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On the preconditioned Jacobi method for solving large linear systems

Zur vorkonditionierten Jacobi-Methode zur Lösung großer linearer Systeme

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Abstract

This paper is concerned with the application of preconditioning techniques to the well known Jacobi iterative method for solving the finite difference equations derived from the discretization of self-adjoint elliptic partial differential equations. The convergence properties of this one parameter preconditioned method are analyzed and the value of the optimum preconditioning parameter and the performance of the method determined for a variety of standard problems.

Zusammenfassung

Die Arbeit behandelt die Anwendung von Vorkonditionierungstechniken auf das bekannte Iterationsverfahren von Jacobi zur Lösung der Differenzengleichungen, die bei der Diskretisierung selbstadjungierter elliptischer partieller Differentialgleichungen entstehen. Die Konvergenzeigenschaften dieses einparametrigen Vorkonditionnierungsverfahrens werden untersucht, die Werte des optimalen Parameters und das Verhalten der Methode werden für verschiedene Standardprobleme bestimmt.

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References

  1. Evans, D. J.: The use of preconditioning in iterative methods for solving linear equations with symmetric positive definite matrices. J.I.M.A.4, 295–314 (1968).

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  2. Young, D. M.: Iterative solution of large linear systems. New York-London: Academic Press 1971.

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  3. Missirlis, N. M.: Preconditioned iterative methods for solving elliptic partial differential equations. Ph. D. Thesis, Loughborough University of Technology, 1978.

  4. Evans, D. J., Missirlis, N. M.: The preconditioned Jacobi method for solving elliptic difference equations. T. R. 160, Computer Studies Dept. Loughborough University, Loughborough, 1980.

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Author notes

  1. N. M. Missirlis

    Present address: Unit of Applied Mathematics, University of Athens, Athens, Greece

Authors and Affiliations

  1. Department of Computer Science, Loughborough University of Technology, LE 11 3 TU, Loughborough, Leicestershire, UK

    D. J. Evans & N. M. Missirlis

Authors
  1. D. J. Evans
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  2. N. M. Missirlis
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Evans, D.J., Missirlis, N.M. On the preconditioned Jacobi method for solving large linear systems. Computing 29, 167–173 (1982). https://doi.org/10.1007/BF02249939

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  • DOI: https://doi.org/10.1007/BF02249939

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