Spectra

Spectra stands for Sparse Eigenvalue Computation Toolkit as a Redesigned ARPACK. It is a C++ library for large scale eigenvalue problems, built on top of Eigen, an open source linear algebra library.

Spectra is implemented as a header-only C++ library, whose only dependency, Eigen, is also header-only. Hence Spectra can be easily embedded in C++ projects that require calculating eigenvalues of large matrices.

The development page of Spectra is at https://github.com/yixuan/spectra/.

Relation to ARPACK

ARPACK is a software package written in FORTRAN for solving large scale eigenvalue problems. The development of Spectra is much inspired by ARPACK, and as the full name indicates, Spectra is a redesign of the ARPACK library using the C++ language.

In fact, Spectra is based on the algorithm described in the ARPACK Users' Guide, the implicitly restarted Arnoldi/Lanczos method. However, it does not use the ARPACK code, and it is NOT a clone of ARPACK for C++. In short, Spectra implements the major algorithms in ARPACK, but Spectra provides a completely different interface, and it does not depend on ARPACK.

Common Usage

Spectra is designed to calculate a specified number ( k) of eigenvalues of a large square matrix ( A). Usually k is much smaller than the size of the matrix ( n), so that only a few eigenvalues and eigenvectors are computed, which in general is more efficient than calculating the whole spectral decomposition. Users can choose eigenvalue selection rules to pick the eigenvalues of interest, such as the largest k eigenvalues, or eigenvalues with largest real parts, etc.

To use the eigen solvers in this library, the user does not need to directly provide the whole matrix, but instead, the algorithm only requires certain operations defined on A. In the basic setting, it is simply the matrix-vector multiplication. Therefore, if the matrix-vector product Ax can be computed efficiently, which is the case when A is sparse, Spectra will be very powerful for large scale eigenvalue problems.

There are two major steps to use the Spectra library:

1. Define a class that implements a certain matrix operation, for example the matrix-vector multiplication y=Ax or the shift-solve operation y=(A-\sigma I)^{-1}x. Spectra has defined a number of helper classes to quickly create such operations from a matrix object. See the documentation of DenseGenMatProd, DenseSymShiftSolve, etc.
2. Create an object of one of the eigen solver classes, for example SymEigsSolver for symmetric matrices, and GenEigsSolver for general matrices. Member functions of this object can then be called to conduct the computation and retrieve the eigenvalues and/or eigenvectors.

Below is a list of the available eigen solvers in Spectra:

• SymEigsSolver: For real symmetric matrices
• GenEigsSolver: For general real- and complex-valued matrices
• SymEigsShiftSolver: For real symmetric matrices using the shift-and-invert mode
• GenEigsRealShiftSolver: For general real matrices using the shift-and-invert mode, with a real-valued shift
• GenEigsComplexShiftSolver: For general real matrices using the shift-and-invert mode, with a complex-valued shift
• SymGEigsSolver: For generalized eigen solver with real symmetric matrices
• SymGEigsShiftSolver: For generalized eigen solver with real symmetric matrices, using the shift-and-invert mode
• DavidsonSymEigsSolver: Jacobi-Davidson eigen solver for real symmetric matrices, with the DPR correction method
• HermEigsSolver: For complex Hermitian matrices

Examples

Below is an example that demonstrates the use of the eigen solver for symmetric matrices.

#include <Eigen/Core>
#include <Spectra/SymEigsSolver.h>
// <Spectra/MatOp/DenseSymMatProd.h> is implicitly included
#include <iostream>
 
using namespace Spectra;
 
int main()
{
    // We are going to calculate the eigenvalues of M
    Eigen::MatrixXd A = Eigen::MatrixXd::Random(10, 10);
    Eigen::MatrixXd M = A + A.transpose();
 
    // Construct matrix operation object using the wrapper class DenseSymMatProd
    DenseSymMatProd<double> op(M);
 
    // Construct eigen solver object, requesting the largest three eigenvalues
    SymEigsSolver<DenseSymMatProd<double>> eigs(op, 3, 6);
 
    // Initialize and compute
    eigs.init();
    int nconv = eigs.compute(SortRule::LargestAlge);
 
    // Retrieve results
    Eigen::VectorXd evalues;
    if(eigs.info() == CompInfo::Successful)
        evalues = eigs.eigenvalues();
 
    std::cout << "Eigenvalues found:\n" << evalues << std::endl;
 
    return 0;
}

Sparse matrix is supported via classes such as SparseGenMatProd and SparseSymMatProd.

