Skip to main content
Log in

Factorised Preconditionings of Successive Approximations in Finite Precision

BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

This paper examines the possibility of using the method of successive approximations for the approximate solution of a large system of linear equations with a dense, noncontraction, and ill-conditioned matrix. Using Krasnosel'skii's method of transformation of linear operator equations and functional calculi, the procedure of factorised preconditionings of successive approximations is developed and analysed in the finite precision arithmetic. Numerical results of computational experiments are presented to demonstrate the practicability of the proposed approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. V. V. Arestov, Approximation of unbounded operators by bounded operators and related extremal problems, Russian Math. Surveys, 51 (1996), pp. 1093-1126.

    Google Scholar 

  2. V. Ya. Arsenin, Yu. A. Kriksin, and A. A. Timonov, On a spectral approach to the construction of local regularization algorithms, Soviet Math. Dokl., 39 (1989), pp. 86-90.

    Google Scholar 

  3. S. Ashby, T. Manteuffel, and J. Otto, A comparison of adaptive Chebyshev and least squares polynomial preconditioning for Hermitian positive definite linear systems, SIAM J. Sci. Stat. Comput., 13 (1992), pp. 1-29.

    Article  Google Scholar 

  4. O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.

    Google Scholar 

  5. A. B. Bakushinsky, A general method of constructing regularizing algorithms for a linear ill-posed equation in a Hilbert space, U.S.S.R. Comput. Maths. Math. Phys., 7 (1967), pp. 279-288.

    Article  Google Scholar 

  6. A. B. Bakushinsky and A. V. Goncharsky, Ill-Posed Problems: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1994.

    Google Scholar 

  7. R. Barrett, M. Berry, T. Chan, et al., Templates for the Solution of Linear Systems, SIAM, Philadelphia, PA, 1994.

    Google Scholar 

  8. F. Chaitin-Chatelin and S. Gratton, Convergence in finite precision of successive iteration methods under high nonnormality, BIT, 36 (1996), pp. 455-469.

    Article  Google Scholar 

  9. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.

    Google Scholar 

  10. J. Cullum, Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problems, BIT, 36 (1996), pp. 470-493.

    Article  Google Scholar 

  11. K. Davey and S. Bounds, A generalized SOR method for dense linear systems of boundary element equations, SIAM J. Sci. Comput., 19 (1998), pp. 953-967.

    Article  Google Scholar 

  12. R. W. Freund, G. H. Golub, and N. N. Nachtigal, Iterative solution of linear systems, Acta Numerica, (1991), pp. 57-100.

  13. V. M. Fridman, The method of successive approximations for a Fredholm integral equation of the first kind, Uspekhi Mat. Nauk, 11 (1956), pp. 233-254 (in Russian).

    Google Scholar 

  14. A. Frommer and P. Maas, Fast CG-based methods for Tikhonov-Phillips regularisation, SIAM J. Sci. Comput., 20 (1999), pp. 1831-1850.

    Article  Google Scholar 

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

    Google Scholar 

  16. A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, PA, 1997.

    Google Scholar 

  17. M. J. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853.

    Article  Google Scholar 

  18. W. W. Hager, Iterative methods for nearly singular linear systems, SIAM J. Sci. Comput., 22 (2000), pp. 747-766.

    Article  Google Scholar 

  19. R. J. Hanson, A numerical method for solving Fredholm integral equations of the first kind using singular values, SIAM J. Numer. Anal., 8 (1971), pp. 616-622.

    Article  Google Scholar 

  20. N. J. Higham, A collection of test matrices in MATLAB, ACM Trans. Math. Software, 17 (1991), pp. 289-305.

    Article  Google Scholar 

  21. A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964.

    Google Scholar 

  22. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964.

    Google Scholar 

  23. W. Kerner, Large-scale complex eigenvalue problems, J. Comp. Phys., 85 (1989), pp. 1-85.

    Article  Google Scholar 

  24. M. Kilmer and G. W. Stewart, Iterative regularisation and MINRES, SIAM J. Sci. Comput., 21 (1999), pp. 613-628.

    Article  Google Scholar 

  25. R. E. Kleinman and G. F. Roach, Iterative solutions of boundary integral equations in acoustics, Proc. Roy. Soc. London A, 417 (1988), pp. 45-57.

    Google Scholar 

  26. R. E. Kleinman and Van der Berg, Iterative methods for solving integral equations, Radio Sci., 26(1) (1991), pp. 175-181.

    Google Scholar 

  27. L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.

    Article  Google Scholar 

  28. M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, et al., Approximate Solution of Operator Equations, Wolters-Noordhoff, Groningen, 1972.

    Google Scholar 

  29. M. M. Lavrent'ev, Some Ill-Posed Problems of Mathematical Physics, Springer-Verlag, 1967.

  30. A. P. Raiche, An integral equation approach to three-dimensional modeling, Geophys. J. R. Astron. Soc., 36 (1974), pp. 363-376.

    Google Scholar 

  31. V. N. Strakhov, The method of successive approximations for linear equations in a Hilbert space, U.S.S.R. Comput. Maths. Math. Phys., 13 (1973), pp. 269-274.

    Article  Google Scholar 

  32. A. N. Tikhonov, On ill-posed problems of linear algebra and the stable method of their solution, Soviet Math. Dokl., 163 (1965), pp. 591-595.

    Google Scholar 

  33. A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems, Winston & Sons, Washington, DC, 1977.

    Google Scholar 

  34. A. N. Tikhonov, V. Ya. Arsenin, and A. A. Timonov, Mathematical Problems of Computer Tomography, Nauka, Moscow, 1987 (in Russian).

    Google Scholar 

  35. A. Timonov, Regularised functional calculi and their application to the numerical solution of linear inverse problems, Proceedings 3rd International Conference on Inverse Problems in Engineering, 13-18 June, Port Ludlow, Washington, USA, 1999.

  36. G. M. Vainikko, The discrepancy principle for a class of regularization methods, U.S.S.R. Comput. Maths. Math. Phys., 22 (1982), pp. 1-19.

    Article  Google Scholar 

  37. R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.

    Google Scholar 

  38. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.

    Google Scholar 

  39. D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Timonov, A. Factorised Preconditionings of Successive Approximations in Finite Precision. BIT Numerical Mathematics 41, 582–598 (2001). https://doi.org/10.1023/A:1021975414321

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021975414321

Navigation