritten with $$x$$.

const auto R = A.upper();
blas::sv(NoTrans, R, _b);

## Compile

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


## Run

$shell> cd flens/examples$shell> ./lapack-orgqr
A =
2             3            -1             0
-6            -5             0             2
2            -5             6            -6
4             6             2            -3
b =
20            -33            -43             49
Q =
-0.258199     -0.182574      0.208237     -0.925547
0.774597             0     -0.535468     -0.336563
-0.258199      0.912871     -0.267734     -0.168281
-0.516398     -0.365148     -0.773453     0.0420703
x =
1              9              9              9