#include <SymEigsShiftSolver.h>

Inheritance diagram for Spectra::SymEigsShiftSolver< OpType >:

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

Public Member Functions inherited from Spectra::HermEigsBase< DenseSymShiftSolve< double >, IdentityBOp >
Index	compute (SortRule selection=SortRule::LargestMagn, Index maxit=1000, RealScalar tol=1e-10, SortRule sorting=SortRule::LargestAlge)

RealVector	eigenvalues () const

virtual Matrix	eigenvectors () const

virtual Matrix	eigenvectors (Index nvec) const

CompInfo	info () const

void	init ()

void	init (const Scalar *init_resid)

Index	num_iterations () const

Index	num_operations () const

Private Types
using	Array = Eigen::Array< Scalar, Eigen::Dynamic, 1 >

using	Base = HermEigsBase< OpType, IdentityBOp >

using	Index = Eigen::Index

using	Scalar = typename OpType::Scalar

Private Member Functions
void	sort_ritzpair (SortRule sort_rule) override

Private Attributes
const Index	m_nev

RealVector	m_ritz_val

const Scalar	m_sigma

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

const Index	m_n

const Index	m_ncv

const Index	m_nev

Index	m_niter

Index	m_nmatop

const DenseSymShiftSolve< double > &	m_op

std::vector< DenseSymShiftSolve< double > >	m_op_container

RealVector	m_ritz_val

Detailed Description

template<typename OpType = DenseSymShiftSolve<double>>
class Spectra::SymEigsShiftSolver< 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

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 the type definition Scalar and 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<DenseSymShiftSolve<double>> eigs(op, 3, 6, 0.0);
 
    eigs.init();
    eigs.compute(SortRule::LargestMagn);
    if (eigs.info() == CompInfo::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:
    using Scalar = double;  // A typedef named "Scalar" is required
    int rows() const { return 10; }
    int cols() const { 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) const
    {
        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<MyDiagonalTenShiftSolve> eigs(op, 3, 6, 3.14);
    eigs.init();
    eigs.compute(SortRule::LargestMagn);
    if (eigs.info() == CompInfo::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 149 of file SymEigsShiftSolver.h.