The Mathematical Institute, University of Oxford, Eprints Archive

Convergence of restarted Krylov subspaces to invariant subspaces

Beattie, Christopher and Embree, Mark and Rossi, John (2001) Convergence of restarted Krylov subspaces to invariant subspaces. Technical Report. Unspecified. (Submitted)

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Abstract

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 non-normality. 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.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1230
Deposited By:Lotti Ekert
Deposited On:20 May 2011 09:20
Last Modified:20 May 2011 09:20

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