Taiwanese Journal of Mathematics

Improving Approximate Singular Triplets in Lanczos Bidiagonalization Method

Datian Niu and Jiana Meng

Full-text: Open access


Lanczos bidiagonalization method is the most popular method for computing some largest singular triplets of large matrices. In this method, $2m+1$ base vectors are generated from the $m$-step Lanczos bidiagonalization process, but only $2m$ of them are used to form the approximate singular vectors and one of them is not used. In this paper, we make two improvements on the classical Lanczos bidiagonalization method. Firstly, following Jia and Elsner's idea for eigenproblems [9], we form the new approximate singular vectors by minimizing the corresponding residual norms in subspaces generated by $2m+1$ base vectors to replace the old approximate singular vectors. Secondly, in the process of implicit restarting, we replace the classical exact shifts by new shifts based on the information of the new approximate singular vectors. The total extra cost of the new method can be neglected. Numerical experiments show that, after two improvements, the new method proposed in this paper performs much better than the classical Lanczos bidiagonalization method. It uses less restarts and CPU time to reach the desired convergence.

Article information

Taiwanese J. Math., Volume 20, Number 4 (2016), 943-956.

First available in Project Euclid: 1 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65F15: Eigenvalues, eigenvectors 15A18: Eigenvalues, singular values, and eigenvectors

Lanczos bidiagonalization method singular triplet minimizing residual norms implicit restart shift


Niu, Datian; Meng, Jiana. Improving Approximate Singular Triplets in Lanczos Bidiagonalization Method. Taiwanese J. Math. 20 (2016), no. 4, 943--956. doi:10.11650/tjm.20.2016.6194. https://projecteuclid.org/euclid.twjm/1498874499

Export citation


  • J. Baglama and L. Reichel, Augmented implicitly restarted Lanczos bidiagonalization methods, SIAM J. Sci. Comput. 27 (2005), no. 1, 19–42.
  • ––––, Restarted block Lanczos bidiagonalization methods, Numer. Algorithms 43 (2006), no. 3, 251–272.
  • ––––, An implicitly restarted block Lanczos bidiagonalization method using Leja shifts, BIT 53 (2013), no. 2, 285–310.
  • Å. Björck, E. Grimme and P. Van Dooren, An implicit shift bidiagonalization algorithm for ill-posed systems, BIT 34 (1994), no. 4, 510–534.
  • B. Boisvert, R. Pozo, K. Remington, B. Miller and R. Lipman, Matrix Market, available online at http://math.nist.gov/MatrixMarket/, 2004.
  • G. H. Golub, F. T. Luk and M. L. Overton, A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix, ACM Trans. Math. Software 7 (1981), no. 2, 149–169.
  • G. H. Golub and C. F. Van Loan, Matrix Computations, Forth edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, 2013.
  • V. Hernandez, J. E. Roman and A. Tomas, A robust and efficient parallel SVD solver based on restarted Lanczos bidiagonalization, Electron. Trans. Numer. Anal. 31 (2008), 68–85.
  • Z. Jia and L. Elsner, Improving eigenvectors in Arnoldi's method, J. Comput. Math. 18 (2000), no. 3, 265–276.
  • Z. Jia and D. Niu, An implicitly restarted refined bidiagonalization Lanczos method for computing a partial singular value decomposition, SIAM J. Matrix Anal. Appl. 25 (2003), no. 1, 246–265.
  • ––––, A refined harmonic Lanczos bidiagonalization method and an implicitly restarted algorithm for computing the smallest singular triplets of large matrices, SIAM J. Sci. Comput. 32 (2010), no. 2, 714–744.
  • E. Kokiopoulou, C. Bekas and E. Gallopoulos, Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization, Appl. Numer. Math. 49 (2004), no. 1, 39–61.
  • R. M. Larsen, Lanczos Bidiagonalization with Partial Reorthogonalization, DAIMI Report Series 27 (1998), no. 537, 1–101.
  • ––––, Combining implicit restarts and partial reorthogonalization in Lanczos bidiagonalization. http://soi.stanford.edu/~rmunk/PROPACK
  • D. Niu and X. Yuan, An implicitly restarted Lanczos bidiagonalization method with refined harmonic shifts for computing smallest singular triplets, J. Comput. Appl. Math. 260 (2014), 208–217.
  • B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998.
  • Y. Saad, Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl. 34 (1980), 269–295.
  • H. D. Simon and H. Zha, Low-rank matrix approximation using the Lanczos bidiagonalization process with applications, SIAM J. Sci. Comput. 21 (2000), no. 6, 2257–2274.
  • D. C. Sorensen, Implicit application of polynomial filters in a $k$-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992), no. 1, 357–385.
  • M. Stoll, A Krylov-Schur approach to the truncated SVD, Linear Algebra Appl. 436 (2012), no. 8, 2795–2806.