ACM Transactions on Mathematical Software (TOMS)
Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
A large, sparse, and indefinite generalized eigenvalue problem from fluid mechanics
SIAM Journal on Scientific and Statistical Computing - Special issue on iterative methods in numerical linear algebra
Quasi-kernel polynomials and their use in non-Hermitian matrix iterations
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
The Convergence of Generalized Lanczos Methods for Large Unsymmetric Eigenproblems
SIAM Journal on Matrix Analysis and Applications
A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
SIAM Journal on Matrix Analysis and Applications
Applied numerical linear algebra
Applied numerical linear algebra
Global FOM and GMRES algorithms for matrix equations
Applied Numerical Mathematics
Implicitly Restarted GMRES and Arnoldi Methods for Nonsymmetric Systems of Equations
SIAM Journal on Matrix Analysis and Applications
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Matrix algorithms
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems
Convergence properties of some block Krylov subspace methods for multiple linear systems
Journal of Computational and Applied Mathematics
| Hi-index | 7.29 |
The global Arnoldi method can be used to compute exterior eigenpairs of a large non-Hermitian matrix A, but it does not work well for interior eigenvalue problems. Based on the global Arnoldi process that generates an F-orthonormal basis of a matrix Krylov subspace, we propose a global harmonic Arnoldi method for computing certain harmonic F-Ritz pairs that are used to approximate some interior eigenpairs. We propose computing the F-Rayleigh quotients of the large non-Hermitian matrix with respect to harmonic F-Ritz vectors and taking them as new approximate eigenvalues. They are better and more reliable than the harmonic F-Ritz values. The global harmonic Arnoldi method inherits convergence properties of the harmonic Arnoldi method applied to a larger matrix whose distinct eigenvalues are the same as those of the original given matrix. Some properties of the harmonic F-Ritz vectors are presented. As an application, assuming that A is diagonalizable, we show that the global harmonic Arnoldi method is able to solve multiple eigenvalue problems both in theory and in practice. To be practical, we develop an implicitly restarted global harmonic Arnoldi algorithm with certain harmonic F-shifts suggested. In particular, this algorithm can be adaptively used to solve multiple eigenvalue problems. Numerical experiments show that the algorithm is efficient for the eigenproblem and is reliable for quite ill-conditioned multiple eigenproblems.