An invert-free Arnoldi method for computing interior eigenpairs of large matrices
International Journal of Computer Mathematics
A numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics
Deflated block Krylov subspace methods for large scale eigenvalue problems
Journal of Computational and Applied Mathematics
An Arnoldi-Extrapolation algorithm for computing PageRank
Journal of Computational and Applied Mathematics
Further Analysis of the Arnoldi Process for Eigenvalue Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Computation of eigenpair partial derivatives by Rayleigh-Ritz procedure
Journal of Computational and Applied Mathematics
Hi-index | 0.00 |
The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired set of eigenvalues is a maximum "reachable invariant subspace" that can be developed from the given starting vector. Convergence for this distinguished subspace is bounded in terms involving a polynomial approximation problem. Elementary results from potential theory lead to convergence rate estimates and suggest restarting strategies based on optimal approximation points (e.g., Leja or Chebyshev points); exact shifts are evaluated within this framework. Computational examples illustrate the utility of these results. Origins of superlinear effects are also described.