# LQ Factorization (Complex Variant)

In this example we again compute the $$LQ$$ factorization and use it for solving a system of linear equations. In this example we setup matrix $$Q$$ explicitly. See lapack-gelqf for an example that avoids the explicit generation of $$Q$$.

## 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), Q;
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(Q, tau, work);

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

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

cout << "Q = " << Q << endl;

DenseVector<Array<Complex> >  x;
x = conjTrans(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(Q, tau, work);

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

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

Explicitly setup $$Q$$.

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

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

DenseVector<Array<Complex> >  x;
x = conjTrans(Q)*b;

## Compile

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


## Run

$shell> cd flens/examples$shell> ./lapack-complex-unglq
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)
Q =
(-2.22045e-16,0.534522)       (5.55112e-17,0.801784)                (0,-0.267261)                        (0,0)
(-2.77556e-17,-0.595961)       (5.55112e-17,0.218519)      (5.55112e-17,-0.536365)        (5.55112e-17,0.55623)
(-1.38778e-16,-0.564904)       (1.11022e-16,0.386513)      (1.11022e-16,0.0297318)      (5.55112e-17,-0.728428)
(-1.80411e-16,-0.2)            (4.16334e-17,0.4)           (-3.40006e-16,0.8)           (-3.33067e-16,0.4)
x =
(1,-5.09657e-16)               (9,2.18043e-16)               (9,-6.3291e-16)               (9,2.83562e-16)