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Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm

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Abstract

In this paper we consider algorithms to compute bounds of the A-norm of the error in the preconditioned conjugate gradient (PCG) algorithm. We extend to PCG formulas that were given in an earlier paper [8]. We give numerical experiments which show that good upper and lower bounds can be obtained provided estimates of the lowest and largest eigenvalues of the preconditioned matrix are given or adaptively computed.

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References

  1. Z. Bai and G.H. Golub, Bounds for the trace of the inverse and the determinant of symmetric positive definite matrices, Ann. Numer. Math. 4 (1997) 29–38.

    MATH  MathSciNet  Google Scholar 

  2. M. Benzi, C.D. Meyer and M. Tuma, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput. 17 (1996) 1135–1149.

    Article  MATH  MathSciNet  Google Scholar 

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

  4. G.H. Golub and G. Meurant, Matrices, moments and quadrature, in: Numerical Analysis 1993, eds. D.F. Griffiths and G.A. Watson, Pitman Research Notes in Mathematics, Vol. 303 (1994) pp. 105–156.

  5. G.H. Golub and G. Meurant, Matrices, moments and quadrature II or how to compute the norm of the error in iterative methods, BIT 37(3) (1997) 687–705.

    MATH  MathSciNet  Google Scholar 

  6. G.H. Golub and Z. Strakoš, Estimates in quadratic formulas, Numer. Algorithms 8(2–4) (1994).

  7. G.H. Golub and C. Van Loan, Matrix Computations (Johns Hopkins Univ. Press, Baltimore, MD, 1989).

    Google Scholar 

  8. G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm, Numer. Algorithms 16 (1997) 77–87.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Meurant, Computer Solution of Large Linear Systems (North-Holland, Amsterdam, 1999).

    MATH  Google Scholar 

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  1. CEA/DIF, BP 12, F-91680, Bruyères le Châtel, France E-mail:

    Gérard Meurant

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  1. Gérard Meurant
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Meurant, G. Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm. Numerical Algorithms 22, 353–365 (1999). https://doi.org/10.1023/A:1019179412560

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  • DOI: https://doi.org/10.1023/A:1019179412560

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