lanczos.hpp File Reference

Lanczos algorithm – k extreme eigenvalues of a large sparse matrix. More...

#include "core/vector.hpp"
#include "core/matrix.hpp"
#include "core/policy.hpp"
#include "linalg/solvers/cg.hpp"
#include "linalg/eigen/jacobi_eig.hpp"

Go to the source code of this file.

Classes
struct	num::LanczosResult
	Result of a Lanczos eigensolver. More...

Namespaces
namespace	num

Functions
LanczosResult	num::lanczos (MatVecFn matvec, idx n, idx k, real tol=1e-10, idx max_steps=0, Backend backend=Backend::seq)
	Lanczos eigensolver for large sparse symmetric matrices.

Detailed Description

Lanczos algorithm – k extreme eigenvalues of a large sparse matrix.

The Lanczos algorithm builds a k-dimensional Krylov subspace

K_k(A, v) = span{v, Av, A^2v, ..., A^{k-1}v}

and finds the k Ritz pairs (approximate eigenpairs) that are the extreme eigenvalues of the projected kxk tridiagonal matrix T_k.

Key properties:

• Matrix-free: A is given as a MatVecFn callable (works with sparse, structured A)
• O(k*nnz) work for k steps (nnz = cost of one matvec)
• Finds the k LARGEST and k SMALLEST eigenvalues accurately

Usage:

lanczos(mv, n, k);                          // sequential
lanczos(mv, n, k, 1e-10, 0, num::omp);     // OMP reorthogonalisation

Definition in file lanczos.hpp.