Hostname: page-component-54dcc4c588-5q6g5 Total loading time: 0 Render date: 2025-09-11T10:41:19.579Z Has data issue: false hasContentIssue false
  • English
  • Français

Lanczos-type solvers for nonsymmetric linear systems of equations

Published online by Cambridge University Press:  07 November 2008

Martin H. Gutknecht
Martin H. Gutknecht
Affiliation:
Swiss Center for Scientific Computing ETH-Zentrum, CH-8092 Zürich, Switzerland E-mail: mhg@scsc.ethz.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by look-ahead are also discussed.

Information

Type
Research Article
Information
Acta Numerica , Volume 6 , January 1997 , pp. 271 - 397
Copyright
Copyright © Cambridge University Press 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Aliaga, J. I., Hernandez, V. and Boley, D. L. (1994), Using the block clustered nonsymmetric Lanczos algorithm to solve control problems for MIMO linear systems, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, pp. 387–389.Google Scholar
Arnoldi, W. E. (1951), ‘The principle of minimized iterations in the solution of the matrix eigenvalue problem’, Quart. Appl. Math. 9, 17–29.CrossRefGoogle Scholar
Ashby, S. F., Manteuffel, T. A. and Saylor, P. E. (1990), ‘A taxonomy for conjugate gradient methods’, SIAM J. Numer. Anal. 27, 1542–1568.CrossRefGoogle Scholar
Bai, Z. (1994), ‘Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem’, Math. Comp. 62, 209–226.CrossRefGoogle Scholar
Bank, R. E. and Chan, T. F. (1993), ‘An analysis of the composite step biconjugate gradient method’, Numer. Math. 66, 295–319.CrossRefGoogle Scholar
Bank, R. E. and Chan, T. F. (1994), ‘A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems’, Numerical Algorithms 7, 1–16.CrossRefGoogle Scholar
Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. and van der Vorst, H. (1994), Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia.CrossRefGoogle Scholar
Barth, T. and Manteuffel, T. (1994), Variable metric conjugate gradient methods, in Advances in Numerical Methods for Large Sparse Sets of Linear Systems (Natori, M. and Nodera, T., eds), number 10 in ‘Parallel Processing for Scientific Computing’, Keio University, Yokahama, Japan, pp. 165–188.Google Scholar
Boley, D. and Golub, G. H. (1991), ‘The nonsymmetric Lanczos algorithm and controllability’, Systems Control Lett. 16, 97–105.CrossRefGoogle Scholar
Boley, D. L. (1994), Krylov space methods in linear control and model reduction: A survey, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 377–379.Google Scholar
Boley, D. L., Elhay, S., Golub, G. H. and Gutknecht, M. H. (1991), ‘Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights’, Numerical Algorithms 1, 21–43.CrossRefGoogle Scholar
Brezinski, C. and Redivo, M. Zaglia (1994), ‘Treatment of near-breakdown in the CGS algorithms’, Numerical Algorithms 7, 33–73.CrossRefGoogle Scholar
Brezinski, C. and Redivo, M. Zaglia (1995), ‘Look-ahead in Bi-CGSTAB and other product methods for linear systems’, BIT 35, 169–201.CrossRefGoogle Scholar
Brezinski, C. and Sadok, H. (1991), ‘Avoiding breakdown in the CGS algorithm’, Numerical Algorithms 1, 199–206.CrossRefGoogle Scholar
Brezinski, C., Redivo Zaglia, M. and Sadok, H. (1991), ‘Avoiding breakdown and near-breakdown in Lanczos type algorithms’, Numerical Algorithms 1, 261–284.CrossRefGoogle Scholar
