Abstract
Lanczos type algorithms form a wide and interesting class of iterative methods for solving systems of linear equations. One of their main interest is that they provide the exact answer in at mostn steps wheren is the dimension of the system. However a breakdown can occur in these algorithms due to a division by a zero scalar product. After recalling the so-called method of recursive zoom (MRZ) which allows to jump over such breakdown we propose two new variants. Then the method and its variants are extended to treat the case of a near-breakdown due to a division by a scalar product whose absolute value is small which is the reason for an important propagation of rounding errors in the method. Programming the various algorithms is then analyzed and explained. Numerical results illustrating the processes are discussed. The subroutines corresponding to the algorithms described can be obtained vianetlib.
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References
D.L. Boley, S. Elhay, G.H. Golub and M.H. Gutknecht, Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights, Numer. Algorithms 1 (1991) 21–44.
C. Brezinski,Padé-type Approximation and General Orthogonal Polynomials. ISNM Vol. 50, (Birkhäuser, Basel, 1980).
C. Brezinski, M. Redivo Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math., to appear.
C. Brezinski and H. Sadok, Lanczos type methods for systems of linear equations, submitted.
C. Brezinski and H. Sadok, Avoiding breakdown in the CGS algorithm, Numer. Algorithms 1 (1991) 199–206.
A. Draux,Polynômes Orthogonaux Formels. Applications., LNM 974 (Springer-Verlag, Berlin, 1983).
R. Fletcher, Conjugate gradient methods for indefinite systems, in:Numerical Analysis, ed. G.A. Watson, LNM 506 (Springer-Verlag, Berlin, 1976) pp. 73–89.
M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, Parts I, II, SIAM J. Matrix Anal. Appl., to appear.
P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comp. 10 (1989) 36–52.
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Brezinski, C., Zaglia, M.R. & Sadok, H. Avoiding breakdown and near-breakdown in Lanczos type algorithms. Numer Algor 1, 261–284 (1991). https://doi.org/10.1007/BF02142326
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DOI: https://doi.org/10.1007/BF02142326