Abstract
We consider the n × n system of linear equations
and restrict ourselves to real systems, the case of complex ones being an easy extension.
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References
W.E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951) 17–29.
G. Auchmuty, A posteriori error estimates for linear equations, Numer. Math., 61 (1992) 1–6.
H. Ayachour, Avoiding look-ahead in the Lanczos method and Padé approximation, Applicationes Mathematicae, 26 (1999) 33–62.
C. Baheux, New implementations of the Lanczos method, J. Comput. Appl. Math., 57 (1995) 3–15.
R.E. Bank, T.F. Chan, An analysis of the composite step bi-conjugate gradient algorithm for solving nonsymmetric systems, Numer. Math., 66 (1993) 295–319.
R.E. Bank, T.F. Chan, A composite step bi-conjugate gradient algorithm for solving nonsymmetric systems, Numer. Algorithms, 7 (1994) 1–16.
R. Barrett et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1993.
D.L. Boley, S. Elhay, G.H. Golub, M.H. Gutknecht, Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights, Numer. Algorithms, 1 (1991) 21–44.
C. Brezinski, Généralisations de la transformation de Shanks, de la table de Padé et de Fe-algorithme, Calcolo, 12 (1975) 317–360.
C. Brezinski, Rational approximation to formal power series, J. Approx. Theory, 25 (1979) 295–317.
C. Brezinski, Padé-Type Approximation and General Orthogonal Polynomials, Birkhäuser-Verlag, Basel, 1980.
C. Brezinski, Biorthogonality and its Applications to Numerical Analysis, Marcel Dekker, New York, 1992.
C. Brezinski, A unified approach to various orthogonalities, Ann. Fac. Sci. Toulouse, sér. 3, vol. I, fasc. 3 (1992), 277–292.
C. Brezinski, Biorthogonality and conjugate gradient-type algorithms, in Contributions in Numerical Mathematics, R.P. Agarwal ed., World Scientific, Singapore, 1993, pp. 55–70.
C. Brezinski, Projection Methods for Systems of Equations, North-Holland, Amsterdam, 1997.
C. Brezinski, A transpose-free Lanczos/Orthodir algorithm for linear systems, C.R. Acad. Sci. Paris, Sér. I, 324 (1997) 349–354.
C. Brezinski, Krylov subspace methods, biorthogonal polynomials and Padé-type approximants, Numer. Algorithms, 21 (1999) 97–107.
C. Brezinski, Error estimates for the solution of linear systems, SIAM J. Sci. Comput., 21 (1999)764–781.
C. Brezinski, C. Musschoot, Biorthogonal polynomials and the bordering method for linear systems, Rend. Sem. Mat. Fis. Milano, 64 (1994) 85–98.
C. Brezinski, M. Redivo-Zaglia, Treatment of near-breakdown in the CGS algorithm, Publication ANO 257, Laboratoire d’Analyse Numérique et d’Optimisation, Université des Sciences et Technologies de Lille, November 1991.
C. Brezinski, M. Redivo-Zaglia, A new presentation of orthogonal polynomials with applications to their computation, Numer. Algorithms, 1 (1991), 207–221.
C. Brezinski, M. Redivo-Zaglia, Treatment of near-breakdown in the CGS algorithm, Numer. Algorithms, 7 (1994) 33–73.
C. Brezinski, M. Redivo-Zaglia, Look-ahead in Bi-CGSTAB and other methods for linear systems, BIT, 35 (1995) 169–201.
C. Brezinski, M. Redivo Zaglia, Transpose-free implementations of Lanczos’ method for nonsymmetric linear systems, Publication ANO-372, Laboratoire d’Analyse Numérique et d’Optimisation, Université des Sciences et Technologies de Lille, June 1997.
C. Brezinski, M. Redivo Zaglia, Breakdowns in the computation of orthogonal polynomials, in Nonlinear Numerical Methods and Rational Approximation, A. Cuyt ed., Kluwer, Dordrecht, 1994, pp. 49–59.