Brezinski, C., Redivo Zaglia, M. and Sadok, H. (1992 a), ‘Addendum to “Avoiding breakdown and near-breakdown in Lanczos type algorithms”’, Numerical Algorithms 2, 133–136.CrossRefGoogle Scholar
Brezinski, C., Redivo Zaglia, M. and Sadok, H. (1992 b), ‘A breakdown-free Lanczos’ type algorithm for solving linear systems’, Numer. Math. 63, 29–38.CrossRefGoogle Scholar
Brezinski, C., Redivo Zaglia, M. and Sadok, H. (1993), Breakdowns in the implementation of the Lánczos method for solving linear systems, Technical Report ANO-320, Université Lille Flandres Artois.Google Scholar
Brown, P. N. (1991), ‘A theoretical comparison of the Arnoldi and GMRES algorithms’, SIAM J. Sci. Statist. Comput. 12, 58–78.CrossRefGoogle Scholar
Cabay, S. and Meleshko, R. (1993), ‘A weakly stable algorithm for Padé approximants and the inversion of Hankel matrices’, SIAM J. Matrix Anal. Appl. 14, 735–765.CrossRefGoogle Scholar
Cao, Z.-H. (1997 a), ‘Avoiding breakdown in variants of the BI-CGSTAB algorithm’, Linear Algebra Appl. To appear.CrossRefGoogle Scholar
Cao, Z.-H. (1997 b), ‘On the QMR approach for iterative methods including coupled three-term recurrences for solving nonsymmetric linear systems’, Int. J. Num. Math. Engin. To appear.Google Scholar
Chan, T. F. and Szeto, T. (1994), ‘A composite step conjugate gradient squared algorithm for solving nonsymmetric linear systems’, Numerical Algorithms 7, 17–32.CrossRefGoogle Scholar
Chan, T. F. and Szeto, T. (1996), ‘Composite step product methods for solving non-symmetric linear systems’, SIAM J. Sci. Comput. 17, 1491–1508.CrossRefGoogle Scholar
Chan, T. F., de Pillis, L. and van der Vorst, H. (1991), A transpose-free squared Lanczos algorithm and application to solving nonsymmetric linear systems, Technical Report CAM 91–17, Dept. of Mathematics, University of California, Los Angeles.Google Scholar
Chan, T. F., Gallopoulos, E., Simoncini, V., Szeto, T. and Tong, C. H. (1994), ‘A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems’, SIAM J. Sci. Comput. 15, 338–347.CrossRefGoogle Scholar
Chronopoulos, A. T. and Ma, S. (1989), On squaring Krylov subspace iterative methods for nonsymmetric linear systems, Technical Report 89–67, Computer Science Department, University of Minnesota.Google Scholar
Cullum, J. (1995), ‘Peaks, plateaus, numerical instabilities in a Galerkin/minimal residual pair of methods for solving Ax = b’, Appl. Numer. Math. 19, 255–278.CrossRefGoogle Scholar
Cullum, J. and Greenbaum, A. (1996), ‘Relations between Galerkin and norm-minimizing iterative methods for solving linear systems’, SIAM J. Matrix Anal. Appl. 17, 223–247.CrossRefGoogle Scholar
Cullum, J. K. (1994), Lanczos algorithms for large scale symmetric and nonsymmetric matrix eigenvalue problems, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 11–31.Google Scholar
Cullum, J. K. and Willoughby, R. A. (1985), Lanczos Algorithms for Large Symmetric Eigenvalue Computations (2 Vols.), Birkhäuser, Boston-Basel-Stuttgart.Google Scholar
Day, D. (1997), ‘An efficient implementation of the non-symmetric Lanczos algorithm’, SIAM J. Matrix Anal. Appl. To appear.CrossRefGoogle Scholar
Day, D. M., III (1993), Semi-duality in the two-sided Lanczos algorithm, PhD thesis, University of California at Berkeley.Google Scholar
Eijkhout, V. (1994), LAPACK Working Note 78: Computational variants of the CGS and BiCGstab methods, Technical Report UTCS-94-241, Computer Science Department, University of Tennessee.Google Scholar
