An implicitly restarted refined bidiagonalization Lanczos method for computing a partial singular value decomposition

SIAM Journal on Matrix Analysis and Applications

Volumn 25, Issue 1, 2004, Pages 246-265

  Jia, Zhongxiao ^a   Niu, Datian ^b  

^a TSINGHUA UNIVERSITY   (China)

^b DALIAN UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Convergence; Exact shifts; Implicit restart; Refined Ritz vector; Refined shifts; Ritz value; Ritz vector; Singular value; Singular vector; The bidiagonalization Lanczos method; The refined bidiagonalization Lanczos method

Indexed keywords

EID: 1342288832     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479802404192     Document Type: Article

References (30)

1. Z. BAI, R. BARRET, D. DAY, J. DEMMEL, AND J. DONGARRA, Test Matrix Collection for Non-Hermitian Eigenvalue Problems, Technical Report CS-97-355, University of Tennessee, Knoxville, 1997. LAPACK Note #123. Data available online from http://math.nist.gov/MarketMatrix.
2. Z. BAI, J. DEMMEL, J. DONGARRA, A. RUHE, AND H. A. VAN DER VORST, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, PA, 2000.
3. Å. BJÖRCK, E. GRIMME, AND P. VAN DOOREN, An implicit shift bidiagonalization algorithm for ill-posed systems, BIT, 34 (1994), pp. 510-534.
4. Å. BJÖRCK, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, PA, 1996.
5. J. W. DEMMEL, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997.
6. I. S. DUFF, R. G. GRIMES, AND J. G. LEWIS, Sparse matrix test problems, ACM Trans. Math. Software, 15 (1989), pp. 1-14.
7. G. H. GOLUB AND W. KAHAN, Calculating the singular values and pseudo-inverse of a matrix, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp. 205-224.
8. G. H. GOLUB, F. T. LUK, AND M. L. OVERTON, A block Lanczos method for computing the singular values and singular vectors of a matrix, ACM Trans. Math. Software, 7 (1981), pp. 149-169.
9. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
10. Z. JIA, Some Numerical Methods for Large Unsymmetric Eigenproblems, Ph.D. thesis, Department of Mathematics, Bielefeld University, Bielefeld, Germany, 1994.
11. Z. JIA, The convergence of generalized Lanczos methods for large unsymmetric eigenproblems, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 843-862.
12. Z. JIA, Refined iterative algorithms based on Arnoldi's process for large unsymmetric eigenproblems, Linear Algebra Appl., 259 (1997), pp. 1-23.
13. Z. JIA, A refined iterative algorithm based on the block Arnoldi process for large unsymmetric eigenproblems, Linear Algebra Appl., 270 (1998), pp. 170-189.
14. Z. JIA, Polynomial characterizations of the approximate eigenvectors by the refined Arnoldi method and an implicitly restarted refined Arnoldi algorithm, Linear Algebra Appl., 287 (1999), pp. 191-214.
15. Z. JIA, Composite orthogonal projection methods for large matrix eigenproblems, Sci. China Ser. A, 42 (1999), pp. 577-585.
16. Z. JIA, A refined subspace iteration algorithm for large sparse eigenproblems, Appl. Numer. Math., 32 (2000), pp. 35-52.
17. Z. JIA, The refined harmonic Arnoldi method and an implicitly restarted refined algorithm for computing interior eigenpairs of large matrices, Appl. Numer. Appl., 42 (2002), pp. 489-512.
18. Z. JIA AND G. W. STEWART, An analysis of the Rayleigh-Ritz method for approximating eigen-spaces, Math. Comp., 70 (2001), pp. 637-647.
19. Z. JIA AND G. W. STEWART, On the Convergence of Ritz Values, Ritz Vectors and Refined Ritz Vectors, Technical Report TR-3986, Department of Computer Science, University of Maryland, College Park, MD, 1999. Also available online from http://www.es.umd.edu/̃stewart.
20. Z. JIA, A Theoretical Comparison of Two Classes of Projection Methods, technical report, Department of Applied Mathematics, Dalian University of Technology, Dalian, People's Republic of China, 2001.
21. R. M. LARSEN, Lanczos Bidiagonalization with Partial Reorthogonalization, technical report, Department of Computer Science, University of Aarhus, Aarhus, Denmark, 1998. Also available online from http://soi.stanford.edu/̃rmunk/PROPACK.
22. R. M. LARSEN, Combining Implicit Restarts and Partial Reorthogonalization in Lanczos Bidiagonalization, available online from http://soi.stanford.edu/̃rmunk/PROPACK.
23. R. B. LEHOUCQ, D. C. SORENSEN, AND C. YANG, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, PA, 1998.
24. R. B. MORGAN, Implicitly restarted GMRES and Arnoldi methods for nonsyrnmetric systems of equations, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1112-1135.
25. C. C. PAIGE AND M. A. SAUNDERS, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software, 8 (1982), pp. 43-71.
26. B. N. PARLETT, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998.
27. D. C. SORENSEN, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357-385.
28. G. W. STEWART, Matrix Algorithms. Volume II: Eigensystems, SIAM, Philadelphia, PA, 2001.
29. H. A. VAN DER VORST, Computational methods for large eigenvalue problems, in Handbook of Numerical Analysis, Vol. III, P. G. Ciarlet and J. L. Lions, eds., Elsevier, North-Holland, Amsterdam, 2002, pp. 3-179.
30. X. WANG AND H. ZHA, An Implicitly Restarted Bidiagonal Lanczos Method for Large-Scale Singular Value Problems, Technical Report 42472, Scientific Computing Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 1998.