Skip to main content
Log in

On the theory and practice of multisplitting methods in parallel computation

Zur Theorie und Praxis von Multisplitting-Verfahren im Parallelen Rechnen

  • Published:
Computing Aims and scope Submit manuscript

Abstract

We present convergence and comparison theorems on parallel iterative multisplitting methods with different weighting schemes. In particular, we show that certain Gauss-Seidel multisplittings cannot converge faster than the usual Gauss-Seidel method. We also give numerical results on a 64 processor local memory computer. These experiments show that the ‘naive’ use of multisplittings can easily produce unsatisfactory results on parallel computers with more than just a few processors.

Zusammenfassung

Wir geben Konvergenz-und Vergleichsaussagen für parallele iterative Multisplitting-Verfahren mit verschiedenen Gewichtungsschemata. Insbesondere zeigen wir, daß bestimmte Gauss-Seidel Multisplitting-Verfahren nicht schneller konvergieren können als das gewöhnliche Gauss-Seidel-Verfahren. Wir berichten darüber hinaus über numerische Experimente auf einem 64-Prozessor-Rechner mit lokalem Speicher. Diese Experimente zeigen, daß der ‘naive’ Einsatz von Multisplittings leicht zu nicht zufriedenstellenden Ergebnissen führen kann, wenn mehr also nur ein paar Prozessoren eingesetzt werden.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Adams, L., Ong, E.: Additive polynomial preconditioners for parallel computing. Parallel Comput.9, 333–345 (1988/89).

    Google Scholar 

  2. Berman, A., Plemmons, R.: Nonnegative matrices in the mathematical sciences. New York: Academic Press 1979.

    Google Scholar 

  3. Block, U., Frommer, A., Mayer, G.: Block colouring schemes for the SOR method on local memory parallel computers. Parallel Comput.14, 61–75 (1990).

    Google Scholar 

  4. Brochard, L.: Efficiency of some parallel numerical algorithms on distributed systems. Parallel Comput.12, 21–44 (1989).

    Google Scholar 

  5. Bru, R., Elsner, L., Neumann, M.: Models of parallel chaotic relaxation method. Linear Algebra Appl.103, 175–192 (1988).

    Google Scholar 

  6. Elsner, L.: Comparisons of weak regular splittings and multisplitting methods. Numer. Math.56, 283–289 (1989).

    Google Scholar 

  7. Fan, Ky: Note onM-matrices. Quart. J. Math.11, 43–49 (1960).

    Google Scholar 

  8. Frommer, A.: Lösung linearer Gleichungssysteme auf Parallelrechnern. Wiesbaden: Vieweg 1990.

    Google Scholar 

  9. Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl.119, 141–152 (1989).

    Google Scholar 

  10. Frommer, A., Mayer, G.: Parallel interval multisplittings. Numer. Math.56, 255–267 (1989).

    Google Scholar 

  11. Frommer, A., Mayer, G.: Theoretische und praktische Ergebnisse zu Multisplitting-Verfahren auf Parallelrechnern. ZAMM70, T600-T602 (1990).

    Google Scholar 

  12. Gentzsch, W.: Programming tree computers. Speedup3, 6–10 (1989).

    Google Scholar 

  13. Gentzsch, W., Block, U.: Numerische Verfahren für den Parallelrechner TX3. ZAMM69, T176-T179 (1989).

    Google Scholar 

  14. Neumann, M., Plemmons, R.: Convergence of parallel multisplitting iterative methods forM-matrices. Linear Algebra Appl.88/89, 559–573 (1987).

    Google Scholar 

  15. Ortega, J.: Introduction to parallel and vector solution of linear systems. New York: Plenum 1988.

    Google Scholar 

  16. Ortega, J., Rheinboldt, W.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.

    Google Scholar 

  17. O'Leary, D., White, R.: Multi-splittings of matrices and parallel solution of linear systems. SIAM J. Alg. Disc. Meth.6, 630–640 (1985).

    Google Scholar 

  18. Papatheodorou, T., Saridakis, Y.: Parallel algorithms and architectures for multisplitting iterative methods. Parallel Comput.12, 171–182 (1989).

    Google Scholar 

  19. Varga, R.: Factorization and normalized iterative methods. In: Langer, R. (ed.) Boundary value problems in differential equations. Madison: The University of Wisconsin Press 1960.

    Google Scholar 

  20. Varga, R.: Matrix iterative analysis. Englewood Cliffs: Prentice Hall 1962.

    Google Scholar 

  21. Wang, D.: On the convergence of the parallel multisplitting AOR algorithm. Linear Algebra Appl.154–156, 473–486 (1991).

    Google Scholar 

  22. White, R.: Multisplitting with different weighting schemes. SIAM J. Matrix Anal. Appl.10, 481–493 (1989).

    Google Scholar 

  23. White, R.: Multisplittings of a symmetric positive definite matrix. SIAM J. Matrix Anal. Appl.11, 69–82 (1990).

    Google Scholar 

  24. Wöst, W.: Wie funktioniert der TX3?, c't Magazin, Heft6, 134–146 (1988).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Frommer, A., Mayer, G. On the theory and practice of multisplitting methods in parallel computation. Computing 49, 63–74 (1992). https://doi.org/10.1007/BF02238650

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02238650

AMS (MOS) Subject Classifications

Key words

Navigation