On GMRES for singular EP and GP systems
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F18%3A00490058" target="_blank" >RIV/67985840:_____/18:00490058 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1137/17M1128216" target="_blank" >http://dx.doi.org/10.1137/17M1128216</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/17M1128216" target="_blank" >10.1137/17M1128216</a>
Alternative languages
Result language
angličtina
Original language name
On GMRES for singular EP and GP systems
Original language description
In this contribution, we study the numerical behavior of the generalized minimal residual (GMRES) method for solving singular linear systems. It is known that GMRES determines a least squares solution without breakdown if the coefficient matrix is range-symmetric (EP) or if its range and nullspace are disjoint (GP) and the system is consistent. We show that the accuracy of GMRES iterates may deteriorate in practice due to three distinct factors: (i) the inconsistency of the linear system, (ii) the distance of the initial residual to the nullspace of the coefficient matrix, and (iii) the extremal principal angles between the ranges of the coefficient matrix and its transpose. These factors lead to poor conditioning of the extended Hessenberg matrix in the Arnoldi decomposition and affect the accuracy of the computed least squares solution. We also compare GMRES with the range restricted GMRES method. Numerical experiments show typical behaviors of GMRES for small problems with EP and GP matrices.
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
10101 - Pure mathematics
Result continuities
Project
—
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM Journal on Matrix Analysis and Applications
ISSN
0895-4798
e-ISSN
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Volume of the periodical
39
Issue of the periodical within the volume
2
Country of publishing house
US - UNITED STATES
Number of pages
16
Pages from-to
1033-1048
UT code for WoS article
000436971900020
EID of the result in the Scopus database
2-s2.0-85049744175