# Solving Systems of Linear Equations

In this example we will solve a system of linear equations $$Ax=b$$. For this purpose we first compute the $$LU$$ factorization of $$A$$ with lapack::trf and then use the triangular solver lapack::trs to finish the job.

## Example Code

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

using namespace std;
using namespace flens;

int
main()
{
GeMatrix<FullStorage<double> >   A(44);
DenseVector<Array<double> >      b(4);
DenseVector<Array<int> >         piv(4);

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::trf(A, piv);

lapack::trs(NoTrans, A, piv, b);
cout << "X = " << b << endl;
}

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

using namespace std;
using namespace flens;

int
main()
{
GeMatrix<FullStorage<double> >   A(44);
DenseVector<Array<double> >      b(4);
DenseVector<Array<int> >         piv(4);

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;

Compute the $$PLU = A$$ factorization with lapack::trf

lapack::trf(A, piv);

Solve the system of linear equation $$PLUx = b$$ using lapack::trs. Vector $$b$$ gets overwritten with vector $$x$$.

lapack::trs(NoTrans, A, piv, b);
cout << "X = " << b << endl;
}

## Compile

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


## Run

$shell> cd flens/examples$shell> ./lapack-getrs
A =
2             3            -1             0
-6            -5             0             2
2            -5             6            -6
4             6             2            -3
b =
20            -33            -43             49
X =
1              9              9              9


## Example with Complex Numbers

### Example Code

#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(44);
DenseVector<Array<Complex> >        b(4);
DenseVector<Array<int> >            piv(4);

A =   2,    3,  -I,    0,
-6,   -5,   02.+I,
2.*I,   -5,   6,   -6,
46.*I,   2,   -3;

b = 20,
-33.*I,
-43,
49;

cout << "A = " << A << endl;
cout << "b = " << b << endl;

lapack::trf(A, piv);

lapack::trs(NoTrans, A, piv, b);
cout << "X = " << b << endl;
}

#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(44);
DenseVector<Array<Complex> >        b(4);
DenseVector<Array<int> >            piv(4);

Setup matrix $$A$$. Note that 2+I or 2*I would not work. That's because the STL does not define the operation int+complex<double> or int*complex<double>.

A =   2,    3,  -I,    0,
-6,   -5,   02.+I,
2.*I,   -5,   6,   -6,
46.*I,   2,   -3;

b = 20,
-33.*I,
-43,
49;

cout << "A = " << A << endl;
cout << "b = " << b << endl;

Compute the $$PLU = A$$ factorization with lapack::trf

lapack::trf(A, piv);

Solve the system of linear equation $$PLUx = b$$ using lapack::trs. Vector $$b$$ gets overwritten with vector $$x$$.

lapack::trs(NoTrans, A, piv, b);
cout << "X = " << b << endl;
}

### Compile

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


### Run

$shell> cd flens/examples$shell> ./lapack-complex-getrs
A =
(2,0)                        (3,0)                      (-0,-1)                        (0,0)
(-6,0)                       (-5,0)                        (0,0)                        (2,1)
(0,2)                       (-5,0)                        (6,0)                       (-6,0)
(4,0)                        (0,6)                        (2,0)                       (-3,0)
b =
(20,0)                      (-0,-33)                       (-43,0)                        (49,0)
X =
(16.5967,25.0647)           (-9.15476,-5.78498)             (32.7744,14.2709)              (39.2151,24.624)