A comparison of iterative methods to solve complex valued linear algebraic systems
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Complex valued linear algebraic systems arise in many important applications. We present analytical and extensive numerical comparisons of some available numerical solution methods. It is advocated, in particular for large scale ill-conditioned problems, to rewrite the complex-valued system in real valued form leading to a two-by-two block system of particular form, for which it is shown that a very efficient and robust preconditioned iterative solution method can be constructed. Alternatively, in many cases it turns out that a simple preconditioner in the form of the sum of the real and the imaginary part of the matrix also works well but involves complex arithmetic.
KeywordsLinear systems Complex symmetric Real valued form Preconditioning
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- 2.Kormann, K.: Efficient and reliable simulation of quantum molecular dynamics. Ph.D. Thesis, Uppsala University. http://uu.diva-portal.org/smash/record.jsf?pid=diva2:549981
- 3.Novikov, S., Manakov, S.V., Pitaevskiĭ, L.P., Zakharov, V.E.: Theory of Solitons. The Inverse Scattering Method. Translated from Russian. Contemporary Soviet Mathematics. Consultants Bureau [Plenum], New York (1984)Google Scholar
- 15.Vassilevski, P.S.: Multilevel Block Factorization Preconditioners: Matrix-Based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York (2008)Google Scholar
- 21.Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994)Google Scholar
- 27.Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1994)Google Scholar
- 28.Axelsson, O., Boyanova, P., Kronbichler, M., Neytcheva, M., Wu, X.: Numerical and computational efficiency of solvers for two-phase problems. Comput. Math. Appl. (2012). Published on line at doi: 10.1016/j.camva.2012.05.020
- 29.Kormann, K., Larsson, E.: An RBF-Galerkin Approach to the Time-Dependent Schrödinger Equation. Department of Information Technology, Uppsala University, TR 2012–024 (2012)Google Scholar
- 30.The University of Florida Sparse Matrix Collection, maintained by T. Davis and Y. Hu, http://www.cise.ufl.edu/research/sparse/matrices/
- 31.Notay, Y.: AGMG software and documentation; see http://homepages.ulb.ac.be/~ynotay/AGMG