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Verified Error Bounds for Linear Systems Through the Lanczos Process

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Reliable Computing

Abstract

We use verified computations and the Lanczos process to obtain guaranteed lower and upper bounds on the 2-norm and the energy-norm error of an approximate solution to a symmetric positive definite linear system. The upper bounds require the a priori knowledge of a lower bound on the smallest eigenvalue.

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References

  1. Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, 1983.

  2. Alefeld, G. and Mayer, G.: The Cholesky Method for Interval Data, Linear Algebra Appl. 194 (1993), pp. 161–182.

    Google Scholar 

  3. Barth, W. and Nuding, E.: Optimale Lösung von Intervallgleichungssystemen, Computing 12 (1974), pp. 117–125.

    Google Scholar 

  4. Frommer, A.: Lösung linearer Gleichungssysteme auf Parallelrechnern. Vieweg-Verlag, 1990.

  5. Frommer, A. and Maass, P.: Fast CG-Based Methods for Thikonov-Phillips Regularization, SIAM J. Sc. Comp., to appear, also available at http://www.math.uniwuppertal.de/SciComp/Preprints.html as preprint BUGHW-SC 96/10.

  6. Golub, G. and Meurant, G.: Matrices, Moments and Quadrature, in: Numerical Analysis 1993 (Dundee 1993) (Pitman Res. Notes Math. Ser. 302), Longman Sci. Tech., Harlow, 1994.

    Google Scholar 

  7. Golub, G. and Meurant, G.: Matrices, Moments and Quadrature II. How to Compute the Error in Iterative Methods, BIT 37 (1997), pp. 687–705.

    Google Scholar 

  8. Hansen, P.: Regularization Tool, a MATLAB Package for the Analysis and Solution of Discrete Ill-Posed Problems; Version 2.0 for MATLAB 4.0, Tech.Rep. UNIC, 92-03, 1993.

  9. Klatte, R., Kulisch, U., Neaga, M., Ratz, D., and Ullrich, Ch.: PASCAL-XSC Sprachbeschreibung mit Beispielen, Springer-Verlag, 1991.

  10. Kulisch, U. and Miranker, W.: The Arithmetic of the Digital Computer, SIAM Rev. 28 (1986).

  11. Matrix Market, http://math.nist.gov/MatrixMarket/.

  12. Neumaier, A.: Interval Methods for Linear Systems of Equations, Cambridge University Press, 1990.

  13. Rump, S.: Vertification Methods for Dense and Sparse Systems of Equations, in: Herzberger, J. (ed.), Topics in Validated Computations, North-Holland, 1993.

  14. Saad, Y.: Iterative Methods for Sparse Linear Systems, PWS, Boston, 1996.

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Bergische Universität Wuppertal, D-42097, Wuppertal, Germany

    Andreas Frommer & Andre Weinberg

Authors
  1. Andreas Frommer
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  2. Andre Weinberg
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Frommer, A., Weinberg, A. Verified Error Bounds for Linear Systems Through the Lanczos Process. Reliable Computing 5, 255–267 (1999). https://doi.org/10.1023/A:1009976221447

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  • DOI: https://doi.org/10.1023/A:1009976221447

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