TBTK::Solver::ArnoldiIterator Class Reference

Solves a Model using Arnoldi iteration. More...

#include <ArnoldiIterator.h>

Inheritance diagram for TBTK::Solver::ArnoldiIterator:

Public Types
enum	Mode { Normal, ShiftAndInvert }

Public Member Functions
	ArnoldiIterator ()
void	setMode (Mode mode)
Mode	getMode () const
void	setNumEigenValues (int numEigenValues)
int	getNumEigenValues () const
void	setCalculateEigenVectors (bool calculateEigenVectors)
bool	getCalculateEigenVectors () const
void	setNumLanczosVectors (int numLanczosVectors)
int	getNumLanczosVectors () const
void	setTolerance (double tolerance)
void	setMaxIterations (int maxIterations)
void	setCentralValue (double centralValue)
void	run ()
const CArray< std::complex< double > > &	getEigenValues () const
const double	getEigenValue (int state) const
const std::complex< double >	getAmplitude (int state, const Index &index)
Public Member Functions inherited from TBTK::Solver::Solver
	Solver ()
virtual	~Solver ()
virtual void	setModel (Model &model)
Model &	getModel ()
const Model &	getModel () const
Public Member Functions inherited from TBTK::Communicator
	Communicator (bool verbose)
void	setVerbose (bool verbose)
bool	getVerbose () const

Additional Inherited Members
Static Public Member Functions inherited from TBTK::Communicator
static void	setGlobalVerbose (bool globalVerbose)
static bool	getGlobalVerbose ()

Detailed Description

Solves a Model using Arnoldi iteration.

The ArnoldiIterator calculates a selected number of eigenvalues and eigenvectors using Arnoldi iteration. Use the PropertyExtractor::ArnoldiIterator to extract Properties.

The ArnoldiIterator is particularly suited for large problems where information from a small energy interval is needed. Compared to the Diagonalizer, the ArnoldiIterator can treat much larger problems as long as only a few eigenvalues or eigenvectors are needed.

Normal mode:
In the normal mode, the ArnoldiIterator calculates eigenvalues and eigenvectors with extremal eigenvalues.

Shift-and-invert mode:
In the shift-and-invert mode, the ArnoldiIterator calculates the eigenvalues and eigenvectors closest to a given "central value".

Example

#include "TBTK/Model.h"
#include "TBTK/PropertyExtractor/ArnoldiIterator.h"
#include "TBTK/Solver/ArnoldiIterator.h"
#include "TBTK/Streams.h"
#include "TBTK/TBTK.h"
 
using namespace TBTK;
 
int main(){
    Initialize();
 
    Model model;
    for(unsigned int x = 0; x < 200; x++)
        model << HoppingAmplitude(1, {x+1}, {x}) + HC;
    model.construct();
 
    Solver::ArnoldiIterator solver;
    solver.setVerbose(true);
    solver.setModel(model);
    solver.setMode(Solver::ArnoldiIterator::Mode::ShiftAndInvert);
    solver.setCentralValue(1.0);
    solver.setNumEigenValues(10);
    solver.setCalculateEigenVectors(true);
    solver.setNumLanczosVectors(20);
    solver.setMaxIterations(100);
    solver.run();
 
    Streams::out << "----------------\n";
 
    //Print the eigenvalues and the amplitude on site 50.
    PropertyExtractor::ArnoldiIterator propertyExtractor(solver);
    for(int n = 0; n < 10; n++){
        Streams::out
            << propertyExtractor.getEigenValue(n) << "\t"
            << propertyExtractor.getAmplitude(n, {50}) << "\n";
    }
}

Output

Running ArnoldiIterator.
.......... .......... .......... .......... .......... 
......
 
Number of accurately converged eigenvalues: 10
----------------
0.871769    (-0.0474405,-0.0232819)
0.899657    (0.017597,-0.0148148)
0.927327    (0.0611058,0.0592616)
0.954773    (0.0141037,-0.095402)
0.981988    (0.050148,0.00223721)
1.00897 (0.0149255,-0.0212923)
1.0357  (0.0244087,0.0831764)
1.06218 (-0.0327649,0.089842)
1.08841 (0.0467869,-0.00820977)
1.11437 (0.0259249,0.0129447)

Member Enumeration Documentation

Mode

enum TBTK::Solver::ArnoldiIterator::Mode

strong

Enum class describing the different modes of operation.

