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.
Similar content being viewed by others
REFERENCES
V. V. Arestov, Approximation of unbounded operators by bounded operators and related extremal problems, Russian Math. Surveys, 51 (1996), pp. 1093-1126.
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.
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.
O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, UK, 1994.
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.
A. B. Bakushinsky and A. V. Goncharsky, Ill-Posed Problems: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1994.
R. Barrett, M. Berry, T. Chan, et al., Templates for the Solution of Linear Systems, SIAM, Philadelphia, PA, 1994.
F. Chaitin-Chatelin and S. Gratton, Convergence in finite precision of successive iteration methods under high nonnormality, BIT, 36 (1996), pp. 455-469.
D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
J. Cullum, Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problems, BIT, 36 (1996), pp. 470-493.
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.
R. W. Freund, G. H. Golub, and N. N. Nachtigal, Iterative solution of linear systems, Acta Numerica, (1991), pp. 57-100.
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).
A. Frommer and P. Maas, Fast CG-based methods for Tikhonov-Phillips regularisation, SIAM J. Sci. Comput., 20 (1999), pp. 1831-1850.
G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins Press, Baltimore, MD, 1996.
A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, PA, 1997.
M. J. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853.
W. W. Hager, Iterative methods for nearly singular linear systems, SIAM J. Sci. Comput., 22 (2000), pp. 747-766.
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.
N. J. Higham, A collection of test matrices in MATLAB, ACM Trans. Math. Software, 17 (1991), pp. 289-305.
A. S. Householder, The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964.
L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1964.
W. Kerner, Large-scale complex eigenvalue problems, J. Comp. Phys., 85 (1989), pp. 1-85.
M. Kilmer and G. W. Stewart, Iterative regularisation and MINRES, SIAM J. Sci. Comput., 21 (1999), pp. 613-628.
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.
R. E. Kleinman and Van der Berg, Iterative methods for solving integral equations, Radio Sci., 26(1) (1991), pp. 175-181.
L. Yu. Kolotilina and A. Yu. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.
M. A. Krasnosel'skii, G. M. Vainikko, P. P. Zabreiko, et al., Approximate Solution of Operator Equations, Wolters-Noordhoff, Groningen, 1972.
M. M. Lavrent'ev, Some Ill-Posed Problems of Mathematical Physics, Springer-Verlag, 1967.
A. P. Raiche, An integral equation approach to three-dimensional modeling, Geophys. J. R. Astron. Soc., 36 (1974), pp. 363-376.
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.
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.
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems, Winston & Sons, Washington, DC, 1977.
A. N. Tikhonov, V. Ya. Arsenin, and A. A. Timonov, Mathematical Problems of Computer Tomography, Nauka, Moscow, 1987 (in Russian).
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.
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.
R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.
D. M. Young, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
Author information
Authors and Affiliations
Rights 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
Issue Date:
DOI: https://doi.org/10.1023/A:1021975414321