Two Quasi-combining Real and Imaginary Parts Iteration Methods for Solving Complex Symmetric System of Linear Equations

Abstract

To solve the large sparse complex symmetric linear equations more efficiently, we introduce a new matrix

H_{\omega }=W+\omega T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_\omega =W+\omega T$$\end{document}

and establish two quasi-combining real and imaginary parts iteration methods, which will be simply called the QCRI1 and QCRI2 iteration methods. We give the upper bounds of the spectral radiuses of the two methods and discuss their convergence conditions that make these upper bounds less than 1. In addition, the theoretical quasi-optimal parameters minimizing the upper bound of the spectral radius of the iteration matrix of the QCRI1 method are presented. Meanwhile, the inexact versions of the proposed methods are also provided, and their convergence properties are given. Finally, numerical results illustrate the effectiveness of our methods.

Keywords

Complex symmetric linear system / Hermitian and skew-Hermitian splitting / Iteration method / Convergence property / Inexact implementation / 65F10 / 65H10

References

[1] Arridge, S.-R.: Optical tomography in medical imaging. Inverse Prob. 15, 41–93 (1999)

[2] Axelsson O, Kucherov A. Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl.. 2000, 7: 197-218.

[3] Bai Z-Z. Quasi-HSS iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts. Numer. Linear Algebr Appl.. 2018, 25. e2116

[4] Bai Z-Z, Benzi M, Chen F. Modified HSS iteration methods for a class of complex symmetric linear systems. Computing. 2010, 87: 93-111.

[5] Bai Z-Z, Benzi M, Chen F. On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms. 2011, 56: 297-317.

[6] Bai Z-Z, Golub GH. Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal.. 2007, 27: 1-23.

[7] Bai, Z.-Z., Golub, G.-H., Ng, M.-K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

[8] Bai, Z.-Z., Golub, G.-H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

[9] Benzi M, Bertaccini D. Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal.. 2008, 28: 598-618.

[10] Bertaccini D. Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Trans. Numer. Anal.. 2004, 18: 49-64

[11] Bill P. Efficient preconditioning scheme for block partitioned matrices with structured sparsity. Numer. Linear Algebra Appl.. 2000, 7: 715-726.

[12] Chen, F., Li, T.-Y., Lu, K.-Y., Muratova, G.-V.: Modified QHSS iteration methods for a class of complex symmetric linear systems. Appl. Numer. Math. 164, 3–14 (2021)

[13] Concus, P., Golub, G.-H.: A generalized conjugate gradient method for nonsymmetric systems of linear equations. In: Computing Methods in Applied Sciences and Engineering, pp. 56–65, Springer, Berlin Heidelberg (1976)

[14] Cui J-J, Huang Z-G, Li B-B, Xie X-F. Single step real-valued iterative method for linear system of equations with complex symmetric matrices. Bull. Korean Math. Soc.. 2023, 60: 1181-1199

[15] Dehghan M, Dehghani-Madiseh M, Hajarian M. A generalized preconditioned MHSS method for a class of complex symmetric linear systems. Math. Model. Anal.. 2013, 18: 561-576.

[16] Van Dijk, W., Toyama, F.-M.: Accurate numerical solutions of the time-dependent Schro¨\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\ddot{{\rm o}}$$\end{document}dinger equation. Phys. Rev. E 75, 036707 (2007)

[17] Feriani A, Perotti F, Simoncini V. Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Eng.. 2000, 190: 1719-1739.

[18] Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical Challenges in Lattice Quantum Chromodynamics. Springer, Berlin (1999)

[19] Greenbaum A. Iterative Methods for Solving Linear Systems. 1997, Philadelphia, Society for Industrial and Applied Mathematics.

[20] Hackbusch, W.: Multi-grid Methods and Applications. Series in Computational Mathematics. Springer, Berlin (2013)

[21] Huang Y-Y, Chen G-L. A relaxed block splitting preconditioner for complex symmetric indefinite linear systems. Open Math.. 2018, 16: 561-573.

[22] Huang Z-G. A new double-step splitting iteration method for certain block two-by-two linear systems. Comput. Appl. Math.. 2020, 39: 1-42.

[23] Huang Z-G. Efficient block splitting iteration methods for solving a class of complex symmetric linear systems. J. Comput. Appl. Math.. 2021, 395. 113574

[24] Huang Z-G. Modified two-step scale-splitting iteration method for solving complex symmetric linear systems. Comput. Appl. Math.. 2021, 40: 122.

[25] Huang, Z.-G., Wang, L.-G., Xu, Z., Cui, J.-J.: Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems. Comput. Math. Appl. 77, 1902–1916 (2019)

[26] Li B-B, Cui J-J, Huang Z-G, Xie X-F. On preconditioned MQHSS iterative method for solving a class of complex symmetric linear systems. Comput. Appl. Math.. 2022, 41250.

[27] Li L, Huang T-Z, Liu X-P. Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Numer. Linear Algebra Appl.. 2007, 14217-235.

[28] Li, X., Yang, A.-L., Wu, Y.-J.: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms 66, 555–568 (2014)

[29] Saad, Y.: Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia (2003)

[30] Shirilord A, Dehghan M. Single step iterative method for linear system of equations with complex symmetric positive semi-definite coefficient matrices. Appl. Math. Comput.. 2022, 426127111

[31] Siahkolaei, T.-S., Salkuyeh, D.-K.: A new double-step method for solving complex Helmholtz equation. Hacet. J. Math. Stat. 49, 1245–1260 (2019)

[32] Wang T, Zheng Q-Q, Lu L-Z. A new iteration method for a class of complex symmetric linear systems. J. Comput. Appl. Math.. 2017, 325: 188-197.

[33] Wesseling, P.: Introduction to Multigrid Methods. Institute for Computer Applications in Science and Engineering, Hampton, Virginia (1995)

[34] Yang A-L, Cao Y, Wu Y-J. Minimum residual Hermitian and skew-Hermitian splitting iteration method for non-Hermitian positive definite linear systems. BIT Numer. Math.. 2019, 59: 299-319.

[35] Zeng M-L. Inexact modified QHSS iteration methods for complex symmetric linear systems of strong skew-Hermitian parts. IAENG Int. J. Appl. Math.. 2021, 51: 109-115

[36] Zhang J-H, Dai H. A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems. Appl. Math. Lett.. 2015, 49: 100-106.

[37] Zhang J-H, Dai H. A new block preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms. 2017, 74: 889-903.

[38] Zhang J-H, Wang Z-W, Zhao J. Double-step scale splitting real-valued iteration method for a class of complex symmetric linear systems. Appl. Math. Comput.. 2019, 353: 338-346

[39] Zhang J-L, Fan H-T, Gu C-Q. An improved block splitting preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms. 2018, 77: 451-478.

[40] Zhang W-H, Yang A-L, Wu Y-J. Minimum residual modified HSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms. 2021, 86: 1543-1559.

[41] Zheng Z, Huang F-L, Peng Y-C. Double-step scale splitting iteration method for a class of complex symmetric linear systems. Appl. Math. Lett.. 2017, 73: 91-97.

Funding

Natural Science Foundation of Guangxi Province(2021JJB110006)

National Natural Science Foundation of China(12361078)

RIGHTS & PERMISSIONS

Shanghai University