Skip to main content
SpringerLink
Log in

A wider convergence area for the MSTMAOR iteration methods for LCP

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In order to solve large sparse linear complementarity problems on parallel multiprocessor systems, modulus-based synchronous two-stage multisplitting iteration methods based on two-stage multisplittings of the system matrices were constructed and investigated by Bai and Zhang (Numer. Algoritm. 62, 59-77 2013). These iteration methods include the multisplitting relaxation methods such as Jacobi, Gauss-Seidel, SOR and AOR of the modulus type as special cases. In the same paper the convergence theory of these methods is developed, under the following assumptions: (i) the system matrix is an H +-matrix and (ii) one acceleration parameter is greater than the other. Here we show that the second assumption can be avoided, thus enabling us to obtain an improved convergence area. The result is obtained using the similar technique proposed by Cvetković and Kostić (Numer. Linear Algebra Appl. 21, 534-539 2014), and its usage is demonstrated by an example of the LCP.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z.-Z., Evans, D.J.: Matrix multisplitting methods with applications to linear complementarity problems: Parallel synchronous and chaotic methods. Reseaux et systemes repartis Calculateurs Paralleles 13, 125–154 (2001)

    Google Scholar 

  3. Bai, Z.-Z., Evans, D.J.: Matrix multisplitting methods with applications to linear complementarity problems: Parallel asynchronous methods. Int. J. Comput. Math. 79, 205–232 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z., Sun, J.-C., Wang, D.-R.: A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations. Comput. Math. Appl. 32, 51–76 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425–439 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algoritm. 62, 59–77 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences, vol. 9. SIAM, Philadelphia (1994)

    Book  Google Scholar 

  8. Cvetković, Lj., Kostić, V.: A note on the convergence of the MSMAOR method for linear complementarity problems. Numer. Linear Algebra Appl. 21, 534–539 (2014)

  9. Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra Appl. 16, 129–143 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hadjidimos, A., Tzoumas, M.: The principle of extrapolation and the Cayley transform. Linear Algebra Appl. 429, 2465–2480 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hadjidimos, A., Tzoumas, M.: Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem. Linear Algebra Appl. 431, 197–210 (2008)

    Article  MathSciNet  Google Scholar 

  12. Leenaerts, D.M.W., van Bokhoven, W.M.G.: Piecewise Linear Modelling and Analysis. Kluwer Academic, Dordrecht (1998)

    Book  Google Scholar 

  13. Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann, Berlin (1988)

    MATH  Google Scholar 

  14. O’Leary, D.P., White, R.E.: Multi-splittings of matrices and parallel solution of linear systems. SIAM J Algebraic Discret. Methods 6, 630–640 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Novi Sad, Serbia

    Ljiljana Cvetković, Vladimir Kostić & Ernest Šanca

Authors
  1. Ljiljana Cvetković
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Kostić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ernest Šanca
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ernest Šanca.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cvetković, L., Kostić, V. & Šanca, E. A wider convergence area for the MSTMAOR iteration methods for LCP. Numer Algor 71, 77–88 (2016). https://doi.org/10.1007/s11075-015-9985-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-015-9985-6

Keywords

Search

Navigation