C. Brezinski, M. Redivo Zaglia, Transpose-free Lanczos-type algorithms for nonsymmetric linear systems, Numer. Algorithms, 17 (1998) 67–103.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, Avoiding breakdown and near-breakdown in Lanczos type algorithms, Numer. Algorithms, 1 (1991), 261–284.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, Addendum to “Avoiding breakdown and near-breakdown in Lanczos type algorithms”, Numer. Algorithms, 2 (1992) 133–136.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, A breakdown-free Lanczos type algorithm for solving linear systems, Numer. Math., 63 (1992) 29–38.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, Breakdowns in the implementation of the Lânczos method for solving linear systems, Inter. J. Comp. Math. with Applics., 33 (1997) 31–44.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, Problems of breakdown and near-breakdown in Lanczos-based algorithms, in Algorithms for Sparse Large Scale Linear Algebraic Systems, G. Winter Althaus and E. Spedicato eds., Kluwer, Dordrecht, 1998, pp. 271–290.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, New look-ahead Lanczos-type algorithms for linear systems, Numer. Math., 83 (1999) 53–85.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, The matrix and polynomial approaches to Lanczos-type algorithms, J. Comput. Appl. Math., 123 (2000) 241–260.
C. Brezinski, M. Redivo-Zaglia, H. Sadok, A review of formal orthogonality in Lanczosbased methods, J. Comput. Appl. Math., to appear.
C. Brezinski, H. Sadok, Avoiding breakdown in the CGS algorithm, Numer. Algorithms, 1 (1991) 199–206.
C. Brezinski, H. Sadok, Lanczos-type algorithms for solving systems of linear equations, Appl. Numer. Math., 11 (1993) 443–473.
C.G. Broyden, Look-ahead block-CG algorithms, Optim. Methods and Soft., to appear.
A.M. Bruaset, A Survey of Preconditioned Iterative Methods, Longman, Harlow, 1995.
T.F. Chan, T. Szeto, A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems, Numer. Algorithms, 7 (1994) 17–32.
T.F. Chan, L. de Pillis, H.A. Van der Vorst, Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems, Numer. Algorithms, 17 (1998) 51–66.
J.-M. Chesneaux, A.C. Matos, Breakdown and near-breakdown control in the CGS algorithm using stochastic arithmetic, Numer. Algorithms, 11 (1996) 99–116.
J. Cullum, A. Greenbaum, Relations between Galerkin and norm-minimizing iterative methods for solving linear systems, SIAM J. Matrix Anal. Appl., 17 (1996) 223–247.
A. Draux, Polynômes Orthogonaux Formels. Applications, LNM vol. 974, Springer-Verlag, Berlin, 1983.
A. Draux, Formal orthogonal polynomials revisited. Applications, Numer. Algorithms, 11 (1996)143–158.
A. El Guennouni, A unified approach to some strategies for the treatment of breakdown in Lanczos-type algorithms, Applicationes Mathematicae, to appear.
R. Fletcher, Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee 1975, G.A. Watson ed., LNM vol. 506, Springer-Verlag, Berlin, 1976, pp.73–89.
D.R. Fokkema G.L.G. Sleijpen, H.A. Van der Vorst, Generalized conjugate gradient squared, J. Comput. Appl. Math., 71 (1996) 125–146.
D.R. Fokkema, Subspace Methods for Linear, Nonlinear, and Eigen Problems,Thesis, University of Utrecht, 1996.
R.W. Freund, Solution of shifted linear systems by quasi-minimal residual iterations, in Numerical Linear Algebra, L. Reichel, A. Ruttan and R.S. Varga eds., W. de Gruyter, Berlin, 1993, pp.101–121.
R.W. Freund, G.H. Golub, N.M. Nachtigal, Iterative solution of linear systems, Acta Numerica, 1 (1991) 57–100.
R.W. Freund, M.H. Gutknecht, N.M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comput., 14 (1993) 137–158.
P.R. Graves-Morris, A “Look-around Lanczos” algorithm for solving a system of linear equations, Numer. Algorithms, 15 (1997) 247–274.
A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM, Philadelphia, 1997.
M.H. Gutknecht, The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, and the qd-algorithm, in Proceedings of the Copper Mountain Conference on Iterative Methods,Copper Mountain, Colorado, April 1–5, 1990, vol. 2, unpublished.