Eisenstat, S. C., Elman, H. C. and Schultz, M. H. (1983), ‘Variational iterative methods for nonsymmetric systems of linear equations’, SIAM J. Numer. Anal. 20, 345–357.CrossRefGoogle Scholar
Faddeev, D. K. and Faddeeva, V. N. (1964), Numerische Verfahren der linearen Algebra, Oldenbourg, München. This is not the same book as Computational Methods of Linear Algebra.Google Scholar
Fernando, K. V. and Parlett, B. N. (1994), ‘Accurate singular values and differential qd algorithms’, Numer. Math. 67, 191–229.CrossRefGoogle Scholar
Fletcher, R. (1976), Conjugate gradient methods for indefinite systems, in Numerical Analysis, Dundee, 1975 (Watson, G. A., ed.), Vol. 506 of Lecture Notes in Mathematics, Springer, Berlin, pp. 73–89.Google Scholar
Fokkema, D. R. (1996 a), Enhanced implementation of BiCGstab(ℓ) for solving linear systems of equations, Preprint 976, Department of Mathematics, Utrecht University.Google Scholar
Fokkema, D. R. (1996 b), Subspace Methods for Linear, Nonlinear, and Eigen Problems, PhD thesis, Utrecht University.Google Scholar
Fokkema, D. R., Sleijpen, G. L. G. and van der Vorst, H. A. (1996), ‘Generalized conjugate gradient squared’, J. Comput. Appl. Math. 71, 125–146.CrossRefGoogle Scholar
Freund, R. W. (1992), ‘Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices’, SIAM J. Sci. Statist. Comput. 13, 425–448.CrossRefGoogle Scholar
Freund, R. W. (1993), ‘A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems’, SIAM J. Sci. Comput. 14, 470–482.CrossRefGoogle Scholar
Freund, R. W. (1994), Lanczos-type algorithms for structured non-Hermitian eigenvalue problems, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 243–245.Google Scholar
Freund, R. W. and Malhotra, M. (1997), ‘A block-QMR algorithm for non-Hermitian linear systems with multiple right-hand sides’, Linear Algebra Appl. To appear.CrossRefGoogle Scholar
Freund, R. W. and Nachtigal, N. M. (1991), ‘QMR: a quasi-minimal residual method for non-Hermitian linear systems’, Numer. Math. 60, 315–339.CrossRefGoogle Scholar
Freund, R. W. and Nachtigal, N. M. (1993), Implementation details of the coupled QMR algorithm, in Numerical Linear Algebra (Reichel, L., Ruttan, A. and Varga, R. S., eds), W. de Gruyter, pp. 123–140.Google Scholar
Freund, R. W. and Nachtigal, N. M. (1994), ‘An implementation of the QMR method based on coupled two-term recurrences’, SIAM J. Sci. Comput. 15, 313–337.CrossRefGoogle Scholar
Freund, R. W. and Nachtigal, N. M. (1996), ‘QMRPACK: a package of QMR algorithms’, ACM Trans. Math. Software 22, 46–77.CrossRefGoogle Scholar
Freund, R. W. and Szeto, T. (1991), A quasi-minimal residual squared algorithm for non-Hermitian linear systems, Technical Report 91.26, RIACS, NASA Ames Research Center, Moffett Field, CA.Google Scholar
Freund, R. W. and Zha, H. (1993), ‘A look-ahead algorithm for the solution of general Hankel systems’, Numer. Math. 64, 295–321.CrossRefGoogle Scholar
Freund, R. W., Gutknecht, M. H. and Nachtigal, N. M. (1993), ‘An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices’, SIAM J. Sci. Comput. 14, 137–158.CrossRefGoogle Scholar
Golub, G. and van Loan, C. (1989), Matrix Computations, 2nd edn, Johns Hopkins University Press, Baltimore, MD.Google Scholar
