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Approximate solutions and eigenvalue bounds from Krylov subspaces
Abstract
Approximations to the solution of a large sparse symmetric system of equations are considered. The conjugate gradient and minimum residual approximations are studied without reference to their computation. Several different bases for the associated Krylov subspace are used, including the usual Lanczos basis. The zeros of the iteration polynomial for the minimum residual approximation (harmonic Ritz values) are characterized in several ways and, in addition, attractive convergence properties are established. The connection of these harmonic Ritz values to Lehmann's optimal intervals for eigenvalues of the original matrix appears to be new.
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Cited by
A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems
G. Sleijpen, G. L., & Van der Vorst, H. A. (1996) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/s0895479894270427 · Cited by 848
Self-citationTime-evolution methods for matrix-product states
Paeckel, S., Köhler, T., Swoboda, A., Manmana, S. R., Schollwöck, U., & Hubig, C. (2019) · Annals of Physics · doi:10.1016/j.aop.2019.167998 · Cited by 768
Eigenvalue computation in the 20th century
Golub, G. H., & van der Vorst, H. A. (2000) · Journal of Computational and Applied Mathematics · doi:10.1016/s0377-0427(00)00413-1 · Cited by 388
Self-citationRecent computational developments in Krylov subspace methods for linear systems
Simoncini, V., & Szyld, D. B. (2007) · Numerical Linear Algebra with Applications · doi:10.1002/nla.499 · Cited by 375
GMRES with Deflated Restarting
Morgan, R. B. (2002) · SIAM Journal on Scientific Computing · doi:10.1137/s1064827599364659 · Cited by 289
Metadata
- Authors
- Paige, C. C., Parlett, B. N., & van der Vorst, H. A.
- Venue
- Numerical Linear Algebra with Applications
- Publisher
- Wiley
- Published
- 1995-03
- Volume
- 2
- Issue
- 2
- Pages
- 115-133
- ISSN
- 1070-5325 (ISSN-L)
- 1099-1506
- OpenAlex
- W2170486042
- Cited by
- 260
- References
- 0
- FWCI
- 12.24
- Type
- journal article
- Language
- en
- Open access
- closed
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- CoChat. (2026). Citation check: Approximate solutions and eigenvalue bounds from Krylov subspaces. https://app.cochat.ai/papers/10.1002/nla.1680020205
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- Paige, C. C., Parlett, B. N., & van der Vorst, H. A. (1995). Approximate solutions and eigenvalue bounds from Krylov subspaces. Numerical Linear Algebra with Applications. https://doi.org/10.1002/nla.1680020205
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