Geodesic
A minimal residual method for linear polynomials in unitary matrices
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 6, pp. 975-982 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{ZVMMF_2006_46_6_a1,
     author = {M. Dana and Kh. D. Ikramov},
     title = {A~minimal residual method for linear polynomials in unitary matrices},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {975--982},
     year = {2006},
     volume = {46},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_6_a1/}
}
TY  - JOUR
AU  - M. Dana
AU  - Kh. D. Ikramov
TI  - A minimal residual method for linear polynomials in unitary matrices
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2006
SP  - 975
EP  - 982
VL  - 46
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_6_a1/
LA  - ru
ID  - ZVMMF_2006_46_6_a1
ER  - 
%0 Journal Article
%A M. Dana
%A Kh. D. Ikramov
%T A minimal residual method for linear polynomials in unitary matrices
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2006
%P 975-982
%V 46
%N 6
%U http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_6_a1/
%G ru
%F ZVMMF_2006_46_6_a1
M. Dana; Kh. D. Ikramov. A minimal residual method for linear polynomials in unitary matrices. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 6, pp. 975-982. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_6_a1/

[1] Dana M., Zykov A. G., Ikramov Kh. D., “Metod minimalnykh nevyazok dlya spetsialnogo klassa lineinykh sistem s normalnymi matritsami koeffitsientov”, Zh. vychisl. matem. i matem. fiz., 45:11 (2005), 1928–1937 | MR | Zbl

[2] Elsner L., Ikramov Kh. D., “On a condensed form for normal matrices under finite sequences of elementary unitary similarities”, Linear Algebra and Appl., 254 (1997), 79–98 | DOI | MR | Zbl

[3] Greenbaum A., Iterative methods for solving linear systems, SIAM, Philadelphia, 1997 | MR