Abstract
The Lanczos method can be generalized to block form to compute multiple eigenvalues without the need of any deflation techniques. The block Lanczos method reduces a general sparse symmetric matrix to a block tridiagonal matrix via a Gram–Schmidt process. During the iterations of the block Lanczos method an off-diagonal block of the block tridiagonal matrix may become singular, implying that the new set of Lanczos vectors are linearly dependent on the previously generated vectors. Unlike the single vector Lanczos method, this occurrence of linearly dependent vectors may not imply an invariant subspace has been computed. This difficulty of a singular off-diagonal block is easily overcome in non-restarted block Lanczos methods, see [12,30]. The same schemes applied in non-restarted block Lanczos methods can also be applied in restarted block Lanczos methods. This allows the largest possible subspace to be built before restarting. However, in some cases a modification of the restart vectors is required or a singular block will continue to reoccur. In this paper we examine the different schemes mentioned in [12,30] for overcoming a singular block for the restarted block Lanczos methods, namely the restarted method reported in [12] and the Implicitly Restarted Block Lanczos (IRBL) method developed by Baglama et al. [3]. Numerical examples are presented to illustrate the different strategies discussed.
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References
J. Baglama, D. Calvetti and L. Reichel, Iterative methods for the computation of a few eigenvalues of a large symmetric matrix, BIT 36 (1996) 400–421.
J. Baglama, D. Calvetti and L. Reichel, Fast Leja points, Electronic Trans. Numer. Anal. 7 (1998) 124–140.
J. Baglama, D. Calvetti, L. Reichel and A. Ruttan, Computation of a few close eigenvalues of a large matrix with application to liquid crystal modeling, J. Comput. Phys. 146 (1998) 203–226.
M.W. Berry, A survey of public domain Lanczos-based software, in: Proc. of the Cornelius Lanczos Internat. Centenary Conference (SIAM, 1994) pp. 332–334.
Å. Björck, Numerics of Gram-Schmidt orthogonalization, Linear Algebra Appl. 197/198 (1994) 297–316.
D. Calvetti, L. Reichel and D.C. Sorensen, An implicitly restarted Lanczos method for large symmetric eigenvalue problems, Electronic Trans. Numer. Anal. 2 (1994) 1–21.
F. Chatelin, Eigenvalues of Matrices (Wiley, Chichester, 1993).
J. Cullum and W.E. Donath, A block Lanczos algorithm for computing the q algebraically largest eigenvalues and a corresponding eigenspace for large, sparse symmetric matrices, in: Proc. of the 1974 IEEE Conf. on Decision and Control, New York, 1974, pp. 505–509.
J. Cullum and R.A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1 (Birkhäuser, Boston, 1985).
J.J. Dongarra, J. DuCroz, I.F. Duff and S. Hammarling, A set of level 3 basic linear algebra subprograms, ACM Trans. Math. Software 16 (1990) 1–17.
R.W. Freund, Templates for band Lanczos methods and for the complex symmetric Lanczos method, Numerical Analysis Manuscript No. 99–3–01, Bell Laboratories (January 1999).
G.H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, in: Mathematical Software, Vol. III, ed. J.R. Rice (Academic Press, New York, 1977) pp. 361–377.
G.H. Golub and C.F. Van Loan, Matrix Computations, 2nd ed. (Johns Hopkins Univ. Press, Baltimore, MD, 1989).
R.G. Grimes, J.L. Lewis and H.D. Simon, A shifted block Lanczos algorithm for solving sparse symmetric generalized eigenproblems, SIAM J. Matrix Anal. 15 (1994) 228–272.
W. Karush, An iterative method for finding characteristic vectors of a symmetric matrix, Pacific J. Math. 1 (1951) 233–248.
C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, National Bureau of Standards 45 (1950) 225–282.
R.B. Lehoucq, Analysis and implementation of an implicitly restarted Arnoldi iteration, Ph.D. thesis, Rice University, Houston (1995).
R.B. Lehoucq, S.K. Gray, D.H. Zhang and J.C. Light, Vibrational eigenstates of four-atom molecules: A parallel strategy employing the implicitly restarted Lanczos method, Comput. Phys. Comm. 109 (1998) 15–26.
R.B. Lehoucq and K.J. Maschhoff, Block Arnoldi Method, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, eds. Z. Bai, J. Demmel, J. Dongarra, A. Ruhe and H. Van der Vorst, SIAM, to be published.
R.B. Lehoucq and D.C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl. 17 (1996) 789–821.
R.B. Lehoucq, D.C. Sorensen, P.A. Vu and C. Wang, ARPACK: An implementation of an implicitly restarted Arnoldi method for computing some of the eigenvalues and eigenvectors of a large sparse matrix (1996), code available from Netlib in directory scalapack.
O.A. Marques, BLZPACK: A Fortran 77 implementation of the block Lanczos algorithm, code available at http://www.nersc.gov/~osni/marques_soitware.html.
R.B. Morgan and D.S. Scott, Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Statist. Comput. 7 (1986) 817–825.
C.C. Paige, Computational variants of the Lanczos method for the eigenproblem, J. Inst. Math. Appl. 10 (1972) 373–381.
B.N. Parlett, The rewards for maintaining semi-orthogonality among Lanczos vectors, Numer. Linear Algebra Appl. 1 (1992) 243–267.
B.N. Parlett, The Symmetric Eigenvalue Problem (SIAM, Philadelphia, PA, 1998).
B.N. Parlett and B. Nour-Omid, Towards a black box Lanczos program, Comput. Phys. Comm. 53 (1989) 169–179.
B.N. Parlett and D.S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comp. 33 (1979) 311–328.
L. Reichel and W.B. Gragg, Algorithm 686: FORTRAN subroutines for updating the QR decomposition of a matrix, ACM Trans. Math. Software 16 (1990) 369–377.
A. Ruhe, Implementation aspect of band Lanczos algorithms for computation of eigenvalues of large sparse symmetric matrices, Math. Comp. 33 (1979) 680–687.
Y. Saad, Numerical Methods for Large Eigenvalue Problems (Halstead Press, New York, 1992).
D.S. Scott, LASO2-FORTRAN implementation of the Lanczos process with selective orthogonalization, code and documentation available from Netlib.
G.L.G. Sleijpen and H.A. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (1996) 401–425.
D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992) 357–385.
K. Wu and H. Simon, Thick-restart Lanczos method for symmetric eigenvalue problems, LBNL Report 41412 (1998).
Q. Ye, An adaptive block Lanczos algorithm, Numer. Algorithms 12 (1996) 97–110.
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Baglama, J. Dealing with linear dependence during the iterations of the restarted block Lanczos methods. Numerical Algorithms 25, 23–36 (2000). https://doi.org/10.1023/A:1016646115432
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DOI: https://doi.org/10.1023/A:1016646115432