Geodesic
A method of congruent type for linear systems with conjugate-normal coefficient matrices
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 211-224 
M. Ghasemi Kamalvand; Kh. D. Ikramov. A method of congruent type for linear systems with conjugate-normal coefficient matrices. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 2, pp. 211-224. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a0/
@article{ZVMMF_2009_49_2_a0,
     author = {M. Ghasemi Kamalvand and Kh. D. Ikramov},
     title = {A~method of congruent type for linear systems with conjugate-normal coefficient matrices},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {211--224},
     year = {2009},
     volume = {49},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a0/}
}
TY  - JOUR
AU  - M. Ghasemi Kamalvand
AU  - Kh. D. Ikramov
TI  - A method of congruent type for linear systems with conjugate-normal coefficient matrices
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2009
SP  - 211
EP  - 224
VL  - 49
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a0/
LA  - ru
ID  - ZVMMF_2009_49_2_a0
ER  - 
%0 Journal Article
%A M. Ghasemi Kamalvand
%A Kh. D. Ikramov
%T A method of congruent type for linear systems with conjugate-normal coefficient matrices
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2009
%P 211-224
%V 49
%N 2
%U http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a0/
%G ru
%F ZVMMF_2009_49_2_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

Minimal residual methods, such as MINRES and GMRES, are well-known iterative versions of direct procedures for reducing a matrix to special condensed forms. The method of reduction used in these procedures is a sequence of unitary similarity transformations, while the condensed form is a tridiagonal matrix (MINRES) or a Hessenberg matrix (GMRES). The algorithm CSYM proposed in the 1990s for solving systems with complex symmetric matrices was based on the tridiagonal reduction performed via unitary congruences rather than similarities. In this paper, we construct an extension of this algorithm to the entire class of conjugate-normal matrices. (Complex symmetric matrices are a part of this class.) Numerical results are presented. They show that, on many occasions, the proposed algorithm has a superior convergence rate compared to GMRES.

[1] Bunse-Gerstner A., Stöver R., “On a conjugate gradient-type method for solving complex symmetric linear systems”, Linear Algebra Appl., 287 (1999), 105–123 | DOI | MR | Zbl

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

[3] Ikramov Kh. D., “Sopryazhenno-normalnye matritsy i matrichnye uravneniya otnositelno $A$, $\overline A$ i $A^T$”, Dokl. AN, 412:3 (2007), 305–307 | MR

[4] Ikramov Kh. D., “O trekhdiagonalnykh sopryazhenno-normalnykh matritsakh”, Zh. vychisl. matem. i matem. fiz., 47:2 (2007), 179–185 | MR | Zbl

[5] 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

[6] Fassbender H., Ikramov Kh. D., “Some observations on the Youla form and conjugate-normal matrices”, Linear Algebra Appl., 422 (2007), 29–38 | DOI | MR | Zbl