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 Zh. Vychisl. Mat. Mat. Fiz., 2006, Volume 46, Number 6, Pages 975–982 (Mi zvmmf452)

A minimal residual method for linear polynomials in unitary matrices

M. Danaa, Kh. D. Ikramovb

a Faculty of Mathematics, University of Kurdistan, Sanandage, 66177, Islamic Republic of Iran
b Faculty of Computational Mathematics and Cybernetics, Moscow State University, Leninskie gory, Moscow, 119992, Russia

Abstract: A minimal residual method, called MINRES-N2, that is based on the use of unconventional Krylov subspaces was previously proposed by the authors for solving a system of linear equations $Ax=b$ with a normal coefficient matrix whose spectrum belongs to an algebraic second-degree curve $\Gamma$. However, the computational scheme of this method does not cover matrices of the form $A=\alpha U+\beta I$, where $U$ is an arbitrary unitary matrix; for such matrices, $\Gamma$ is a circle. Systems of this type are repeatedly solved when the eigenvectors of a unitary matrix are calculated by inverse iteration. In this paper, a modification of MINRES-N2 suitable for linear polynomials in unitary matrices is proposed. Numerical results are presented demonstrating the significant superiority of the modified method over GMRES as applied to systems of this class.

Key words: linear polynomials in unitary matrices, minimal residual method, modification of the MINRES-N2 algorithm.

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English version:
Computational Mathematics and Mathematical Physics, 2006, 46:6, 930–936

Bibliographic databases:

UDC: 519.614

Citation: M. Dana, Kh. D. Ikramov, “A minimal residual method for linear polynomials in unitary matrices”, Zh. Vychisl. Mat. Mat. Fiz., 46:6 (2006), 975–982; Comput. Math. Math. Phys., 46:6 (2006), 930–936

Citation in format AMSBIB
\Bibitem{DanIkr06} \by M.~Dana, Kh.~D.~Ikramov \paper A~minimal residual method for linear polynomials in unitary matrices \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2006 \vol 46 \issue 6 \pages 975--982 \mathnet{http://mi.mathnet.ru/zvmmf452} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2285358} \transl \jour Comput. Math. Math. Phys. \yr 2006 \vol 46 \issue 6 \pages 930--936 \crossref{https://doi.org/10.1134/S0965542506060029} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33746133903}