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A Stochastic Approach to Error Estimates for Iterative Linear Solvers: Part 1

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Abstract

We consider the problem of estimating the magnitude of the error of an iterative linear solver after k iterations. Assuming that the initial error can be described using a probability distribution we derive L2-estimates for the magnitude of the error in the average case. In Part 1 the ideas are presented and applied to a simple splitting method, while Part 2 extends the same ideas to the conjugate gradient method.

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REFERENCES

  • 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), pp. 29–38.

    Google Scholar 

  • P. Concus, G. H. Golub, and D. P. O'Leary, A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, J. R. Bunch and D. J. Rose, eds., Academic Press, New York, 1976, pp. 309–332.

    Google Scholar 

  • P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1984.

    Google Scholar 

  • G. H. Golub, Bounds for the round-off errors in the Richardson second order method, BIT, 2:4 (1962), pp. 212–223.

    Google Scholar 

  • G. H. Golub and G. Meurant, Matrices, moments and quadrature, in Numerical Analysis 1993, Proceedings of the 15th Dundee Conference, D. F. Griffiths and G. A. Watson, eds., Pitman Res. Notes Math. Ser., Vol. 303, Longman Scientific & Technical, Harlow, Essex, 1994.

    Google Scholar 

  • G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.

    Google Scholar 

  • N. L. Johnson and S. Kotz, Encyclopedia of Statistical Sciences, Vol. 2, John Wiley, New York, 1982, pp. 386–387.

    Google Scholar 

  • H. Melbø, Preconditioning and Error Estimates for Iterative Linear Solvers, Masters thesis, Department of Mathematics, NTNU, Trondheim, Norway.

  • B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998.

    Google Scholar 

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Authors and Affiliations

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

    G. H. Golub

  2. Department of Mathematics, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    H. Melbø

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  1. G. H. Golub
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  2. H. Melbø
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Golub, G.H., Melbø, H. A Stochastic Approach to Error Estimates for Iterative Linear Solvers: Part 1. BIT Numerical Mathematics 41, 977–985 (2001). https://doi.org/10.1023/A:1021985111111

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