Summary
The Chebyshev and second-order Richardson methods are classical iterative schemes for solving linear systems. We consider the convergence analysis of these methods when each step of the iteration is carried out inexactly. This has many applications, since a preconditioned iteration requires, at each step, the solution of a linear system which may be solved inexactly using an “inner” iteration. We derive an error bound which applies to the general nonsymmetric inexact Chebyshev iteration. We show how this simplifies slightly in the case of a symmetric or skew-symmetric iteration, and we consider both the cases of underestimating and overestimating the spectrum. We show that in the symmetric case, it is actually advantageous to underestimate the spectrum when the spectral radius and the degree of inexactness are both large. This is not true in the case of the skew-symmetric iteration. We show how similar results apply to the Richardson iteration. Finally, we describe numerical experiments which illustrate the results and suggest that the Chebyshev and Richardson methods, with reasonable parameter choices, may be more effective than the conjugate gradient method in the presence of inexactness.
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References
Concus, P., Golub, G.H.: A generalized conjugate gradient method for nonsymmetric systems of equations. Proc. Second Internat. Symp. on Computing Methods in Applied Sciences and Engineering, IRIA (Paris, Dec. 1975), Lecture Notes in Economics and Mathematical Systems, vol. 134, R. Glowinski and J.L. Lions, eds.), Springer, Berlin Heidelberg New York 1976
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Number. Anal.19, 400–408 (1982)
Eiermann, M., Niethammer, W., Varga, R.S.: A study of semiiterative methods for nonsymmetric systems of linear equations. Numer. Math.47, 505–533 (1985)
Freund, R., Ruscheweyh, S.: On a class of Chebyshev approximation problems which arise in connection with a conjugate gradient type method. Numer. Math.48, 525–542 (1986)
Glowinski, R., Lions, J., Trémolières, R.: Analyse numerique des inéquations variationelles, Vol. I. Paris: Dunod 1976
Golub, G.H.: The use of Chebyshev matrtix polynomials in the iterative solution of linear equations compared with the method of successive overrelaxation. Ph.D. thesis, University of Illinois 1959
Golub, G.H.: Bounds for the round-off errors in the Richardson second-order method. BIT2, 212–223 (1962)
Golub, G.H., Overton, M.L.: Convergence of a two-stage Richardson iterative procedure for solving systems of linear equations. In: Numerical analysis (Proceedings of the Ninth Biennial Conference, Dundee, Scotland, 1981) (G.A. Watson, ed.). Lect. Notes Math. 912, Springer, New York Heidelberg Berlin, pp. 128–139
Golub, G.H., Varga, R.S.: Chebyshev semi-iterative methods, successive overrelaxation iterative methods, and second order Richardson iterative methods, Parts I and II. Numer. Math.3, 147–168 (1961)
Greenbaum, A.: Behavior of slightly perturbed Lanczos and conjugate gradient recurrences. Linear Algebra Appl. 1988 (to appear)
Gunn, J.E.: The numerical solution ofV·aV u=f by a semi-explicit alternating-direction iterative technique. Numer. Math.6, 181–184 (1964)
Manteuffel, T.A.: The Tchebyshev iteration for nonsymmetric linear systems. Numer. Math.28, 307–327 (1977)
Nichols, N.K.: On the convergence of two-stage iterative processes for solving linear equations. SIAM J. Numer. Anal.10, 460–469 (1973)
NAG Library Manual (1984). Numerical Algorithms Group, 256 Banbury Rd., Oxford
Nicolaides, R.A.: On the local convergence of certain two step iterative procedures. Numer. Math.24, 95–101 (1975)
Niethammer, W., Varga, R.S.: The analysis ofk-step iterative methods for linear systems from summability theory. Numer. Math.41, 177–206 (1983)
Pereyra, V.: Accelerating the convergence of discretization algorithms. SIAM J. Numer. Anal.4, 508–533 (1967)
Varga, R.S.: Matrix iterative analysis. Englewood Cliffs, NJ: Prentice-Hall 1962
Widlund, O.: A Lanczos method for a class of non symmetric systems of linear equations. SIAM J. Numer. Anal.15, 801–812 (1978)
Woźniakowski, H.: Numerical stability of the Chebyshev method for the solution of large linear systems. Numer. Math.28, 191–209 (1977)
Young, D.M.: Iterative solution of large linear systems. New York, London: Academic Press 1971
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This work was supported in part by National Science Foundation Grants DCR-8412314 and DCR-8502014
The work of this author was completed while he was on sabbatical leave at the Centre for Mathematical Analysis and Mathematical Sciences Research Institute at the Australian National University, Canberra, Australia
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Golub, G.H., Overton, M.L. The convergence of inexact Chebyshev and Richardson iterative methods for solving linear systems. Numer. Math. 53, 571–593 (1988). https://doi.org/10.1007/BF01397553
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DOI: https://doi.org/10.1007/BF01397553