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

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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 formsu T A −1 u 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.

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References

  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.

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  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.

  3. B. Fischer and G. H. Golub,On the error computation for polynomial based iteration methods, Tech. Report NA 92-21, Stanford University, 1992.

  4. G. H. Golub,Matrix computation and the theory of moments, in Proceedings of the International Congress of Mathematicians, Birkhäuser, 1995.

  5. 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.

  6. G. H. Golub and Z. Strakoš,Estimates in quadratic formulas, Tech. Report SCCM-93-08, Stanford University, to appear in Numerical Algorithms.

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Authors and Affiliations

  1. Stanford University, Computer Science Department, 94305, Stanford, CA, USA

    G. H. Golub

  2. CEA/Limeil-Valenton, Villeneuve, F-94195, St Georges Cedex, France

    G. Meurant

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  1. G. H. Golub
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  2. G. Meurant
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Additional information

The work of the first author was partially supported by NSF Grant CCR-950539.

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Golub, G.H., Meurant, G. Matrices, moments and quadrature II; How to compute the norm of the error in iterative methods. Bit Numer Math 37, 687–705 (1997). https://doi.org/10.1007/BF02510247

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  • DOI: https://doi.org/10.1007/BF02510247

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