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Solving a Symmetric Positive Definite System of Linear Equations

In this example we solve a system of linear equations \(Sx = b\) were the coefficient matrix is symmetric and positiv definite. We first compute the Cholesky factorization \(S = L L^T\) with lapack::potrf which is the FLENS port of LAPACK's dpotrf. We then solve \(Lu=b\) and then \(L^T x = u\) using lapack::potrs which is the port of dpotrs.

Example Code

#include <flens/flens.cxx>
#include <iostream>

using namespace flens;
using namespace std;

int
main()
{
    GeMatrix<FullStorage<double> >   A(33);
    DenseVector<Array<double> >      b(3);

    A = 210,
        021,
        002;

    b = 1,
        2,
        3;

    auto S = A.upper().symmetric();

    cout << "S = " << S << endl;
    cout << "b = " << b << endl;

    int info = lapack::potrf(S);

    if (info!=0) {
        cerr << "S is singular" << endl;
        return info;
    }

    lapack::potrs(S, b);
    cout << "inv(S)*b = " << b << endl;

    return 0;
}

Comments on Example Code

Setup the raw data.

    A = 210,
        021,
        002;

\(S\) is a symmetric matrix view constructed from the upper triangular part of \(A\). Note that \(S\) only references data from \(A\).

    auto S = A.upper().symmetric();

Computes the Cholesky factorization \(S = L L^T\) where \(L\) is upper triangular. Matrix \(S\) (i.e. the lower triangular part of \(A\)) gets overwritten with \(L^T\).

    int info = lapack::potrf(S);

If info is not zero the factorization could not be computed as the matrix was singular.

    if (info!=0) {
        cerr << "S is singular" << endl;
        return info;
    }

Compute \(x = S^{-1}b\). Vector \(b\) gets overwritten with the solution vector \(x\).

    lapack::potrs(S, b);
    cout << "inv(S)*b = " << b << endl;

Compile

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

Run

$shell> cd flens/examples                                                      
$shell> ./lapack-potrs                                                         
S = 
            2             1             0 
            1             2             1 
            0             1             2 
b = 
            1              2              3 
inv(S)*b = 
          0.5              0            1.5 

Example with Complex Numbers

Example Code

#include <flens/flens.cxx>
#include <iostream>

using namespace flens;
using namespace std;

int
main()
{
    typedef complex<double>             Complex;
    const Complex                       I(0,1);

    GeMatrix<FullStorage<Complex> >     A(33);
    DenseVector<Array<Complex> >        b(3);

    A = 2, I,    0,
        021.+I,
        00,    3;

    b = 1,
        2,
        3;

    auto H = A.upper().hermitian();

    cout << "H = " << H << endl;
    cout << "b = " << b << endl;

    int info = lapack::potrf(H);

    if (info!=0) {
        cerr << "H is singular" << endl;
        return info;
    }

    lapack::potrs(H, b);
    cout << "inv(H)*b = " << b << endl;

    return 0;
}

Comments on Example Code

Setup the raw data.

    A = 2, I,    0,
        021.+I,
        00,    3;

\(H\) is a symmetric matrix view constructed from the upper triangular part of \(A\). Note that \(H\) only references data from \(A\).

    auto H = A.upper().hermitian();

Computes the Cholesky factorization \(H = L L^T\) where \(L\) is upper triangular. Matrix \(H\) (i.e. the lower triangular part of \(A\)) gets overwritten with \(L^T\).

    int info = lapack::potrf(H);

If info is not zero the factorization could not be computed as the matrix was singular.

    if (info!=0) {
        cerr << "H is singular" << endl;
        return info;
    }

Compute \(x = H^{-1}b\). Vector \(b\) gets overwritten with the solution vector \(x\).

    lapack::potrs(H, b);
    cout << "inv(H)*b = " << b << endl;

Compile

Note that an external LAPACK implementation is needed for the complex variant of lapack::potrs

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

Run

$shell> cd flens/examples                                                       
$shell> ./lapack-complex-potrs                                                  
H = 
                       (2,0)                       (0,1)                       (0,0)
                      (0,-1)                       (2,0)                       (1,1)
                      (0,-0)                      (1,-1)                       (3,0)
b = 
                       (1,0)                         (2,0)                         (3,0) 
inv(H)*b = 
                  (0.2,-0.6)                    (1.2,-0.6)                     (0.8,0.6)