Abstract
The method of Lanczos for solving systems of linear equations is implemented by using recurrence relationships between formal orthogonal polynomials. A drawback is that the computation of the coefficients of these recurrence relationships usually requires the use of the transpose of the matrix of the system. Due to the indirect addressing, this is a costly operation. In this paper, a new procedure for computing these coefficients is proposed. It is based on the recursive computation of the products of polynomials appearing in their expressions and it does not involve the transpose of the matrix. Moreover, our approach allows to implement simultaneously and at a low price a Lanczos-type product method such as the CGS or the BiCGSTAB.
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References
C. Baheux, New implementations of Lanczos method, J. Comput. Appl. Math. 57 (1995) 3–15.
C. Brezinski, Padé-type Approximation and General Orthogonal Polynomials, International Series of Numerical Mathematics 50 (Birkhäuser, Basel, 1980).
C. Brezinski, CGM: a whole class of Lanczos-type solvers for linear systems, Note ANO-253, Laboratoire d'Analyse Numérique et d'Optimisation, Université des Sciences et Technologies de Lille (November 1991).
C. Brezinski, A transpose-free Lanczos/Orthodir algorithm for linear systems, C. R. Acad. Sci. Paris Sér. I 324 (1997) 349–354.
C. Brezinski and M. Redivo-Zaglia, Hybrid procedures for solving systems of linear systems, Numer. Math. 67 (1994) 1–19.
C. Brezinski and M. Redivo-Zaglia, Treatment of near-breakdown in the CGS algorithm, Numer. Algorithms 7 (1994) 33–73.
C. Brezinski and M. Redivo-Zaglia, Look-ahead in Bi-CGSTAB and other product methods for linear systems, BIT 35 (1995) 169–201.
C. Brezinski and M. Redivo-Zaglia, Transpose-free implementations of Lanczos's method for non-symmetric linear systems, Note ANO-372, Laboratoire d'Analyse Numérique et d'Optimisation, Université des Sciences et Technologies de Lille (1997).
C. Brezinski, M. Redivo-Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms 1 (1991) 261–284.
C. Brezinski, M. Redivo-Zaglia and H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math. 63 (1992) 29–38.
C. Brezinski, M. Redivo-Zaglia and H. Sadok, New look-ahead Lanczos-type algorithms for linear systems, submitted.
C. Brezinski and H. Sadok, Avoiding breakdown in the CGS algorithms, Numer. Algorithms 1 (1991) 207–221.
C. Brezinski and H. Sadok, Some vector sequence transformations with applications to systems of equations, Numer. Algorithms 3 (1992) 75–80.
C. Brezinski and H. Sadok, Lanczos-type algorithms for solving systems of linear equations, Appl. Numer. Math. 11 (1993) 443–473.
T.F. Chan, L. de Pillis and H. van der Vorst, Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems, Numer. Algorithms 17 (1998), this issue.
R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numerical Analysis, Dundee 1975, ed. G.A. Watson, Lecture Notes in Mathematics 506 (Springer, Berlin, 1976) pp. 73–89.
D.R. Fokkema, Subspace methods for linear, nonlinear, and eigen problems, thesis, University of Utrecht (1996).
D.R. Fokkema, G.L.G. Sleijpen and H.A. van der Vorst, Generalized conjugate gradient squared, J. Comput. Appl. Math. 71 (1996) 125–146.
W. Gander, G.H. Golub and D. Gruntz, Solving linear equations by extrapolation, in: Supercom-puting, ed. J.S. Kovalik (Springer, Berlin, 1989) pp. 279–293.
M.H. Gutknecht, The unsymmetric Lanczos algorithms and their relations to Pad´ e approximation, continued fractions, and the qd-algorithm, in: Proc.of the Copper Mountain Conf.on Iterative Methods, Vol. 2, Copper Mountain, CO (April 1–5, 1990) unpublished.
M.H. Gutknecht, Variants of BiCGSTAB for matrices with complex spectrum, SIAM J. Sci. Comput. 14 (1993) 1020–1033.
N.J. Higham, The test matrix toolbox for MATLAB (version 3.0), Numer. Analysis Report No. 276, Department of Mathematics, The University of Manchester (September 1995).
C. Lanczos, An iteration method for the solution of the eignevalue problem of linear differential and integral operators, J. Res. Natl. Bur. Stand. 45 (1950) 255–282.
C. Lanczos, Solution of systems of linear equations by minimized iterations, J. Res. Natl. Bur. Stand. 49 (1952) 33–53.
K.J. Ressel and M.H. Gutknecht, QMR-smoothing for Lanczos-type product methods based on three-term recurrences, to appear.
H. Rutishauser, Der Quotienten–Differenzen-Algorithmus, Z. Angew. Math. Phys. 5 (1954) 233-251.
P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10 (1989) 35–52.
H.A. van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631–644.
P.K.W. Vinsome, Orthomin, an iterative method for solving sparse sets of simultaneous linear equations, in: Proc.4th Symp.on Reservoir Simulation, Society of Petroleum Engineers of AIME (1976) pp. 149–159.
D.M. Young and K.C. Jea, Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods, Linear Algebra Appl. 34 (1980) 159–194.
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Brezinski, C., Redivo-Zaglia, M. Transpose-free Lanczos-type algorithms for nonsymmetric linear systems. Numerical Algorithms 17, 67–103 (1998). https://doi.org/10.1023/A:1012085428800
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DOI: https://doi.org/10.1023/A:1012085428800