# Quick Start

### 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 up 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, and in the basic setting, it is simply the matrix-vector multiplication. Therefore, if the matrix-vector product A * x can be computed efficiently, which is the case when A is sparse, Spectra will be very powerful for large scale eigenvalue problems.

### Key steps

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 = A * x or the shift-solve operation y = inv(A - σ * I) * 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 to retrieve the eigenvalues and/or eigenvectors.

### Solvers

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 <SymEigsSolver.h>  // Also includes <MatOp/DenseSymMatProd.h>
#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< double, LARGEST_ALGE, DenseSymMatProd<double> > eigs(&op, 3, 6);

// Initialize and compute
eigs.init();
int nconv = eigs.compute();

// Retrieve results
Eigen::VectorXd evalues;
if(eigs.info() == SUCCESSFUL)
evalues = eigs.eigenvalues();

std::cout << "Eigenvalues found:\n" << evalues << std::endl;

return 0;
}


Sparse matrix is supported via the SparseGenMatProd class.

#include <Eigen/Core>
#include <Eigen/SparseCore>
#include <GenEigsSolver.h>
#include <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< double, LARGEST_MAGN, SparseGenMatProd<double> > eigs(&op, 3, 6);

// Initialize and compute
eigs.init();
int nconv = eigs.compute();

// Retrieve results
Eigen::VectorXcd evalues;
if(eigs.info() == 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 <SymEigsSolver.h>
#include <iostream>

using namespace Spectra;

// M = diag(1, 2, ..., 10)
class MyDiagonalTen
{
public:
int rows() { return 10; }
int cols() { return 10; }
// y_out = M * x_in
void perform_op(double *x_in, double *y_out)
{
for(int i = 0; i < rows(); i++)
{
y_out[i] = x_in[i] * (i + 1);
}
}
};

int main()
{
MyDiagonalTen op;
SymEigsSolver<double, LARGEST_ALGE, MyDiagonalTen> eigs(&op, 3, 6);
eigs.init();
eigs.compute();
if(eigs.info() == SUCCESSFUL)
{
Eigen::VectorXd evalues = eigs.eigenvalues();
std::cout << "Eigenvalues found:\n" << evalues << std::endl;
}

return 0;
}


To compile and run these examples, simply download the source code of Spectra and Eigen, and let the compiler know about their paths. For example:

g++ -I/path/to/eigen -I/path/to/spectra/include -O2 example.cpp