#include <SymEigsShiftSolver.h>

Inheritance diagram for Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >:

Public Member Functions
	SymEigsShiftSolver (OpType *op, Index nev, Index ncv, Scalar sigma)

Public Member Functions inherited from Spectra::SymEigsBase< double, LARGEST_MAGN, DenseSymShiftSolve< double >, IdentityBOp >
Index	compute (Index maxit=1000, double tol=1e-10, int sort_rule=LARGEST_ALGE)

Vector	eigenvalues () const

virtual Matrix	eigenvectors () const

virtual Matrix	eigenvectors (Index nvec) const

int	info () const

void	init ()

void	init (const double *init_resid)

Index	num_iterations () const

Index	num_operations () const

Private Types
typedef Eigen::Array< Scalar, Eigen::Dynamic, 1 >	Array

typedef Eigen::Index	Index

Private Member Functions
void	sort_ritzpair (int sort_rule)

Private Attributes
const Scalar	m_sigma

Additional Inherited Members
Protected Attributes inherited from Spectra::SymEigsBase< double, LARGEST_MAGN, DenseSymShiftSolve< double >, IdentityBOp >
LanczosFac	m_fac

const Index	m_n

const Index	m_ncv

const Index	m_nev

Index	m_niter

Index	m_nmatop

DenseSymShiftSolve< double > *	m_op

Vector	m_ritz_val

Detailed Description

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>
class Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >

This class implements the eigen solver for real symmetric matrices using the shift-and-invert mode. The background information of the symmetric eigen solver is documented in the SymEigsSolver class. Here we focus on explaining the shift-and-invert mode.

The shift-and-invert mode is based on the following fact: If $\lambda$ and $x$ are a pair of eigenvalue and eigenvector of matrix $A$ , such that $Ax=\lambda x$ , then for any $\sigma$ , we have

$(A-\sigma I)^{-1}x=\nu x$

where

$\nu=\frac{1}{\lambda-\sigma}$

which indicates that $(\nu, x)$ is an eigenpair of the matrix $(A-\sigma I)^{-1}$ .

Therefore, if we pass the matrix operation $(A-\sigma I)^{-1}y$ (rather than $Ay$ ) to the eigen solver, then we would get the desired values of $\nu$ , and $\lambda$ can also be easily obtained by noting that $\lambda=\sigma+\nu^{-1}$ .

The reason why we need this type of manipulation is that the algorithm of Spectra (and also ARPACK) is good at finding eigenvalues with large magnitude, but may fail in looking for eigenvalues that are close to zero. However, if we really need them, we can set $\sigma=0$ , find the largest eigenvalues of $A^{-1}$ , and then transform back to $\lambda$ , since in this case largest values of $\nu$ implies smallest values of $\lambda$ .

To summarize, in the shift-and-invert mode, the selection rule will apply to $\nu=1/(\lambda-\sigma)$ rather than $\lambda$ . So a selection rule of LARGEST_MAGN combined with shift $\sigma$ will find eigenvalues of $A$ that are closest to $\sigma$ . But note that the eigenvalues() method will always return the eigenvalues in the original problem (i.e., returning $\lambda$ rather than $\nu$ ), and eigenvectors are the same for both the original problem and the shifted-and-inverted problem.

Template Parameters

Scalar	The element type of the matrix. Currently supported types are `float`, `double` and `long double`.
SelectionRule	An enumeration value indicating the selection rule of the shifted-and-inverted eigenvalues. The full list of enumeration values can be found in Enumerations.
OpType	The name of the matrix operation class. Users could either use the wrapper classes such as DenseSymShiftSolve and SparseSymShiftSolve, or define their own that implements all the public member functions as in DenseSymShiftSolve.

Below is an example that illustrates the use of the shift-and-invert mode:

#include <Eigen/Core>
#include <Spectra/SymEigsShiftSolver.h>
// <Spectra/MatOp/DenseSymShiftSolve.h> is implicitly included
#include <iostream>
 
using namespace Spectra;
 
int main()
{
    // A size-10 diagonal matrix with elements 1, 2, ..., 10
    Eigen::MatrixXd M = Eigen::MatrixXd::Zero(10, 10);
    for(int i = 0; i < M.rows(); i++)
        M(i, i) = i + 1;
 
    // Construct matrix operation object using the wrapper class
    DenseSymShiftSolve<double> op(M);
 