Golub, G. H. and Gutknecht, M. H. (1990), ‘Modified moments for indefinite weight functions’, Numer. Math. 57, 607–624.CrossRefGoogle Scholar
Golub, G. H. and O'Leary, D. P. (1989), ‘Some history of the conjugate gradient and Lanczos algorithms: 1949–1976’, SIAM Rev. 31, 50–102.CrossRefGoogle Scholar
Golub, G., Kågström, B. and Van Dooren, P. (1992), ‘Direct block tridiagonalization of single-input single-output systems’, Systems Control Lett. 18, 109–120.CrossRefGoogle Scholar
Gragg, W. B. (1974), ‘Matrix interpretations and applications of the continued fraction algorithm’, Rocky Mountain J. Math. 4, 213–225.CrossRefGoogle Scholar
Gragg, W. B. and Lindquist, A. (1983), ‘On the partial realization problem’, Linear Algebra Appl. 50, 277–319.CrossRefGoogle Scholar
Greenbaum, A. (1989), ‘Predicting the behavior of finite precision Lanczos and conjugate gradient computations’, Linear Algebra Appl. 113, 7–63.CrossRefGoogle Scholar
Greenbaum, A. (1994 a), Accuracy of computed solutions from conjugate-gradient-like methods, in Advances in Numerical Methods for Large Sparse Sets of Linear Systems (Natori, M. and Nodera, T., eds), number 10 in ‘Parallel Processing for Scientific Computing’, Keio University, Yokahama, Japan, pp. 126–138.Google Scholar
Greenbaum, A. (1994 b), The Lanczos and conjugate gradient algorithms in finite precision arithmetic, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 49–60.Google Scholar
Greenbaum, A. (1997), ‘Estimating the attainable accuracy of recursively computed residual methods’, SIAM J. Matrix Anal. Appl. To appear.CrossRefGoogle Scholar
Greenbaum, A. and Strakos, Z. (1992), ‘Predicting the behavior of finite precision Lanczos and conjugate gradient computations’, SIAM J. Matrix Anal. Appl. 13, 121–137.CrossRefGoogle Scholar
Gutknecht, M. H. (1989 a), ‘Continued fractions associated with the Newton–Padé table’, Numer. Math. 56, 547–589.CrossRefGoogle Scholar
Gutknecht, M. H. (1989 b), ‘Stationary and almost stationary iterative (k, l)-step methods for linear and nonlinear systems of equations’, Numer. Math. 56, 179–213.CrossRefGoogle Scholar
Gutknecht, M. H. (1990), ‘The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, and the qd algorithm’, in Preliminary Proceedings of the Copper Mountain Conference on Iterative Methods,April 1–5, 1990,http://www.scsc.ethz.ch/~mgh/pub/CopperMtn90.ps.Z and http://www.scsc.ethz.ch/~mgh/pub/CopperMtn90/7.ps.Z.Google Scholar
Gutknecht, M. H. (1992), ‘A completed theory of the unsymmetric Lanczos process and related algorithms, Part I’, SIAM J. Matrix Anal. Appl. 13, 594–639.CrossRefGoogle Scholar
Gutknecht, M. H. (1993 a), ‘Changing the norm in conjugate gradient type algorithms’, SIAM J. Numer. Anal. 30, 40–56.CrossRefGoogle Scholar
Gutknecht, M. H. (1993 b), ‘Stable row recurrences in the Padé table and generically superfast lookahead solvers for non-Hermitian Toeplitz systems’, Linear Algebra Appl. 188/189, 351–421.CrossRefGoogle Scholar
Gutknecht, M. H. (1993 c), ‘Variants of BiCGStab for matrices with complex spectrum’, SIAM J. Sci. Comput. 14, 1020–1033.CrossRefGoogle Scholar
Gutknecht, M. H. (1994 a), ‘A completed theory of the unsymmetric Lanczos process and related algorithms, Part II’, SIAM J. Matrix Anal. Appl. 15, 15–58.CrossRefGoogle Scholar
Gutknecht, M. H. (1994 b), The Lanczos process and Padé approximation, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 61–75.Google Scholar
