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 Γ. However, the computational scheme of this method does not cover matrices of the form A = αU + βI, where U is an arbitrary unitary matrix; for such matrices, Γ 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Dana, A. G. Zykov, and Kh. D. Ikramov, “A Minimal Residual Method for a Special Class of Linear Systems with Normal Coefficients Matrices,” Zh. Vychisl. Mat. Mat. Fiz. 45, 1928–1937 (2005) [Comput. Math. Math. Phys. 45, 1854–1863 (2005)].
L. Elsner and Kh. D. Ikramov, “On a Condensed Form for Normal Matrices under Finite Sequences of Elementary Unitary Similarities,” Linear Algebra Appl. 254, 79–98 (1997).
A. Greenbaum, Iterative Methods for Solving Linear Systems (SIAM, Philadelphia, 1997).
Author information
Authors and Affiliations
Additional information
Original Russian Text © M. Dana, Kh.D. Ikramov, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 6, pp. 975–982.
Rights and permissions
About this article
Cite this article
Dana, M., Ikramov, K.D. A minimal residual method for linear polynomials in unitary matrices. Comput. Math. and Math. Phys. 46, 930–936 (2006). https://doi.org/10.1134/S0965542506060029
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0965542506060029