Normal: Lanczos like iteration for finding extreme eigenvalues and corresponding eigen vectors.

ShiftAndInvert: Shift and invert iteration for finding eigen values around a given value and the corresponding eigen vectors.

Constructor & Destructor Documentation

ArnoldiIterator()

TBTK::Solver::ArnoldiIterator::ArnoldiIterator

(

)

Constructs a Solver::ArnoldiIterator.

Member Function Documentation

getAmplitude()

const std::complex< double > TBTK::Solver::ArnoldiIterator::getAmplitude ( int  state, const Index &  index  )

inline

Get amplitude for given eigen vector \(n\) and physical index \(x\): \(\Psi_{n}(x)\).

Parameters

state	Eigen state number \(n\).
index	Physical index \(\).

getCalculateEigenVectors()

bool TBTK::Solver::ArnoldiIterator::getCalculateEigenVectors ( ) const

inline

Get wether eigen vectors are calculated or not.

Returns

True if eigenvectors are set to be calculated.

getEigenValue()

const double TBTK::Solver::ArnoldiIterator::getEigenValue ( int  state ) const

inline

Get eigen value.

getEigenValues()

const CArray< std::complex< double > > & TBTK::Solver::ArnoldiIterator::getEigenValues ( ) const

inline

Get eigenValues.

getMode()

ArnoldiIterator::Mode TBTK::Solver::ArnoldiIterator::getMode ( ) const

inline

Get mode of operation.

Returns

The mode of operation.

getNumEigenValues()

int TBTK::Solver::ArnoldiIterator::getNumEigenValues ( ) const

inline

Get number of eigenvalues.

Returns

The number of eigenvalues that are calculated.

getNumLanczosVectors()

int TBTK::Solver::ArnoldiIterator::getNumLanczosVectors ( ) const

inline

Get the number of Lanczos vectors to use. (Dimension of the Krylov space).

Returns

The dimension of the Krylov space used.

run()

void TBTK::Solver::ArnoldiIterator::run

(

)

Run the implicitly restarted Arnoldi algorithm.

setCalculateEigenVectors()

void TBTK::Solver::ArnoldiIterator::setCalculateEigenVectors ( bool  calculateEigenVectors )

inline

Set whether eigen vectors should be calculated.

Parameters

calculateEigenVectors

True to enable the calculation of eigenvectors.

setCentralValue()

void TBTK::Solver::ArnoldiIterator::setCentralValue ( double  centralValue )

inline

Set amount by which eigenvalues will be shifted during calculations. I.e. the calculation is caried out on \(H -\sigma I\) rather than \(H\). Eigenvalues are shifted back, so this does not affect the eigenvalues. However, since Arnoldi iteration calculates extreme eigenvalues in the normal mode and eigenvalues closest to zero in shift-and-invert mode, it does affect which eigenvalues that are calculated.

Note: The shift is not yet implemented in for the normal mode.

Parameters

centralValue

The value \(\sigma\) by which the Hamiltonian is shifted.

setMaxIterations()

void TBTK::Solver::ArnoldiIterator::setMaxIterations ( int  maxIterations )

inline

Set the maixmum number of iterations in the implicitly restarted Arnoldi algorithm.

Parameters

maxIterations

The maximum number of iterations performed in the implicitly restarted Arnoldi algorithm.

setMode()

void TBTK::Solver::ArnoldiIterator::setMode ( Mode  mode )

inline

Set mode of operation.

Parameters

mode	The mode of operation to use.

setNumEigenValues()

void TBTK::Solver::ArnoldiIterator::setNumEigenValues ( int  numEigenValues )

inline

Set the number of eigenvalues to calculate.

Parameters

numEigenValues

The number of eigenvalues to calculate.

setNumLanczosVectors()

void TBTK::Solver::ArnoldiIterator::setNumLanczosVectors ( int  numLanczosVectors )

inline

Set the number of Lanczos vectors to use. (Dimension of the Krylov space).

Parameters

numLanczosVectors

The dimension of the Krylov space used.

setTolerance()

void TBTK::Solver::ArnoldiIterator::setTolerance ( double  tolerance )

inline

Set the accpeted tolerance. Passed to the ARPACK functions [d/z]naupd and [d/z]neupd. See the TOL parameter in the ARPACK documentation for more information.

Parameters

tolerance

Tolerance used by ARPACK.

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The documentation for this class was generated from the following file:

• /TBTK/Lib/include/TBTK/Solver/ArnoldiIterator.h