Gutknecht, M. H. (1994 c), ‘Local minimum residual smoothing’, Talk at Oberwolfach, Germany.Google Scholar
Gutknecht, M. H. and Gragg, W. B. (1994), Stable look-ahead versions of the Euclidean and Chebyshev algorithms, IPS Research Report 94–04, IPS, ETH Zürich.Google Scholar
Gutknecht, M. H. and Hochbruck, M. (1995), ‘Look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems’, Numer. Math. 70, 181–227.CrossRefGoogle Scholar
Gutknecht, M. H. and Ressel, K. J. (1996), Look-ahead procedures for Lanczos-type product methods based on three-term recurrences, Tech. Report TR-96-19, Swiss Center for Scientific Computing.Google Scholar
Gutknecht, M. H. and Ressel, K. J. (1997), Attempts to enhance the BiCGStab family, Technical report, Swiss Center for Scientific Computing. In preparation.Google Scholar
Hanke, M. (1997), ‘Superlinear convergence rates for the Lanczos method applied to elliptic operators’, Numer. Math. To appear.CrossRefGoogle Scholar
Hayes, R. M. (1954), Iterative methods of solving linear problems on Hilbert space, in Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues (Taussky, O., ed.), Vol. 49 of Applied Mathematics Series, National Bureau of Standards, pp. 71–103.Google Scholar
Henrici, P. (1974), Applied and Computational Complex Analysis, Vol. 1, Wiley, New York.Google Scholar
Hestenes, M. R. (1951), Iterative methods for solving linear equations, NAML Report 52–9, National Bureau of Standards, Los Angeles, CA. Reprinted in J. Optim. Theory Appl. 11, 323–334 (1973).CrossRefGoogle Scholar
Hestenes, M. R. (1980), Conjugate Direction Methods in Optimization, Springer, Berlin.CrossRefGoogle Scholar
Hestenes, M. R. and Stiefel, E. (1952), ‘Methods of conjugate gradients for solving linear systems’, J. Res. Nat. Bur. Standards 49, 409–435.CrossRefGoogle Scholar
Hochbruck, M. (1992), Lanczos- und Krylov-Verfahren für nicht-Hermitesche lineare Systeme, PhD thesis, Fakultät für Mathematik, Universität Karlsruhe.Google Scholar
Hochbruck, M. (1996), ‘The Padé table and its relation to certain numerical algorithms’, Habilitationsschrift, Universität T¨bingen, Germany.Google Scholar
Hochbruck, M. and Lubich, C. (1997 a), ‘Error analysis of Krylov methods in a nutshell’, SIAM J. Sci. Comput. To appear.Google Scholar
Hochbruck, M. and Lubich, C. (1997 b), ‘On Krylov subspace approximations to the matrix exponential operator’, SIAM J. Numer. Anal. To appear.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (1985), Matrix Analysis, Cambridge University Press.CrossRefGoogle Scholar
Householder, A. S. (1964), The Theory of Matrices in Numerical Analysis, Dover, New York.Google Scholar
Jea, K. C. and Young, D. M. (1983), ‘On the simplification of generalized conjugate-gradient methods for nonsymmetrizable linear systems’, Linear Algebra Appl. 52, 399–417.CrossRefGoogle Scholar
Joubert, W. D. (1990), Generalized conjugate gradient and Lanczos methods for the solution of nonsymmetric systems of linear equations, PhD thesis, Center for Numerical Analysis, University of Texas at Austin. Tech. Rep. CNA-238.Google Scholar
Joubert, W. D. (1992), ‘Lanczos methods for the solution of nonsymmetric systems of linear equations’, SIAM J. Matrix Anal. Appl. 13, 926–943.CrossRefGoogle Scholar
