On the Convergence of Some Generalized Preconditioned Iterative Methods

Abstract

Next article On the Convergence of Some Generalized Preconditioned Iterative MethodsNikolaos M. Missirlis and David J. EvansNikolaos M. Missirlis and David J. Evanshttps://doi.org/10.1137/0718037PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis paper generalizes the preconditioning techniques introduced by Evans [2] for the numerical solution of the linear system $Au = b$ and defines two extrapolated versions of the Gauss–Siedel method as well as an extrapolated version of the successive overrelaxation method. Finally, it establishes some convergence theorems for the considered iterative schemes when A has particular properties such as being consistently ordered, positive definite having weak diagonal dominance, etc...[1] D. J. Evans, The extrapolated modified Aitken iteration method for solving elliptic difference equations, Comput. J., 6 (1963/1964), 193–201 27:2128 0119.33404 CrossrefISIGoogle Scholar[2] D. J. Evans, The use of preconditioning in iterative methods for solving linear equations with symmetric positive definite matrices, J. Inst. Math. Appl., 4 (1968), 295–314 38:4012 CrossrefGoogle Scholar[3] D. J. Evans, Comparison of the convergence rates of iterative methods for solving linear equations with preconditioning, Proceedings of the C. Carathéodory International Symposium (Athens, 1973), Greek Math. Soc., Athens, 1974, 106–135 57:14372 0357.65026 Google Scholar[4] D. J. Evans and , N. M. Missirlis, The modified alternating direction preconditioning method for the numerical solution of the elliptic self-adjoint second order and biharmonic equations, BIT, 19 (1979), 172–185 80h:65080 0419.65065 CrossrefGoogle Scholar[5] George E. Forsythe and , Wolfgang R. Wasow, Finite Difference Methods For Partial Differential Equations, Applied Mathematics Series, John Wiley & Sons Inc., New York, 1960x+444 23:B3156 0099.11103 Google Scholar[6] Apostolos Hadjidimos, Accelerated overrelaxation method, Math. Comp., 32 (1978), 149–157 58:3353 0382.65015 CrossrefISIGoogle Scholar[7] N. M. Missirlis, Ph.D. Thesis, Preconditioned iterative methods for solving elliptic partial differential equations, Loughborough University, Department of Computer Studies, 1978 Google Scholar[8] Richard S. Varga, A comparison of the successive overrelaxation method and semi-iterative methods using Chebyshev polynomials, J. Soc. Indust. Appl. Math., 5 (1957), 39–46 19,772d 0080.10701 LinkISIGoogle Scholar[9] Richard S. Varga, Matrix Iterative Analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1962xiii+322 28:1725 Google Scholar[10] David M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971xxiv+570, London 46:4698 0231.65034 Google Scholar Next article FiguresRelatedReferencesCited byDetails A comparison of the Extrapolated Successive Overrelaxation and the Preconditioned Simultaneous Displacement methods for augmented linear systems14 January 2015 | Numerische Mathematik, Vol. 131, No. 3 Cross Ref Is modified PSD equivalent to modified SOR for two-cyclic matrices?Linear Algebra and its Applications, Vol. 432, No. 11 Cross Ref Optimal Parameters for 2-cyclic AORApplied Mathematics and Computation, Vol. 216, No. 5 Cross Ref The impact of eigenvalue locality on the convergence behavior of the PSD method for two-cyclic matricesLinear Algebra and its Applications, Vol. 430, No. 8-9 Cross Ref Solution of the fully fuzzy linear systems using iterative techniquesChaos, Solitons & Fractals, Vol. 34, No. 2 Cross Ref Iterative solution of fuzzy linear systemsApplied Mathematics and Computation, Vol. 175, No. 1 Cross Ref Comparison results between Jacobi and other iterative methodsJournal of Computational and Applied Mathematics, Vol. 169, No. 1 Cross Ref Relationship of eigenvalues for USAOR iterative method applied to a class of p-cyclic matricesLinear Algebra and its Applications, Vol. 362 Cross Ref Iterative schemes for the neutron diffusion equationComputers & Mathematics with Applications, Vol. 44, No. 10-11 Cross Ref Semiconvergence of extrapolated iterative methods for singular linear systemsJournal of Computational and Applied Mathematics, Vol. 106, No. 1 Cross Ref The extrapolated ω-double jacobi (Eω-DOJ) method19 March 2007 | International Journal of Computer Mathematics, Vol. 70, No. 2 Cross Ref Error Bounds for some overrelaxation methods19 March 2007 | International Journal of Computer Mathematics, Vol. 70, No. 2 Cross Ref Stein-rosenberg type theorems for the ssor and ussor, psd and mpsd methodsInternational Journal of Computer Mathematics, Vol. 64, No. 1-2 Cross Ref Extensions of the symmetric successive overrelaxation theory ∗20 March 2007 | International Journal of Computer Mathematics, Vol. 55, No. 3-4 Cross Ref Extensions of the Ostrowski–Reich theoremLinear Algebra and its Applications, Vol. 207 Cross Ref Convergence domains for the symmetric successive overrelaxation methodInternational Journal of Computer Mathematics, Vol. 52, No. 3-4 Cross Ref Construction and efficient implementation of implicit preconditioning methods. 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