THREE-PRECISION GMRES-BASED ITERATIVE REFINEMENT FOR LEAST SQUARES PROBLEMS
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10420371" target="_blank" >RIV/00216208:11320/20:10420371 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=epgdYWDpjD" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=epgdYWDpjD</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/20M1316822" target="_blank" >10.1137/20M1316822</a>
Alternative languages
Result language
angličtina
Original language name
THREE-PRECISION GMRES-BASED ITERATIVE REFINEMENT FOR LEAST SQUARES PROBLEMS
Original language description
The standard iterative refinement procedure for improving an approximate solution to the least squares problem min(x) parallel to b - Ax parallel to(2), where A is an element of R-mxn with m >= n has full rank, is based on solving the (m n) x (m n) augmented system with the aid of a QR factorization. In order to exploit multiprecision arithmetic, iterative refinement can be formulated to use three precisions, but the resulting algorithm converges only for a limited range of problems. We build an iterative refinement algorithm called GMRES-LSIR, analogous to the GMRES-IR algorithm developed for linear systems [E. Carson and N. J. Higham, SIAM T. Set. Comput., 40 (2018), pp. A817-A8471, that solves the augmented system using GMRES preconditioned by a matrix based on the computed QR factors. We explore two left preconditioners; the first has full off-diagonal blocks, and the second is block diagonal and can be applied in either left-sided or split form. We prove that for a wide range of problems the first preconditioner yields backward and forward errors for the augmented system of order the working precision under suitable assumptions on the precisions and the problem conditioning. Our proof does not extend to the block diagonal preconditioner, but our numerical experiments show that with this preconditioner the algorithm performs about as well in practice. The experiments also show that if we use MINRES in place of GMRES then the convergence is similar for sufficiently well conditioned problems but worse for the most ill conditioned ones.
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
—
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2020
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 of Scientific Computing
ISSN
1064-8275
e-ISSN
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Volume of the periodical
42
Issue of the periodical within the volume
6
Country of publishing house
US - UNITED STATES
Number of pages
21
Pages from-to
"A4063"-"A4083"
UT code for WoS article
000600650400024
EID of the result in the Scopus database
2-s2.0-85099012171