When does the Lanczos algorithm compute exactly?
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F22%3A10446641" target="_blank" >RIV/00216208:11320/22:10446641 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=2jn0kV86XK" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=2jn0kV86XK</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1553/etna_vol55s547" target="_blank" >10.1553/etna_vol55s547</a>
Alternative languages
Result language
angličtina
Original language name
When does the Lanczos algorithm compute exactly?
Original language description
In theory, the Lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic the orthogonality and linear independence of the computed Lanczos vectors is usually lost quickly. In this paper we study a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the Lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results are formulated also for an implementation of the conjugate gradient method called cgLanczos. This implementation then computes approximations that agree with their exact counterparts to a relative accuracy given by the machine precision and the condition number of the system matrix. The results are extended to the Arnoldi algorithm, the nonsymmetric Lanczos algorithm, the Golub-Kahan bidiagonalization, the block-Lanczos algorithm, and their counterparts for solving linear systems.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10102 - Applied mathematics
Result continuities
Project
<a href="/en/project/GA20-01074S" target="_blank" >GA20-01074S: Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Electronic Transactions on Numerical Analysis
ISSN
1068-9613
e-ISSN
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Volume of the periodical
55
Issue of the periodical within the volume
June 9, 2022
Country of publishing house
US - UNITED STATES
Number of pages
21
Pages from-to
547-567
UT code for WoS article
000813353900020
EID of the result in the Scopus database
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