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, SIAM J. Matrix Anal. Appl., 15 (1994) 15–58.
M.H. Gutknecht, K.J. Ressel, Look-ahead procedures for Lanczos-type product methods based on three-term recurrences, in Preliminary Proceedings of the Copper Mountain Conference on Iterative Methods, 1996.
M.H. Gutknecht, Z. Strakos, Accuracy of the three-term and two-term recurrences for Krylov space solvers, to appear.
Cs.J. Hegedüs, Generating conjugate directions for arbitrary matrices by matrix equations, Computers Math. Applic., 21 (1991) 71–85; 87–94.
M.R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., 49 (1952) 409–436.
N.J. Higham, The Test Matrix Toolbox for MATLAB (Version 3.0), Numerical Analysis Report 276, Dept. of Mathematics, The University of Manchester, 1995.
T. Kailath, A. Vieira, M. Morf, Inverses of Toeplitz operators, innovations, and orthogonal polynomials, SIAM Rev., 20 (1978) 106–119.
M. Khelifi, Lanczos maximal algorithm for unsymmetric eigenvalue problems, Appl. Numer. Math., 7 (1991) 179–193.
C. Lanczos, An iteration method for the solution of the eigenvalue 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.
H. Le Ferrand, Vector Padé approximants and the Lanczos method for solving a system of linear equations, May 1994, unpublished.
V.Y. Pan, Structured Matrices and Polynomials, Birkhäuser, Boston, 2001.
B.N. Parlett, D.R. Taylor, Z.A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comput., 44 (1985) 105–124.
M.A. Pinar, V. Ramirez, Recursive inversion of Hankel matrices, Monogr. Acad. Ciencias Zaragoza, 1 (1988) 119–128.
M.A. Pinar, V. Ramirez, Inversion of Toeplitz matrices, in Orthogonal Polynomials and their Applications, J. Vinuesa ed., Marcel Dekker, New York, 1989, pp.171–177.
Qiang Ye, A breakdown-free variation of the nonsymmetric Lanczos algorithm, Math. Comput., 62 (1994) 179–207.
Qiang Ye, An adaptative block Lanczos algorithm, Numer. Algorithms, 12 (1996) 97–110.
J. Rissanen, Solution of linear equations with Hankel and Toeplitz matrices, Numer. Math., 22 (1974) 361–366.
Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publ. Co., Boston, 1996.
Y. Saad, M.H. Schultz, Conjugate gradient-like algorithms for solving nonsymmetric linear systems, Math. Comput., 44 (1985) 417–424.
Y. Saad, M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986) 856–869.
P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 10 (1989) 36–52.
D.R. Taylor, Analysis of the Look-Ahead Lanczos Algorithm, Ph.D. Thesis, Dept. of Mathematics, University of California, Berkeley, Nov. 1982.
C.H. Tong, Qiang Ye, A linear system solver based on a modified Krylov subspace method for breakdown recovery, Numer. Algorithms, 12 (1996) 233–251.
W. Trench, An algorithm for the inversion of finite Toeplitz matrices, SIAM J. Appl. Math., 12 (1964) 515–522.
W. Trench, An algorithm for the inversion of finite Hankel matrices, SIAM J. Appl. Math., 13 (1965)1102–1107.
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. Stat. Comput., 13 (1992) 631–644.
J. van Iseghem, Vector Padé approximants, in Numerical Mathematics and Applications, R. Vichnevetsky and J. Vignes eds., North-Holland, Amsterdam, 1986, pp. 73–77.
P.K.W. Vinsome, ORTHOMIN, an iterative method for solving sparse sets of simultaneous linear equations, in Proc. Fourth Symposium on Reservoir Simulation, Society of Petroleum Engineers of AIME, 1976, pp. 149–159.
D.M. Young, K.C. Jea, Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods, Linear Algebra Appl., 34 (1980) 159–194.
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Brezinski, C. (2002). Systems of Linear Algebraic Equations. In: Brezinski, C. (eds) Computational Aspects of Linear Control. Numerical Methods and Algorithms, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0261-2_8
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DOI: https://doi.org/10.1007/978-1-4613-0261-2_8
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