Abstract
Despite its usefulness in solving eigenvalue problems and linear systems of equations, the nonsymmetric Lanczos method is known to suffer from a potential breakdown problem. Previous and recent approaches for handling the Lanczos exact and near-breakdowns include, for example, the look-ahead schemes by Parlett-Taylor-Liu [23], Freund-Gutknecht-Nachtigal [9], and Brezinski-Redivo Zaglia-Sadok [4]; the combined look-ahead and restart scheme by Joubert [18]; and the low-rank modified Lanczos scheme by Huckle [17]. In this paper, we present yet another scheme based on a modified Krylov subspace approach for the solution of nonsymmetric linear systems. When a breakdown occurs, our approach seeks a modified dual Krylov subspace, which is the sum of the original subspace and a new Krylov subspaceK m (w j ,A T) wherew j is a newstart vector (this approach has been studied by Ye [26] for eigenvalue computations). Based on this strategy, we have developed a practical algorithm for linear systems called the MLAN/QM algorithm, which also incorporates the residual quasi-minimization as proposed in [12]. We present a few convergence bounds for the method as well as numerical results to show its effectiveness.
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References
D. Boley, S. Elhay, G. Golub and M. Gutknecht, Nonsymmetric Lanczos algorithms and finding orthogonal polynomials with indefinite weight, Numer. Algo. 1 (1991) 21–43.
C. Brezinski and H. Sadok, Avoiding breakdown in the CGS algorithm, Numer. Algo. 1 (1991) 199–206.
C. Brezinski and H. Sadok, Lanczos-type algorithms for solving systems of linear equations, Appl. Numer. Math. 11 (1993) 443–473.
C. Brezinski, M. Redivo-Zaglia and H. Sadok, Avoiding breakdown and near-breakdown in Lanczos-type algorithms, Numer. Algo. 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.
I.S. Duff, R.G. Grimes and J.G. Lewis, Sparse matrix test problems, ACM Trans. Math. Softw. 15 (1989) 1–14.
V. Faber and T. Manteuffel, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. Numer. Anal. 21 (1984) 352–362.
R. Fletcher, Conjugate gradient methods for indefinite systems, in:Proc. Dundee Conference on Numerical Analysis, Lecture Notes in Mathematics 506, ed. G.A. Watson (Springer, Berlin, 1976) pp. 73–89.
R.W. Freund, M.H. Gutknecht and N.M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Statist. Comp. 14 (1993) 137–158.
R.W. Freund and N.M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices Part I, Technical Report 90-45, RIACS, NASA Ames Research Center, Moffett Field, CA (1990).
R.W. Freund and N.M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices Part II, Technical Report 90-46, RIACS, NASA Ames Research Center, Moffett Field, CA (1990).
R.W. Freund and N.M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315–339.
G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, 1989).
W.B. Gragg, Matrix interpretations and applications of the continued fraction algorithm, Rocky Mountain J. Math. 4 (1974) 213–225.
M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, Part I, SIAM J. Matrix Anal. Appl. 13 (1992) 594–639.
M.H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms, Part II, IPS (Interdisciplinary Project Center for Supercomputing) Research Report 90-16, ETH-Zentrum, CH-8092 Zurich, Switzerland (September 1990).
T. Huckle, Low rank modification of the unsymmetric Lanczos algorithm, preprint (January 1994).
W. Joubert, Generalized conjugate gradient and Lanczos methods for the solution of nonsymmetric systems of linear equations, PhD thesis, Center for Numerical Analysis, The University of Texas at Austin (1990).
W. Joubert, Lanczos methods for the solution of nonsymmetric systems of linear equations, SIAM J. Matrix Anal. Appl. 13 (1992) 926–943.
C. Lanczos, Solution of systems of linear equations by minimized iterations, J. Res. Natl. Bur. Stand. 49 (1952) 33–53.
N.M. Nachtigal, A look-ahead variant of the Lanczos algorithm and its application to the quasiminimal residual method for non-Hermitian linear systems, PhD dissertation, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 (Aug. 1991).
B.N. Parlett, Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl. 13 (1992) 567–593.
B.N. Parlett, D.R. Taylor and Z.A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comp. 44 (1985) 105–124.
Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7 (1986) 856–869.
D.R. Taylor, Analysis of the lookahead Lanczos algorithm, PhD dissertation, University of California, Berkeley (1982).
Q. Ye, A breakdown-free variation of the nonsymmetric Lanczos algorithms, Math Comp. 62 (1994) 179–207.
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Communicated by M.H. Gutknecht
Research supported by Natural Sciences and Engineering Research Council of Canada.
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Tong, C.H., Ye, Q. A linear system solver based on a modified Krylov subspace method for breakdown recovery. Numer Algor 12, 233–251 (1996). https://doi.org/10.1007/BF02141750
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DOI: https://doi.org/10.1007/BF02141750