BIT Numerical Mathematics

, Volume 37, Issue 3, pp 687–705 | Cite as

Matrices, moments and quadrature II; How to compute the norm of the error in iterative methods

  • G. H. Golub
  • G. Meurant
Article

Abstract

In this paper, we study the numerical computation of the errors in linear systems when using iterative methods. This is done by using methods to obtain bounds or approximations of quadratic formsuTA−1u whereA is a symmetric positive definite matrix andu is a given vector. Numerical examples are given for the Gauss-Seidel algorithm.

Moreover, we show that using a formula for theA-norm of the error from Dahlquist, Golub and Nash [1978] very good bounds of the error can be computed almost for free during the iterations of the conjugate gradient method leading to a reliable stopping criterion.

AMS subject classification

65F50 

Key words

Iterative methods error computation conjugate gradient 

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References

  1. 1.
    G. Dahlquist, S. C. Eisenstant, and G. H. Golub,Bounds for the error of linear systems of equations using the theory of moments, J. Math. Anal. Appl., 37 (1972), pp. 151–166.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Dahlquist, G. H. Golub, and S. G. Nash,Bounds for the error in linear systems, in Proceedings of the Workshop on Semi-Infinite Programming, R. Hettich, ed., Springer, 1978, pp. 154–172.Google Scholar
  3. 3.
    B. Fischer and G. H. Golub,On the error computation for polynomial based iteration methods, Tech. Report NA 92-21, Stanford University, 1992.Google Scholar
  4. 4.
    G. H. Golub,Matrix computation and the theory of moments, in Proceedings of the International Congress of Mathematicians, Birkhäuser, 1995.Google Scholar
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    G. H. Golub and G. Meurant,Matrices moments and quadrature, in Numerical Analysis 1993, D. F. Griffiths and G. A. Watson, eds., Pitman Research Notes in Mathematics, v 303, 1994, pp. 105–156.Google Scholar
  6. 6.
    G. H. Golub and Z. Strakoš,Estimates in quadratic formulas, Tech. Report SCCM-93-08, Stanford University, to appear in Numerical Algorithms.Google Scholar

Copyright information

© BIT Foundation 1997

Authors and Affiliations

  • G. H. Golub
    • 1
  • G. Meurant
    • 2
  1. 1.Stanford UniversityComputer Science DepartmentStanfordUSA
  2. 2.CEA/Limeil-ValentonSt Georges CedexFrance

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