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

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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. 
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     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},
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     publisher = {mathdoc},
     volume = {49},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_2_a0/}
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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/