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QR Factorization (Complex Variant)

In this example we again compute the \(QR\) factorization and use it for solving a system of linear equations. In this example we setup matrix \(Q\) explicitly. See lapack-unmqr for an example that avoids the explicit generation of \(Q\).

Example Code

flens/examples/lapack-complex-ungqr.cc
#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), 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::qrf(A, tau);
    //lapack::qrf(A, tau, work);

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

    cout << "Q = " << Q << endl;

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

    const auto R = A.upper();
    blas::sv(NoTrans, R, x);

    cout << "x = " << x << endl;
}

Comments on Example Code

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);

Explicitly setup \(Q\).

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

Compute \(\tilde{b} = Q^T b\).

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

Solve \(R x = \tilde{b}\). Vector \(b\) gets overwritten with \(x\).

    const auto R = A.upper();
    blas::sv(NoTrans, R, x);

Compile

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

Run

$shell> cd flens/examples                                                       
$shell> ./lapack-complex-ungqr                                                  
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 = 
      (1.11022e-16,0.258199)      (-1.38778e-17,0.182574)      (9.71445e-17,-0.208237)     (-5.55112e-16,-0.925547) 
     (2.77556e-17,-0.774597)   (-8.32667e-17,-5.3323e-17)      (-2.77556e-17,0.535468)                (0,-0.336563) 
                (0,0.258199)     (-3.46945e-17,-0.912871)        (-1.249e-16,0.267734)                (0,-0.168281) 
                (0,0.516398)      (-6.93889e-17,0.365148)      (-1.11022e-16,0.773453)     (-3.33067e-16,0.0420703) 
x = 
            (1,-4.92361e-14)               (9,3.97047e-15)              (9,-1.10517e-13)              (9,-1.30365e-13)
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