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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

flens/examples/lapack-complex-unglq.cc
#include <cxxstd/iostream.h>
#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;
}

Comments on Example Code

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> clang++ -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 = 
               (0,-0.534522)                (0,-0.801784)                 (0,0.267261)                      (-0,-0) 
    (-5.55112e-17,-0.595961)       (1.11022e-16,0.218519)     (-1.11022e-16,-0.536365)                  (0,0.55623) 
      (9.71445e-16,0.564904)     (-8.88178e-16,-0.386513)    (-2.22045e-16,-0.0297318)       (8.60423e-16,0.728428) 
         (-4.30211e-16,-0.2)            (1.11022e-16,0.4)           (-6.03684e-16,0.8)           (-8.60423e-16,0.4) 
x = 
            (1,-5.20006e-15)                (9,1.6922e-15)              (9,-5.23752e-15)              (9,-8.81201e-15)
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