Spectra  1.0.1 Header-only C++ Library for Large Scale Eigenvalue Problems
Spectra::SymGEigsSolver< OpType, BOpType, GEigsMode::Cholesky > Class Template Reference

#include <Spectra/SymGEigsSolver.h>

Inheritance diagram for Spectra::SymGEigsSolver< OpType, BOpType, GEigsMode::Cholesky >:

## Public Member Functions

SymGEigsSolver (OpType &op, BOpType &Bop, Index nev, Index ncv)

Public Member Functions inherited from Spectra::SymEigsBase< SymGEigsCholeskyOp< OpType, BOpType >, IdentityBOp >
void init (const Scalar *init_resid)

void init ()

Index compute (SortRule selection=SortRule::LargestMagn, Index maxit=1000, Scalar tol=1e-10, SortRule sorting=SortRule::LargestAlge)

CompInfo info () const

Index num_iterations () const

Index num_operations () const

Vector eigenvalues () const

virtual Matrix eigenvectors (Index nvec) const

virtual Matrix eigenvectors () const

## Detailed Description

### template<typename OpType, typename BOpType> class Spectra::SymGEigsSolver< OpType, BOpType, GEigsMode::Cholesky >

This class implements the generalized eigen solver for real symmetric matrices using Cholesky decomposition, i.e., to solve $$Ax=\lambda Bx$$ where $$A$$ is symmetric and $$B$$ is positive definite with the Cholesky decomposition $$B=LL'$$.

This solver requires two matrix operation objects: one for $$A$$ that implements the matrix multiplication $$Av$$, and one for $$B$$ that implements the lower and upper triangular solving $$L^{-1}v$$ and $$(L')^{-1}v$$.

If $$A$$ and $$B$$ are stored as Eigen matrices, then the first operation can be created using the DenseSymMatProd or SparseSymMatProd classes, and the second operation can be created using the DenseCholesky or SparseCholesky classes. If the users need to define their own operation classes, then they should implement all the public member functions as in those built-in classes.

Template Parameters
 OpType The name of the matrix operation class for $$A$$. Users could either use the wrapper classes such as DenseSymMatProd and SparseSymMatProd, or define their own that implements the type definition Scalar and all the public member functions as in DenseSymMatProd. BOpType The name of the matrix operation class for $$B$$. Users could either use the wrapper classes such as DenseCholesky and SparseCholesky, or define their own that implements all the public member functions as in DenseCholesky. Mode Mode of the generalized eigen solver. In this solver it is Spectra::GEigsMode::Cholesky.

Below is an example that demonstrates the usage of this class.

#include <Eigen/Core>
#include <Eigen/SparseCore>
#include <Eigen/Eigenvalues>
#include <Spectra/SymGEigsSolver.h>
#include <Spectra/MatOp/DenseSymMatProd.h>
#include <Spectra/MatOp/SparseCholesky.h>
#include <iostream>
using namespace Spectra;
int main()
{
// We are going to solve the generalized eigenvalue problem A * x = lambda * B * x
const int n = 100;
// Define the A matrix
Eigen::MatrixXd M = Eigen::MatrixXd::Random(n, n);
Eigen::MatrixXd A = M + M.transpose();
// Define the B matrix, a band matrix with 2 on the diagonal and 1 on the subdiagonals
Eigen::SparseMatrix<double> B(n, n);
B.reserve(Eigen::VectorXi::Constant(n, 3));
for (int i = 0; i < n; i++)
{
B.insert(i, i) = 2.0;
if (i > 0)
B.insert(i - 1, i) = 1.0;
if (i < n - 1)
B.insert(i + 1, i) = 1.0;
}
// Construct matrix operation objects using the wrapper classes
DenseSymMatProd<double> op(A);
SparseCholesky<double> Bop(B);
// Construct generalized eigen solver object, requesting the largest three generalized eigenvalues
SymGEigsSolver<DenseSymMatProd<double>, SparseCholesky<double>, GEigsMode::Cholesky>
geigs(op, Bop, 3, 6);
// Initialize and compute
geigs.init();
int nconv = geigs.compute(SortRule::LargestAlge);
// Retrieve results
Eigen::VectorXd evalues;
Eigen::MatrixXd evecs;
if (geigs.info() == CompInfo::Successful)
{
evalues = geigs.eigenvalues();
evecs = geigs.eigenvectors();
}
std::cout << "Generalized eigenvalues found:\n" << evalues << std::endl;
std::cout << "Generalized eigenvectors found:\n" << evecs.topRows(10) << std::endl;
// Verify results using the generalized eigen solver in Eigen
Eigen::MatrixXd Bdense = B;
std::cout << "Generalized eigenvalues:\n" << es.eigenvalues().tail(3) << std::endl;
std::cout << "Generalized eigenvectors:\n" << es.eigenvectors().rightCols(3).topRows(10) << std::endl;
return 0;
}
@ Cholesky
Using Cholesky decomposition to solve generalized eigenvalues.
@ Successful
Computation was successful.

Definition at line 149 of file SymGEigsSolver.h.

## ◆ SymGEigsSolver()

template<typename OpType , typename BOpType >
 Spectra::SymGEigsSolver< OpType, BOpType, GEigsMode::Cholesky >::SymGEigsSolver ( OpType & op, BOpType & Bop, Index nev, Index ncv )
inline

Constructor to create a solver object.

Parameters
 op The $$A$$ matrix operation object that implements the matrix-vector multiplication operation of $$A$$: calculating $$Av$$ for any vector $$v$$. Users could either create the object from the wrapper classes such as DenseSymMatProd, or define their own that implements all the public members as in DenseSymMatProd. Bop The $$B$$ matrix operation object that represents a Cholesky decomposition of $$B$$. It should implement the lower and upper triangular solving operations: calculating $$L^{-1}v$$ and $$(L')^{-1}v$$ for any vector $$v$$, where $$LL'=B$$. Users could either create the object from the wrapper classes such as DenseCholesky, or define their own that implements all the public member functions as in DenseCholesky. $$B$$ needs to be positive definite. nev Number of eigenvalues requested. This should satisfy $$1\le nev \le n-1$$, where $$n$$ 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 $$ncv \ge 2\cdot nev$$.

Definition at line 188 of file SymGEigsSolver.h.

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