# QR Factorization (Complex Variant)

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

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

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);

Compute $$\tilde{b} = Q^T b$$. Note that we do not setup $$A$$ explicitly.

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

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++ -DUSE_CXXLAPACK -framework vecLib -std=c++11 -Wall -I../.. -o lapack-complex-unmqr lapack-complex-unmqr.cc


## Run

$shell> cd flens/examples$shell> ./lapack-complex-unmqr
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,-2.3617e-14)                (9,1.8825e-15)              (9,-5.29762e-14)              (9,-6.22797e-14)