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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 470-485, 2023/07/03
In this paper, we consider a family of algorithms, called IDR, based on the induced dimension reduction theorem. IDR is a family of efficient short recurrence methods introduced by Sonneveld and Van Gijzen for solving large systems of nonsymmetric linear equations. These methods generate residual vectors that live in a sequence of nested subspaces. We present the IDR(s) method and give two improvements of its convergence. We also define and give a global version of the IDR(s) method and describe a partial and a complete improvement of its convergence. Moreover, we recall the block version and state its improvements. Numerical experiments are provided to illustrate the performances of the derived algorithms compared to the well-known classical GMRES method and the bi-conjugate gradient stabilized method for systems with a single right-hand side, as well as the global GMRES, the global bi-conjugate gradient stabilized, the block GMRES, and the block bi-conjugate gradient stabilized methods for systems with multiple right-hand sides.
Keywords: linear equations, iterative methods, IDR method, Krylov subspace, global and block Krylov subspace methods