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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. 127-152, 2017/11/15
Today the most popular iterative methods for solving nonsymmetric linear systems are Krylov methods. In this paper we show how to construct a non-orthogonal basis of the Krylov subspace such that the quasi-orthogonal residual (Q-OR) Krylov method using this basis yields the same residual norms as GMRES up to the final stagnation phase, provided GMRES is not stagnating. In many examples this new Krylov method gives a better maximum attainable accuracy than GMRES with a modified Gram-Schmidt (MGS) implementation. Even though the number of floating point operations per iteration is larger than for GMRES, the optimal Q-OR method offers more potential for parallelism than GMRES with MGS.
Keywords: linear systems, Krylov methods, Q-OR algorithm