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 Restarted GMRES Method Augmented with Eigenvectors
SIAM Journal on Matrix Analysis and Applications
A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The symmetric eigenvalue problem
The symmetric eigenvalue problem
A refined subspace iteration algorithm for large sparse eigenproblems
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
An analysis of the Rayleigh—Ritz method for approximating eigenspaces
Mathematics of Computation
Matrix algorithms
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems
A Test Matrix Collection for Non-Hermitian Eigenvalue Problems
A modified harmonic block Arnoldi algorithm with adaptive shifts for large interior eigenproblems
Journal of Computational and Applied Mathematics
An invert-free Arnoldi method for computing interior eigenpairs of large matrices
International Journal of Computer Mathematics
Journal of Computational and Applied Mathematics
The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs
Journal of Computational and Applied Mathematics
A new shift scheme for the harmonic Arnoldi method
Mathematical and Computer Modelling: An International Journal
Mathematical and Computer Modelling: An International Journal
Journal of Computational and Applied Mathematics
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The harmonic Arnoldi method can be used to compute some eigenpairs of a large matrix, and it is more suitable for finding interior eigenpairs. However, the harmonic Ritz vectors obtained by the method may converge erratically and may even fail to converge, so that resulting algorithms may not perform well. To improve convergence, a refined harmonic Arnoldi method is proposed that replaces the harmanonic Ritz vectors by certain refined eigenvector approximations. The implicit restart technique advocated by Sorensen is applied to the refined harmonic Arnoldi method and an implicitly restarted refined harmonic Arnoldi algorithm (IRRHA) is presented. For the refined algorithm a new shift scheme, called refined shifts, is proposed that is shown to be better than the exact shifts scheme used in literature. The refined shifts can be computed reliably and cheaply. Extensive numerical experiments are made on real world problems, and comparisons are drawn on IRRHA and the implicitly restarted harmonic Arnoldi algorithm (IRHA) of Morgan as well as the implicitly restarted Arnoldi algorithm (IRA) of Sorensen and the implicitly restarted refined Arnoldi algorithm (IRRA) of Jia. They indicate that the refined algorithms outperform their standard counterparts considerably. Furthermore, for exterior eigenvalue problems IRRA is preferable and more powerful, and for interior eigenvalue problems IRRHA is more attractive and can be much more efficient.