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LQ Factorization

In this example we again compute the \(LQ\) factorization and use it for solving a system of linear equations. However, in this example we do not setup matrix \(Q\) explicitly.

Example Code

flens/examples/lapack-orglq.cc
#include <cxxstd/iostream.h>
#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::lqf(A, tau);
    //lapack::lqf(A, tau, work);

    const auto L = A.lower();
    blas::sv(NoTrans, L, b);

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

    cout << "Q = " << Q << endl;

    DenseVector<Array<double> >  x;
    x = transpose(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(A, tau, work);

Solve \(L u = b\). Vector \(b\) gets overwritten with \(u\).

    const auto L = A.lower();
    blas::sv(NoTrans, L, b);

Explicitly setup \(Q\).

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

Compute \(x = Q^T u\).

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

Compile

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

Run

$shell> cd flens/examples                                                       
$shell> ./lapack-orglq                                                          
A = 
            2             3            -1             0 
           -6            -5             0             2 
            2            -5             6            -6 
            4             6             2            -3 
b = 
           20            -33            -43             49 
Q = 
    -0.534522     -0.801784      0.267261            -0 
     0.595961     -0.218519      0.536365      -0.55623 
     0.564904     -0.386513    -0.0297318      0.728428 
         -0.2           0.4           0.8           0.4 
x = 
            1              9              9              9
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