    // Construct eigen solver object with shift 0
    // This will find eigenvalues that are closest to 0
    SymEigsShiftSolver< double, LARGEST_MAGN,
                        DenseSymShiftSolve<double> > eigs(&op, 3, 6, 0.0);
 
    eigs.init();
    eigs.compute();
    if(eigs.info() == SUCCESSFUL)
    {
        Eigen::VectorXd evalues = eigs.eigenvalues();
        // Will get (3.0, 2.0, 1.0)
        std::cout << "Eigenvalues found:\n" << evalues << std::endl;
    }
 
    return 0;
}

Also an example for user-supplied matrix shift-solve operation class:

#include <Eigen/Core>
#include <Spectra/SymEigsShiftSolver.h>
#include <iostream>
 
using namespace Spectra;
 
// M = diag(1, 2, ..., 10)
class MyDiagonalTenShiftSolve
{
private:
    double sigma_;
public:
    int rows() { return 10; }
    int cols() { return 10; }
    void set_shift(double sigma) { sigma_ = sigma; }
    // y_out = inv(A - sigma * I) * x_in
    // inv(A - sigma * I) = diag(1/(1-sigma), 1/(2-sigma), ...)
    void perform_op(double *x_in, double *y_out)
    {
        for(int i = 0; i < rows(); i++)
        {
            y_out[i] = x_in[i] / (i + 1 - sigma_);
        }
    }
};
 
int main()
{
    MyDiagonalTenShiftSolve op;
    // Find three eigenvalues that are closest to 3.14
    SymEigsShiftSolver<double, LARGEST_MAGN,
                       MyDiagonalTenShiftSolve> eigs(&op, 3, 6, 3.14);
    eigs.init();
    eigs.compute();
    if(eigs.info() == SUCCESSFUL)
    {
        Eigen::VectorXd evalues = eigs.eigenvalues();
        // Will get (4.0, 3.0, 2.0)
        std::cout << "Eigenvalues found:\n" << evalues << std::endl;
    }
 
    return 0;
}

Definition at line 158 of file SymEigsShiftSolver.h.

Member Typedef Documentation

◆ Array

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>

typedef Eigen::Array<Scalar, Eigen::Dynamic, 1> Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >::Array

private

Definition at line 162 of file SymEigsShiftSolver.h.

◆ Index

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>

typedef Eigen::Index Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >::Index

private

Definition at line 161 of file SymEigsShiftSolver.h.

Constructor & Destructor Documentation

◆ SymEigsShiftSolver()

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>

Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >::SymEigsShiftSolver	(	OpType *	op,
		Index	nev,
		Index	ncv,
		Scalar	sigma
	)

inline

Constructor to create a eigen solver object using the shift-and-invert mode.

Parameters

op	Pointer to the matrix operation object, which should implement the shift-solve operation of : calculating $(A-\sigma I)^{-1}v$ for any vector . Users could either create the object from the wrapper class such as DenseSymShiftSolve, or define their own that implements all the public member functions as in DenseSymShiftSolve.
nev	Number of eigenvalues requested. This should satisfy $1\le nev \le n-1$ , where is the size of matrix.
ncv	Parameter that controls the convergence speed of the algorithm. Typically a larger `ncv_` means faster convergence, but it may also result in greater memory use and more matrix operations in each iteration. This parameter must satisfy $nev < ncv \le n$ , and is advised to take $ncv \ge 2\cdot nev$ .
sigma	The value of the shift.

Definition at line 193 of file SymEigsShiftSolver.h.

Member Function Documentation

◆ sort_ritzpair()

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>

void Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >::sort_ritzpair ( int sort_rule )

inlineprivatevirtual

Reimplemented from Spectra::SymEigsBase< double, LARGEST_MAGN, DenseSymShiftSolve< double >, IdentityBOp >.

Definition at line 167 of file SymEigsShiftSolver.h.

Member Data Documentation

◆ m_sigma

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>

const Scalar Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >::m_sigma

private

Definition at line 164 of file SymEigsShiftSolver.h.

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

SymEigsShiftSolver.h

Public Member Functions

Private Types

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>> class Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >

Member Typedef Documentation

◆ Array

◆ Index

Constructor & Destructor Documentation

◆ SymEigsShiftSolver()

Member Function Documentation

◆ sort_ritzpair()

Member Data Documentation

◆ m_sigma

template<typename Scalar = double, int SelectionRule = LARGEST_MAGN, typename OpType = DenseSymShiftSolve<double>>
class Spectra::SymEigsShiftSolver< Scalar, SelectionRule, OpType >