Abstract
Complex valued systems with an indefinite matrix term arise in important applications such as for certain time-harmonic partial differential equations such as the Maxwell’s equation and for the Helmholtz equation. Complex systems with symmetric positive definite matrices can be solved readily by rewriting the complex matrix system in two-by-two block matrix form with real matrices which can be efficiently solved by iteration using the preconditioned square block (PRESB) preconditioning method and preferably accelerated by the Chebyshev method. The appearances of an indefinite matrix term causes however some difficulties. To handle this we propose different forms of matrix splitting methods, with or without any parameters involved. A matrix spectral analyses is presented followed by extensive numerical comparisons of various forms of the methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Axelsson, O.: Iterative solution methods. Cambridge University Press, Cambridge (1994)
Axelsson, O., Vassilevski, P.S.: A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning. SIAM J. Matrix Anal. Appl. 12, 625–644 (1991)
Axelsson, O., Farouq, S., Neytcheva, M.: Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems, Poisson and convection-diffusion control. Numer. Alg. 73, 631–663 (2016)
Axelsson, O., Liang, Z.-Z.: Parameter modified versions of preconditioning and iterative inner product free refinement methods for two-by-two block matrices. Lin. Algebra Appl. 582, 403–429 (2019)
Axelsson, O., Karátson, J., Magoulès, F.: Superlinear convergence using block preconditioners for the real system formulation of complex Helmholtz equations. J. Comput. Appl. Math. 340, 424–431 (2018)
Axelsson, O., Neytcheva, M., Ström, A.: An efficient preconditioning method for the state box-constrained optimal control problem. J. Num. Math. 26, 185–207 (2018)
Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms 66, 811–841 (2014)
Axelsson, O., Lukáš, D.: Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems. J. Numer. Math. 27, 1–21 (2019)
Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)
Axelsson, O., Salkuyeh, D.K.: A new version of a preconditioning method for certain two-by-two block matrices with square blocks. BIT Numer. Math. 59, 321–342 (2018)
Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)
Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)
Edalatpour, V., Hezari, D., Salkuyeh, D.K.: Two efficient inexact algorithms for a class of large sparse complex linear systems. Mediterr. J. Math. 13, 2301–2318 (2016)
Edalatpour, V., Hezari, D., Salkuyeh, D.K.: Accelerated generalized SOR method for a class of complex systems of linear equations. Math. Commun. 20, 37–52 (2015)
Hezari, D., Salkuyeh, D.K., Edalatpour, V.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22, 761–776 (2015)
Hezari, D., Salkuyeh, D.K., Edalatpour, V.: A new iterative method for solving a class of complex symmetric system of linear equations. Numer. Algorithms 73, 927–955 (2016)
Kollmann, M., Kolmbauer, M.: A preconditioned MinRes solver for time-periodic parabolic optimal control problems. Numer. Lin. Algebra Appl. 20, 761–784 (2013)
Liang, Z.-Z., Axelsson, O., Zhang, G.-F.: Efficient iterative solvers for a complex valued two-by-two block linear system with application to parabolic optimal control problems. Appl. Numer. Math. 152, 422–445 (2020)
Notay, Y.: An aggregation-based algebraic multigrid method. Electron. Trans. Numer. Anal. 37, 123–146 (2010)
Pourbagher, M., Salkuyeh, D.K.: A new two-parameter iteration method for indefinite complex symmetric linear systems. Japan J. Indust. Appl. Math. 39, 145-163 (2022)
Pourbagher, M., Salkuyeh, D.K.: On the solution of a class of complex symmetric linear systems. Appl. Math. Lett. 76, 14–20 (2017)
Salkuyeh, D.K.: Two-step scale-splitting method for solving complex symmetric system of linear equations, math. NA. (2017) arXiv:1705.02468
Salkuyeh, D.K., Hezari, D., Edalatpour, V.: Generalized SOR iterative method for a class of complex symmetric linear system of equations, Intern. J. Comput. Math. 92, 802–815 (2015)
Salkuyeh, D.K., Siahkolaei, T.S.: Two-parameter TSCSP method for solving complex symmetric system of linear equations. Calcolo 55, 8 (2018)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)
Siahkolaei, T.S., Salkuyeh, D.K.: A new double-step method for solving complex Helmholtz equation, Hacet. J. Math. Stat. 49, 1245–1260 (2020)
Wu, S.-L., Li, C.-X.: A splitting method for complex symmetric indefinite linear system. J. Comput. Appl. Math. 313, 343–354 (2017)
Yousept, I.: Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. 52, 559–581 (2012)
Acknowledgements
The authors would like to thank the referee for his/her useful suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Axelsson, O., Pourbagher, M. & Salkuyeh, D.K. Efficient iteration methods for complex systems with an indefinite matrix term. Calcolo 59, 15 (2022). https://doi.org/10.1007/s10092-022-00461-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10092-022-00461-w