# LQ Factorization

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

## Example Code

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

using namespace std;
using namespace flens;

typedef double   T;

int
main()
{
GeMatrix<FullStorage<double> >     A(4,4);
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);

blas::sv(NoTrans, A.lower(), b);

lapack::ormlq(Left, Trans, A, tau, b);
// lapack::ormlq(Left, Trans, A, tau, b, work);

cout << "x = " << b << 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 $$u$$.

blas::sv(NoTrans, A.lower(), b);

Compute $$x = Q^T u$$. Vector $$b$$ gets overwritten with $$x$$.

lapack::ormlq(Left, Trans, A, tau, b);
// lapack::ormlq(Left, Trans, A, tau, b, work);

## Compile

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


## Run

$shell> cd flens/examples$shell> ./lapack-gelqf
A =
2             3            -1             0
-6            -5             0             2
2            -5             6            -6
4             6             2            -3
b =
20            -33            -43             49
x =
1              9              9              9


## Example with Complex Numbers

### Example Code

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

using namespace std;
using namespace flens;

typedef double   T;

int
main()
{
typedef complex<double>             Complex;
const Complex                       I(0,1);

GeMatrix<FullStorage<Complex> >     A(4,4);
DenseVector<Array<Complex> >        b(4);
DenseVector<Array<Complex> >        tau;
//DenseVector<Array<Complex> >      work;

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

b = 20,
-33,
-43,
49;
b *= I;

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

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

blas::sv(NoTrans, A.lower(), b);

lapack::unmlq(Left, ConjTrans, A, tau, b);
// lapack::unmlq(Left, ConjTrans, A, tau, b, work);

cout << "x = " << b << 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 $$u$$.

blas::sv(NoTrans, A.lower(), b);

Compute $$x = Q^T u$$. Vector $$b$$ gets overwritten with $$x$$.

lapack::unmlq(Left, ConjTrans, A, tau, b);
// lapack::unmlq(Left, ConjTrans, A, tau, b, work);

### Compile

$shell> cd flens/examples$shell> g++ -DUSE_CXXLAPACK -framework vecLib -std=c++11 -Wall -I../.. -o lapack-complex-gelqf lapack-complex-gelqf.cc


### Run

$shell> cd flens/examples$shell> ./lapack-complex-gelqf
A =
(0,2)                        (0,3)                      (-0,-1)                        (0,0)
(-0,-6)                      (-0,-5)                        (0,0)                        (0,2)
(0,2)                      (-0,-5)                        (0,6)                      (-0,-6)
(0,4)                        (0,6)                        (0,2)                      (-0,-3)
b =
(0,20)                      (-0,-33)                      (-0,-43)                        (0,49)
x =
(1,8.88178e-16)              (9,-1.30045e-15)              (9,-1.94289e-15)              (9,-2.02977e-16)