The main feature of the algorithms presented is that for large ill-conditioned linear systems of equations, the algorithm works with a relative fast convergence speed.

The algorithm is based on a residual minimisation technique [4,5] working with subspaces both for residuals and the original linear spaces. The structure of the parallel algorithm is a master-worker type [6] and the algorithm has some genetic features [7].

Computer tests have confirmed the theoretical results. The parallel implementation realized shows much better efficiency than any sequential one. Considerable speed-up effects were obtained using a parallel computer. The computer tests were obtained both in parallel and cluster computing environments.

The goal of creating these algorithms is to provide an efficient parallel method for general large ill-conditioned linear systems of equations. Therefore it is a useful method for several practical and realistic problems such as optimization, finite element methods, control or simulation problems etc. even if it leads to generally large, dense ill-conditioned linear systems of equations.

We have to note that the parallel algorithm proposed can easily be adjusted to the large scale of different parallel architectures, such as distributed and heterogeneous clusters or supercomputers. The test with more effective algorithms and new topological architectures will be the subject of forthcoming work.

1

L.A. Hageman, D.M. Joung, "Applied Iterative Methods", Computer Science and Applied Mathematics, Academic Press, 1981.

2

P.G. Ciarlet, "Introduction á l'analyse numérique matricielle et à l'optimisation", Masson, Paris, 1982.

3

G. Golub, A. Greenbaum, M. Luskin, (Editors), "Recent Advances in Iterative Methods", The IMA Volumes in Math. and its Applications, 60., Springer Verlag, 1994.

4

G. Molnárka, N. Varjasi, "Parallel algorithm for solution of general linear systems of equations", Informatika a felsooktatásban, 2005.

5

G. Molnárka, "A scalable parallel algorithm for solving general linear system equations", 77th GAMM annual meeting 2006, Berlin, 2006.

6

N. Varjasi, "Parallel Algorithm for linear equations with different network topologies", Proceedings of International e-Conference on Computer Science (IeCCS) 2006, Lecture Series on Computer and Computational Sciences, Brill Academic Publishers, 502-505, 2007.

7

G. Molnárka, N. Varjasi, "A simultaneous solution for general linear equations ona ring or hierarchical cluster", Acta Technica Jaurinensis, 3(1), 65-73, 2010.

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