#include <Eigen/Core>
#include <Eigen/SparseCore>
#include <Spectra/GenEigsSolver.h>
#include <Spectra/MatOp/SparseGenMatProd.h>
#include <iostream>
 
using namespace Spectra;
 
int main()
{
    // A band matrix with 1 on the main diagonal, 2 on the below-main subdiagonal,
    // and 3 on the above-main subdiagonal
    const int n = 10;
    Eigen::SparseMatrix<double> M(n, n);
    M.reserve(Eigen::VectorXi::Constant(n, 3));
    for(int i = 0; i < n; i++)
    {
        M.insert(i, i) = 1.0;
        if(i > 0)
            M.insert(i - 1, i) = 3.0;
        if(i < n - 1)
            M.insert(i + 1, i) = 2.0;
    }
 
    // Construct matrix operation object using the wrapper class SparseGenMatProd
    SparseGenMatProd<double> op(M);
 
    // Construct eigen solver object, requesting the largest three eigenvalues
    GenEigsSolver<SparseGenMatProd<double>> eigs(op, 3, 6);
 
    // Initialize and compute
    eigs.init();
    int nconv = eigs.compute(SortRule::LargestMagn);
 
    // Retrieve results
    Eigen::VectorXcd evalues;
    if(eigs.info() == CompInfo::Successful)
        evalues = eigs.eigenvalues();
 
    std::cout << "Eigenvalues found:\n" << evalues << std::endl;
 
    return 0;
}

And here is an example for user-supplied matrix operation class.

#include <Eigen/Core>
#include <Spectra/SymEigsSolver.h>
#include <iostream>
 
using namespace Spectra;
 
// M = diag(1, 2, ..., 10)
class MyDiagonalTen
{
public:
    using Scalar = double;  // A typedef named "Scalar" is required
    int rows() const { return 10; }
    int cols() const { return 10; }
    // y_out = M * x_in
    void perform_op(const double *x_in, double *y_out) const
    {
        for(int i = 0; i < rows(); i++)
        {
            y_out[i] = x_in[i] * (i + 1);
        }
    }
};
 
int main()
{
    MyDiagonalTen op;
    SymEigsSolver<MyDiagonalTen> eigs(op, 3, 6);
    eigs.init();
    eigs.compute(SortRule::LargestAlge);
    if(eigs.info() == CompInfo::Successful)
    {
        Eigen::VectorXd evalues = eigs.eigenvalues();
        std::cout << "Eigenvalues found:\n" << evalues << std::endl;
    }
 
    return 0;
}

Shift-and-invert Mode

When it is needed to find eigenvalues that are closest to a number \sigma, for example to find the smallest eigenvalues of a positive definite matrix (in which case \sigma=0), it is advised to use the shift-and-invert mode of eigen solvers.

In the shift-and-invert mode, selection rules are applied to 1/(\lambda-\sigma) rather than \lambda, where \lambda are eigenvalues of A. To use this mode, users need to define the shift-solve matrix operation. See the documentation of SymEigsShiftSolver for details.

Complex-valued Matrices

Spectra provides the HermEigsSolver solver for complex-valued Hermitian matrices, and the GenEigsSolver solver for general complex-valued matrices. See the example below.

#include <Eigen/Core>
#include <Spectra/HermEigsSolver.h>
#include <Spectra/GenEigsSolver.h>
#include <iostream>
 
using namespace Spectra;
 
int main()
{
    std::srand(0);
 
    // We are going to calculate the eigenvalues of H and G
    Eigen::MatrixXcd G = Eigen::MatrixXcd::Random(10, 10);
    // H is Hermitian
    Eigen::MatrixXcd H = G + G.adjoint();
 
    // Construct matrix operation objects using the wrapper
    // classes DenseHermMatProd and DenseGenMatProd
    using OpHType = DenseHermMatProd<std::complex<double>>;
    using OpGType = DenseGenMatProd<std::complex<double>>;
    OpHType opH(H);
    OpGType opG(G);
 
    // Construct solver object for H, requesting the largest three eigenvalues
    HermEigsSolver<OpHType> eigsH(opH, 3, 6);
 
    // Initialize and compute
    eigsH.init();
    int nconvH = eigsH.compute(SortRule::LargestAlge);
 
    // Retrieve results
    // Eigenvalues are real-valued, and eigenvectors are complex-valued
    if (eigsH.info() == CompInfo::Successful)
    {
        Eigen::VectorXd evaluesH = eigsH.eigenvalues();
        std::cout << "Eigenvalues of H found:\n" << evaluesH << std::endl;
        Eigen::MatrixXcd evecsH = eigsH.eigenvectors();
        std::cout << "Eigenvectors of H:\n" << evecsH << std::endl;
    }
 
    // Similar procedure for matrix G
    GenEigsSolver<OpGType> eigsG(opG, 3, 6);
    eigsG.init();
    int nconvG = eigsG.compute(SortRule::LargestMagn);
    if (eigsG.info() == CompInfo::Successful)
    {
        Eigen::VectorXcd evaluesG = eigsG.eigenvalues();
        std::cout << "Eigenvalues of G found:\n" << evaluesG << std::endl;
        Eigen::MatrixXcd evecsG = eigsG.eigenvectors();
        std::cout << "Eigenvectors of G:\n" << evecsG << std::endl;
    }
 
    return 0;
}

License

Spectra is an open source project licensed under MPL2, the same license used by Eigen.