Kaniel, S. (1966), ‘Estimates for some computational techniques in linear algebra’, Math. Comp. 20, 369–378.CrossRefGoogle Scholar
Khelifi, M. (1991), ‘Lanczos maximal algorithm for unsymmetric eigenvalue problems’, Appl. Numer. Math. 7, 179–193.CrossRefGoogle Scholar
Kreuzer, K., Miller, H. and Berger, W. (1981), ‘The Lanczos algorithm for self-adjoint operators’, Physics Letters 81A, 429–432.CrossRefGoogle Scholar
Kreyszig, E. (1978), Introductory Functional Analysis with Applications, Wiley.Google Scholar
Lanczos, C. (1950), ‘An iteration method for the solution of the eigenvalue problem of linear differential and integral operators’, J. Res. Nat. Bur. Standards 45, 255–281.CrossRefGoogle Scholar
Lanczos, C. (1952), ‘Solution of systems of linear equations by minimized iterations’, J. Res. Nat. Bur. Standards 49, 33–53.CrossRefGoogle Scholar
Luenberger, D. G. (1979), ‘Hyperbolic pairs in the method of conjugate gradients’, SIAM J. Appl. Math. 17, 1263–1267.CrossRefGoogle Scholar
Manteuffel, T. A. (1977), ‘The Tchebyshev iteration for nonsymmetric linear systems’, Numer. Math. 28, 307–327.CrossRefGoogle Scholar
Meijerink, J. A. and van der Vorst, H. A. (1977), ‘An iterative solution method for linear equations systems of which the coefficient matrix is a symmetric M-matrix’, Math. Comp. 31, 148–162.Google Scholar
Meijerink, J. A. and van der Vorst, H. A. (1981), ‘Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems’, J. Comput. Phys. 44, 134–155.CrossRefGoogle Scholar
Murray, W., ed. (1972), Numerical Methods for Unconstrained Optimization, Academic, London.Google Scholar
Nachtigal, N. M. (1991), A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems, PhD thesis, Department of Mathematics, MIT.Google Scholar
Neumaier, A. (1994), ‘Iterative regularization for large-scale ill-conditioned linear systems’, Talk at Oberwolfach.Google Scholar
Nevanlinna, O. (1993), Convergence of Iterations for Linear Equations, Birkhäuser, Basel.CrossRefGoogle Scholar
Paige, C. C. (1971), The computations of eigenvalues and eigenvectors of very large sparse matrices, PhD thesis, University of London.Google Scholar
Paige, C. C. (1976), ‘Error analysis of the Lanczos algorithms for tridiagonalizing asymmetric matrix’, J. Inst. Math. Appl. 18, 341–349.CrossRefGoogle Scholar
Paige, C. C. (1980), ‘Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem’, Linear Algebra Appl. 34, 235–258.CrossRefGoogle Scholar
Paige, C. C. and Saunders, M. A. (1975), ‘Solution of sparse indefinite systems of linear equations’, SIAM J. Numer. Anal. 12, 617–629.CrossRefGoogle Scholar
Parlett, B. N. (1980), The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Parlett, B. N. (1992), ‘Reduction to tridiagonal form and minimal realizations’, SIAM J. Matrix Anal. Appl. 13, 567–593.CrossRefGoogle Scholar
Parlett, B. N. (1994), Do we fully understand the symmetric Lanczos algorithm yet?, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 93–107.Google Scholar
Parlett, B. N. (1995), ‘The new qd algorithms’, in Acta Numerica, Vol. 4, Cambridge University Press, pp. 459–491.Google Scholar
Parlett, B. N. and Nour-Omid, B. (1989), ‘Towards a black box Lanczos program’, Computer Physics Comm. 53, 169–179.CrossRefGoogle Scholar
Parlett, B. N. and Reid, J. K. (1981), ‘Tracking the progress of the Lanczos algorithm for large symmetric matrices’, IMA J. Numer. Anal. 1, 135–155.CrossRefGoogle Scholar
