Spectra  1.0.1 Header-only C++ Library for Large Scale Eigenvalue Problems
Spectra Documentation

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 written in FORTRAN for solving large scale eigenvalue problems. The development of Spectra is much inspired by ARPACK, and as the whole 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 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 Spectra::DenseGenMatProd, Spectra::DenseSymShiftSolve, etc.
2. Create an object of one of the eigen solver classes, for example Spectra::SymEigsSolver for symmetric matrices, and Spectra::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:

## 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;
}
@ Successful
Computation was successful.

Sparse matrix is supported via classes such as Spectra::SparseGenMatProd and Spectra::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 Spectra::SymEigsShiftSolver for details.