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.
References
J. Baglama and L. Reichel, Augmented implicitly restarted Lanczos bidiagonalization methods, SIAM J. Sci. Comput. 27 (2005), no. 1, 19–42. 10.1137/04060593x MR2201173 1087.65039 J. Baglama and L. Reichel, Augmented implicitly restarted Lanczos bidiagonalization methods, SIAM J. Sci. Comput. 27 (2005), no. 1, 19–42. 10.1137/04060593x MR2201173 1087.65039
––––, Restarted block Lanczos bidiagonalization methods, Numer. Algorithms 43 (2006), no. 3, 251–272. 10.1007/s11075-006-9057-z MR2310941 1110.65027 ––––, Restarted block Lanczos bidiagonalization methods, Numer. Algorithms 43 (2006), no. 3, 251–272. 10.1007/s11075-006-9057-z MR2310941 1110.65027
––––, An implicitly restarted block Lanczos bidiagonalization method using Leja shifts, BIT 53 (2013), no. 2, 285–310. 10.1007/s10543-012-0409-x MR3123847 1269.65038 ––––, An implicitly restarted block Lanczos bidiagonalization method using Leja shifts, BIT 53 (2013), no. 2, 285–310. 10.1007/s10543-012-0409-x MR3123847 1269.65038
Å. Björck, E. Grimme and P. Van Dooren, An implicit shift bidiagonalization algorithm for ill-posed systems, BIT 34 (1994), no. 4, 510–534. 10.1007/bf01934265 Å. Björck, E. Grimme and P. Van Dooren, An implicit shift bidiagonalization algorithm for ill-posed systems, BIT 34 (1994), no. 4, 510–534. 10.1007/bf01934265
B. Boisvert, R. Pozo, K. Remington, B. Miller and R. Lipman, Matrix Market, available online at http://math.nist.gov/MatrixMarket/, 2004. http://math.nist.gov/MatrixMarket/ B. Boisvert, R. Pozo, K. Remington, B. Miller and R. Lipman, Matrix Market, available online at http://math.nist.gov/MatrixMarket/, 2004. http://math.nist.gov/MatrixMarket/
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. 10.1145/355945.355946 MR618511 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. 10.1145/355945.355946 MR618511
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. 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. 1171.65382 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. 1171.65382
Z. Jia and L. Elsner, Improving eigenvectors in Arnoldi's method, J. Comput. Math. 18 (2000), no. 3, 265–276. MR1765280 0958.65039 Z. Jia and L. Elsner, Improving eigenvectors in Arnoldi's method, J. Comput. Math. 18 (2000), no. 3, 265–276. MR1765280 0958.65039
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. 10.1137/s0895479802404192 MR2002911 1063.65030 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. 10.1137/s0895479802404192 MR2002911 1063.65030
––––, 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. 10.1137/080733383 ––––, 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. 10.1137/080733383
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. 10.1016/j.apnum.2003.11.011 MR2069388 1049.65027 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. 10.1016/j.apnum.2003.11.011 MR2069388 1049.65027
R. M. Larsen, Lanczos Bidiagonalization with Partial Reorthogonalization, DAIMI Report Series 27 (1998), no. 537, 1–101. 10.7146/dpb.v27i537.7070 R. M. Larsen, Lanczos Bidiagonalization with Partial Reorthogonalization, DAIMI Report Series 27 (1998), no. 537, 1–101. 10.7146/dpb.v27i537.7070
––––, Combining implicit restarts and partial reorthogonalization in Lanczos bidiagonalization. http://soi.stanford.edu/~rmunk/PROPACK http://soi.stanford.edu/~rmunk/PROPACK ––––, Combining implicit restarts and partial reorthogonalization in Lanczos bidiagonalization. http://soi.stanford.edu/~rmunk/PROPACK 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. 10.1016/j.cam.2013.09.066 MR3133342 1293.65057 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. 10.1016/j.cam.2013.09.066 MR3133342 1293.65057
B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998. 10.1137/1.9781611971163 0885.65039 B. N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 1998. 10.1137/1.9781611971163 0885.65039
Y. Saad, Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl. 34 (1980), 269–295. 10.1016/0024-3795(80)90169-x MR591435 0456.65017 Y. Saad, Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices, Linear Algebra Appl. 34 (1980), 269–295. 10.1016/0024-3795(80)90169-x MR591435 0456.65017
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. 10.1137/s1064827597327309 0962.65038 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. 10.1137/s1064827597327309 0962.65038
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. 10.1137/0613025 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. 10.1137/0613025
M. Stoll, A Krylov-Schur approach to the truncated SVD, Linear Algebra Appl. 436 (2012), no. 8, 2795–2806. 10.1016/j.laa.2011.07.022 MR2908597 1241.65040 M. Stoll, A Krylov-Schur approach to the truncated SVD, Linear Algebra Appl. 436 (2012), no. 8, 2795–2806. 10.1016/j.laa.2011.07.022 MR2908597 1241.65040
