Skip to main content
SpringerLink
Log in

Computation of Sharp Rigorous Componentwise Error Bounds for the Approximate Solutions of Systems of Linear Equations

  • Published:
Reliable Computing

Abstract

This paper is concerned with the problem of verifying the accuracy of approximate solutions of systems of linear equations. Recently, fast algorithms for calculating guaranteed error bounds of computed solutions of systems of linear equations have been proposed using the rounding mode controlled verification method and the residual iterative verification method. In this paper, a new verification method for systems of linear equations is proposed. Using this verification method, componentwise verified error bounds of approximate solutions of systems of linear equations can be calculated. Numerical results are presented to illustrate that it is possible to get very sharp error bounds of computed solutions of systems of linear equations whose coefficient matrices are symmetric and positive definite.

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. Golub, G. H. and Van Loan, C. F.: Matrix Computations, Third ed., The Johns Hopkins University Press, Baltimore and London, 1996.

    Google Scholar 

  2. Ogita, T., Oishi, S., and Ushiro, Y.: Fast Inclusion and Residual Iteration for Solutions of Matrix Equations, Computing Supplement 16 (2002), pp. 171-184.

    Google Scholar 

  3. Ogita, T., Oishi, S., and Ushiro, Y.: Fast Verification of Solutions for Sparse Monotone Matrix Equations, Computing Supplement 15 (2001), pp. 175-187.

    Google Scholar 

  4. Oishi, S.: Fast Enclosure of Matrix Eigenvalues and Singular Values via Rounding Mode Controlled Computation, Linear Alg. Appl. 324 (2001), pp. 133-146.

    Google Scholar 

  5. Oishi, S. and Rump, S. M.: Fast Verification of Solutions of Matrix Equations, Numer. Math. 90 (2002), pp. 755-773.

    Google Scholar 

  6. Rump, S. M.: Verified Computation of the Solution of Large Sparse Systems, ZAMM 75 (1995), pp. S439-S442.

    Google Scholar 

  7. Stewart, G. W. and Sun, J.-G.: Matrix Perturbation Theory, Academic Press, New York, 1990.

    Google Scholar 

  8. Sun, J.-G.: Rounding-Error and Perturbation Bounds for the Cholesky and LDLT Factorizations, Linear Alg. Appl. 173 (1992), pp. 77-97.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Education, Waseda University, Tokyo, 169-8050, Japan

    Takeshi Ogita & Shin'ichi Oishi

  2. Enterprise Server Division, Hitachi Ltd., Kanagawa, 243-0435, Japan

    Yasunori Ushiro

Authors
  1. Takeshi Ogita
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shin'ichi Oishi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yasunori Ushiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ogita, T., Oishi, S. & Ushiro, Y. Computation of Sharp Rigorous Componentwise Error Bounds for the Approximate Solutions of Systems of Linear Equations. Reliable Computing 9, 229–239 (2003). https://doi.org/10.1023/A:1024655416554

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024655416554

Keywords

Search

Navigation