Parlett, B. N. and Scott, D. S. (1979), ‘The Lanczos algorithm with selective reorthogonalization’, Math. Comp. 33, 217–238.CrossRefGoogle Scholar
Parlett, B. N., Taylor, D. R. and Liu, Z. A. (1985), ‘A look-ahead Lanczos algorithm for unsymmetric matrices’, Math. Comp. 44, 105–124.Google Scholar
Ressel, K. J. and Gutknecht, M. H. (1996), QMR-smoothing for Lanczos-type product methods based on three-term recurrences, Tech. Report TR-96-18, Swiss Center for Scientific Computing.Google Scholar
Rudin, W. (1973), Functional Analysis, McGraw-Hill, New York.Google Scholar
Ruhe, A. (1979), ‘Implementation aspects of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices’, Math. Comp. 33, 680–687.CrossRefGoogle Scholar
Rutishauser, H. (1953), ‘Beiträge zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos’, Z. Angew. Math. Phys. 4, 35–56.CrossRefGoogle Scholar
Rutishauser, H. (1957), Der Quotienten-Differenzen-Algorithmus, Mitt. Inst. angew. Math. ETH, Nr. 7, Birkhäuser, Basel.CrossRefGoogle Scholar
Rutishauser, H. (1990), Lectures on Numerical Mathematics, Birkhäuser, Boston.CrossRefGoogle Scholar
Saad, Y. (1980), ‘Variations on Arnoldi's method for computing eigenelements of large unsymmetric systems’, Linear Algebra Appl. 34, 269–295.CrossRefGoogle Scholar
Saad, Y. (1981), ‘Krylov subspace methods for solving large unsymmetric systems’, Math. Comp. 37, 105–126.CrossRefGoogle Scholar
Saad, Y. (1982), ‘The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems’, SIAM J. Numer. Anal. 2, 485–506.CrossRefGoogle Scholar
Saad, Y. (1994), Theoretical error bounds and general analysis of a few Lanczos-type algorithms, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 123–134.Google Scholar
Saad, Y. (1996), Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston.Google Scholar
Saad, Y. and Schultz, M. H. (1986), ‘GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems’, SIAM J. Sci. Statist. Comput. 7, 856–869.CrossRefGoogle Scholar
Schönauer, W. (1987), Scientific Computing on Vector Computers, Elsevier, Amsterdam.Google Scholar
Simon, H. D. (1984 a), ‘Analysis of the symmetric Lanczos algorithm with reorthogonalization methods’, Linear Algebra Appl. 61, 101–131.CrossRefGoogle Scholar
Simon, H. D. (1984 b), ‘The Lanczos algorithm with partial reorthogonalization’, Math. Comp. 42, 115–142.CrossRefGoogle Scholar
Sleijpen, G. L. G. and Fokkema, D. R. (1993), ‘BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum’, Electronic Trans. Numer. Anal. 1, 11–32.Google Scholar
Sleijpen, G. L. G. and van der Vorst, H. A. (1995 a), ‘Maintaining convergence properties of bicgstab methods in finite precision arithmetic’, Numerical Algorithms 10, 203–223.CrossRefGoogle Scholar
Sleijpen, G. L. G. and van der Vorst, H. A. (1995 b), ‘An overview of approaches for the stable computation of hybrid BiCG methods’, Appl. Numer. Math. 19, 235–254.CrossRefGoogle Scholar
Sleijpen, G. L. G. and van der Vorst, H. A. (1996), ‘Reliable updated residuals in hybrid Bi-CG methods’, Computing 56, 141–163.CrossRefGoogle Scholar
Sleijpen, G. L. G., van der Vorst, H. A. and Fokkema, D. R. (1994), ‘BiCGstab(l) and other hybrid Bi-CG methods’, Numerical Algorithms 7, 75–109.CrossRefGoogle Scholar
Sonneveld, P. (1989), ‘CGS, a fast Lanczos-type solver for nonsymmetric linear systems’, SIAM J. Sci. Statist. Comput. 10, 36–52.CrossRefGoogle Scholar
