Question
Answer and Explanation
Finding eigenvectors of a sparse matrix using the Eigen3 library involves utilizing its sparse matrix functionalities and eigenvalue solvers. Here's a breakdown of how to accomplish this:
1. Include Necessary Headers:
- Begin by including the necessary Eigen3 headers, particularly those related to sparse matrices and eigenvalue computations. Specifically, #include <Eigen/Sparse> for sparse matrices and #include <Eigen/Eigenvalues> or other specific solvers for eigenvalue computations.
2. Create a Sparse Matrix:
- Construct a sparse matrix object using Eigen's SparseMatrix<double> template. You'll likely need to populate this matrix with your data. Common methods for populating a sparse matrix include creating a Triplet list or directly inserting elements using the sparse matrix's insert or coeffRef methods.
3. Choose an Eigenvalue Solver:
- Eigen3 offers various eigenvalue solvers. For sparse matrices, common options include Eigen::SparseSelfAdjointView which can be used with solvers like Spectra::SymEigsSolver, specifically tailored for sparse self-adjoint matrices, or Arnoldi methods if you have a very large matrix.
-If the matrix is non self-adjoint you may use Eigen::ArpackSupport or a similar library that provides Arnoldi based methods.
4. Use the Solver:
- Create an instance of your chosen solver (e.g., Spectra::SymEigsSolver) and configure it with your sparse matrix.
Set the parameters for how many eigenvalues/vectors you want to find, and other convergence properties.
5. Compute Eigenvalues and Eigenvectors:
- Call the compute method of the solver, which takes care of the computation itself. After successful computation the eigenvalues and eigenvectors will be available through the solver's methods.
6. Access the Results:
- Extract the computed eigenvalues and eigenvectors from the solver object. The eigenvectors will likely be stored as columns in a matrix.
7. Example Code Snippet:
#include <Eigen/Sparse>
#include <Eigen/Eigenvalues>
#include <Spectra/SymEigsSolver.h>
#include <iostream>
int main() {
// Sample Sparse Matrix (replace with your actual matrix)
Eigen::SparseMatrix<double> sparseMatrix(5, 5);
sparseMatrix.insert(0, 0) = 2.0;
sparseMatrix.insert(1, 1) = 3.0;
sparseMatrix.insert(2, 2) = 4.0;
sparseMatrix.insert(0, 1) = 1.0;
sparseMatrix.insert(1, 0) = 1.0;
sparseMatrix.makeCompressed();
//Define SelfAdjointView to access the matrix for the solver
Eigen::SparseSelfAdjointView<Eigen::SparseMatrix<double>> op(sparseMatrix);
// Define a lambda function for the solver that does the multiplication
auto mv_mul = [&op] (double out, const double in) {
Eigen::Map<const Eigen::VectorXd> vec_in(in, op.rows());
Eigen::Map<Eigen::VectorXd> vec_out(out, op.rows());
vec_out = op vec_in;
};
// Create an instance of the eigenvalue solver
Spectra::SymEigsSolver<double, Spectra::SelectionRule::LargestAlge, decltype(mv_mul)> eigs(mv_mul,sparseMatrix.rows(),3, 6); // 3 eigenvalues, 6 max iterations
// Compute eigenvalues and eigenvectors
eigs.init();
int nconv = eigs.compute(Spectra::SortRule::LargestAlge);
if(nconv > 0){
Eigen::VectorXd eigenvalues = eigs.eigenvalues();
Eigen::MatrixXd eigenvectors = eigs.eigenvectors();
std::cout << "Eigenvalues:\\n" << eigenvalues << std::endl;
std::cout << "Eigenvectors:\\n" << eigenvectors << std::endl;
} else {
std::cout << "Error, no eigenvalues found" << std::endl;
}
return 0;
}
Key Considerations:
- Ensure that Eigen3 and a suitable sparse solver library like Spectra are correctly installed and linked in your project.
- The choice of solver might be driven by your specific matrix properties (e.g., size, whether it's self-adjoint, etc.)
- It's crucial to make sure your matrix is correctly initialized, as a non correctly initialised sparse matrix could cause undefined behaviours.
By following these steps and adapting the provided code snippets to your use case, you should be able to efficiently compute eigenvectors of your sparse matrices using the Eigen3 library.