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An Extension of the COCR Method to Solving Shifted Linear Systems with Complex Symmetric Matrices

Published online by Cambridge University Press:  28 May 2015

Tomohiro Sogabe  and
Shao-Liang Zhang
Tomohiro Sogabe*
Affiliation:
Graduate School of Information Science & Technology, Aichi Prefectural University, 1522-3 Ibaragabasama, Kumabari, Nagakute-cho, Aichi-gun, Aichi, 480-1198, Japan
Shao-Liang Zhang*
Affiliation:
Department of Computational Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
*
Corresponding author. Email: sogabe@ist.aichi-pu.ac.jp
Corresponding author. Email: zhang@na.cse., nagoya-u.ac.jp
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Abstract

The Conjugate Orthogonal Conjugate Residual (COCR) method [T. Sogabe and S.-L. Zhang, JCAM, 199 (2007), pp. 297-303.] has recently been proposed for solving complex symmetric linear systems. In the present paper, we develop a variant of the COCR method that allows the efficient solution of complex symmetric shifted linear systems. Some numerical examples arising from large-scale electronic structure calculations are presented to illustrate the performance of the variant.

Keywords

Shifted linear systemscomplex symmetric matricesCOCRKrylov subspace methodselectronic structure calculation
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 2 , May 2011 , pp. 97 - 107
Copyright
Copyright © Global-Science Press 2011

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References

[1]van den Eshof, J., Sleijpen, G.L.G., Accurate conjugate gradient methods for families of shifted systems, Appl. Numer. Math., 49 (2004), pp. 17–37.Google Scholar
[2]Freund, R.W., Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 425–448.Google Scholar
[3]Freund, R.W., Solution of shifted linear systems by quasi-minimal residual iterations, in Numerical Linear Algebra, eds. Reichel, L., Ruttan, A. and Varga, R.S., de Gruyter, W., Berlin, 1993, pp. 101–121.Google Scholar
[4]Frommer, A., Bi-CGSTABr(ℓ) for families of shifted linear systems, Computing, 70 (2003), pp. 87–109.Google Scholar
[5]Frommer, A., Grässner, U., Restarted GMRES for shifted linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 15–26.Google Scholar
[6]Frommer, A., Maass, P., Fast CG-based methods for Tikhonov-Phillips regularization, SIAM J. Sci. Comput., 20 (1999), pp. 1831–1850.Google Scholar
[7]Meijerink, J.A., van der Vorst, H.A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comput., 31 (1977), pp. 148–162.Google Scholar
[8]Pommerell, C., Solution of large unsymmetric systems of linear equations, vol. 17 of Series in Micro-electronics, volume 17, Hartung-Gorre Verlag, Konstanz, 1992.Google Scholar
[9]Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, PA, 2003.Google Scholar
[10]Simoncini, V., Szyld, D.B., Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 14 (2007), pp. 1–59.Google Scholar
[11]Sogabe, T. and Zhang, S.-L., An iterative method based on an A-biorthogonalization process for nonsymmetric linear systems, in: Proceedings of The 7th China-Japan Seminar on Numerical Mathematics, eds. Shi, Z.-C. and Okamoto, H., Science Press, Beijing, 2006, pp. 120–130.Google Scholar
[12]Sogabe, T., Zhang, S.-L., A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math., 199 (2007), pp. 297–303.Google Scholar
[13]Sogabe, T., Hoshi, T., Zhang, S.-L., and Fujiwara, T., On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems, Electron. Trans. Numer. Anal., 31 (2008), pp. 126–140.Google Scholar
[14]Stiefel, E., Relaxationsmethoden bester Strategie zur Lösung linearer Gleichungssysteme, Comment. Math. Helv., 29 (1955), pp. 157–179.Google Scholar
[15]Takayama, R., Hoshi, T., Sogabe, T., Zhang, S.-L., Fujiwara, T., Linear algebraic calculation of Green's function for large-scale electronic structure theory, Phys. Rev. B, 73:165108 (2006), pp. 1–9.Google Scholar
[16]van der Vorst, H.A., Melissen, J.B.M., A Petrov-Galerkin type method for solving Ax = b, where A is symmetric complex, IEEE Trans. Mag., 26:2 (1990), pp. 706–708.Google Scholar