Stewart, G. W. (1994), Lanczos and linear systems, in Proceedings of the Cornelius Lanczos International Centenary Conference (Brown, J. D., Chu, M. T., Ellison, D. C. and Plemmons, R. J., eds), SIAM, Philadelphia, PA, pp. 135–139.Google Scholar
Stieltjes, T. J. (1884), ‘Quelques recherches sur la théorie des quadratures dites mécaniques’, Ann. Sci. École Norm. Paris Sér. 3 1, 409–426. [Oeuvres, vol. 1, pp. 377–396].CrossRefGoogle Scholar
Strako, Z.š (1991), ‘On the real convergence rate of the conjugate gradient method’, Linear Algebra Appl. 154–156, 535–549.Google Scholar
Taylor, D. R. (1982), Analysis of the Look Ahead Lanczos Algorithm, PhD thesis, Dept. of Mathematics, University of California, Berkeley.Google Scholar
Tong, C. H. (1994), ‘A family of quasi-minimal residual methods for nonsymmetric linear systems’, SIAM J. Sci. Comput. 15, 89–105.CrossRefGoogle Scholar
Tong, C. H. and Ye, Q. (1995), ‘Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems’. Preprint.Google Scholar
Tong, C. H. and Ye, Q. (1996), ‘A linear system solver based on a modified Krylov subspace method for breakdown recovery’, Numerical Algorithms 12, 233–251.CrossRefGoogle Scholar
van der Sluis, A. (1992), The convergence behaviour of conjugate gradients and Ritz values in various circumstances, in Iterative Methods in Linear Algebra (Beauwens, R. and de Groen, P., eds), Elsevier (North-Holland), Proceedings IMACS Symposium, Brussels, 1991, pp. 49–66.Google Scholar
van der Sluis, A. and van der Vorst, H. A. (1986), ‘The rate of convergence of conjugate gradients’, Numer. Math. 48, 543–560.CrossRefGoogle Scholar
van der Sluis, A. and van der Vorst, H. A. (1987), ‘The convergence behavior of Ritz values in the presence of close eigenvalues’, Linear Algebra Appl. 88/89, 651–694.CrossRefGoogle Scholar
van der Vorst, H. A. (1992), ‘Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems’, SIAM J. Sci. Statist. Comput. 13, 631–644.CrossRefGoogle Scholar
van der Vorst, H. A. and Vuik, C. (1993), ‘The superlinear convergence behaviour of GMRES’, J. Comput. Appl. Math. 48, 327–341.CrossRefGoogle Scholar
Walker, H. F. (1988), ‘Implementation of the GMRES method using Householder transformations’, SIAM J. Sci. Statist. Comput. 9, 152–163CrossRefGoogle Scholar
Walker, H. F. (1995), ‘Residual smoothing and peak/plateau behavior in Krylov subspace methods’, Appl. Numer. Math. 19, 279–286.CrossRefGoogle Scholar
Weiss, R. (1990), Convergence behavior of generalized conjugate gradient methods, PhD thesis, University of Karlsruhe.Google Scholar
Weiss, R. (1994), ‘Properties of generalized conjugate gradient methods’, J. Numer. Linear Algebra Appl. 1, 45–63.CrossRefGoogle Scholar
Wilkinson, J. H. (1965), The Algebraic Eigenvalue Problem, Clarendon Press, Oxford.Google Scholar
Ye, Q. (1991), ‘A convergence analysis for nonsymmetric Lanczos algorithms’, Math. Comp. 56, 677–691.CrossRefGoogle Scholar
Ye, Q. (1994), ‘A breakdown-free variation of the nonsymmetric Lanczos algorithms’, Math. Comp. 62, 179–207.CrossRefGoogle Scholar
Zhang, S.-L. (1997), ‘GPBI-CG: generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems’, SIAM J. Sci. Comput. 18, 537–551.CrossRefGoogle Scholar
Zhou, L. and Walker, H. F. (1994), ‘Residual smoothing techniques for iterative methods’, SIAM J. Sci. Comput. 15, 297–312.CrossRefGoogle Scholar