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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 &gt;= 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

  • Czech description

Classification

  • Type

    J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database

  • CEP classification

  • 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

  • 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