zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Restarted block Lanczos bidiagonalization methods. (English) Zbl 1110.65027
Summary: The problem of computing a few of the largest or smallest singular values and associated singular vectors of a large matrix arises in many applications. This paper describes restarted block Lanczos bidiagonalization methods based on augmentation of Ritz vectors or harmonic Ritz vectors by block Krylov subspaces.
MSC:
65F15Eigenvalues, eigenvectors (numerical linear algebra)
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
Software:
TRLan
References:
[1]Baglama, J., Calvetti, D., Reichel, L.: IRBL: An implicitly restarted block Lanczos method for large-scale Hermitian eigenproblems. SIAM J. Sci. Comput. 24, 1650–1677 (2003) · Zbl 1044.65027 · doi:10.1137/S1064827501397949
[2]Baglama, J., Reichel, L.: Augmented implicitly restarted Lanczos bidiagonalization methods. SIAM J. Sci. Comput. 27, 19–42 (2005) · Zbl 1087.65039 · doi:10.1137/04060593X
[3]Berry, M.W., Dumais, S.T., O’Brien, G.W.: Using linear algebra for intelligent information retrieval. SIAM Rev. 37, 573–595 (1995) · Zbl 0842.68026 · doi:10.1137/1037127
[4]Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia, PA (1996)
[5]Björck, Å., Grimme, E., Van Dooren, P.: An implicit shift bidiagonalization algorithm for ill-posed systems. BIT 34, 510–534 (1994) · Zbl 0821.65023 · doi:10.1007/BF01934265
[6]Golub, G.H., Luk, F.T., Overton, M.L.: A block Lanczos method for computing the singular values and corresponding vectors of a matrix. ACM Trans. Math. Softw. 7, 149–169 (1981) · Zbl 0466.65022 · doi:10.1145/355945.355946
[7]Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd ed. Johns Hopkins University Press, Baltimore, MD (1996)
[8]Hochstenbach, M.E.: A Jacobi–Davidson type SVD method. SIAM J. Sci. Comput. 23, 606–628 (2001) · Zbl 1002.65048 · doi:10.1137/S1064827500372973
[9]Hochstenbach, M.E.: Harmonic and refined extraction methods for the singular value problem, with applications in least-squares problems. BIT 44, 721–754 (2004) · Zbl 1079.65047 · doi:10.1007/s10543-004-5244-2
[10]Jia, Z., Niu, D.: An implicitly restarted refined bidiagonalization Lanczos method for computing a partial singular value decomposition. SIAM J. Matrix Anal. Appl. 25, 246–265 (2003) · Zbl 1063.65030 · doi:10.1137/S0895479802404192
[11]Kokiopoulou, E., Bekas, C., Gallopoulos, E.: Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization. Appl. Numer. Math. 49, 39–61 (2004) · Zbl 1049.65027 · doi:10.1016/j.apnum.2003.11.011
[12]Lehoucq, R.B.: Implicitly restarted Arnoldi methods and subspace iteration. SIAM J. Matrix Anal. Appl. 23, 551–562 (2001) · Zbl 1004.65044 · doi:10.1137/S0895479899358595
[13]Luk, F.T., Qiao, S.: Rank-revealing decomposition of symmetric Toeplitz matrices. In: Luk, F.T. (ed.) Proc. of SPIE. Advanced Signal Processing Algorithms, vol. 2563, pp. 293–301 (1995)
[14]Morgan, R.B.: Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154–156, 289–309 (1991) · Zbl 0734.65029 · doi:10.1016/0024-3795(91)90381-6
[15]Morgan, R.B.: Restarted block GMRES with deflation of eigenvalues. Appl. Numer. Math. 54, 222–236 (2005) · Zbl 1074.65043 · doi:10.1016/j.apnum.2004.09.028
[16]Nagy, J.G.: Fast algorithms for the regularization of banded Toeplitz least squares problems. In: Luk, F.T. (ed.) Proc. of SPIE. Advanced Signal Processing Algorithms, Architecture, and Implementations. IV, vol. 2295, pp. 566–575 (1994)
[17]Paige, C.C., Parlett, B.N., van der Vorst, H.A.: Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2, 115–134 (1995) · Zbl 0831.65036 · doi:10.1002/nla.1680020205
[18]Parlett, B.: Misconvergence in the Lanczos algorithm. In: Cox, M.G., Hammarling, S. (eds.) Reliable Numerical Computation, pp. 7–24. Clarendon, Oxford (1990)
[19]Simon, H.D., Zha, H.: Low rank matrix approximation using the Lanczos bidiagonalization process with applications. SIAM J. Sci. Comput. 21, 2257–2274 (2000) · Zbl 0962.65038 · doi:10.1137/S1064827597327309
[20]Sorensen, D.C.: Numerical methods for large eigenvalue problems. Acta Numer. 11, 519–584 (2002) · Zbl 1105.65325 · doi:10.1017/S0962492902000089
[21]Vudoc, R., Im, E.-J., Yellick, K.A.: SPARSITY: Optimization framework for sparse matrix kernels. Int. J. High Perform. Comput. Appl. 18, 135–158 (2004) · doi:10.1177/1094342004041296
[22]Wijshoff, H.A.G.: Implementing sparse BLAS primitives on concurrent/vector processors: a case study. In: Gibbons, A., Spirakis, P.(eds.) Lectures in Parallel Computation, pp. 405–437. Cambridge University Press, Cambridge, UK (1993)
[23]Wu, K., Simon, H.: Thick-restarted Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 602–616 (2000) · Zbl 0969.65030 · doi:10.1137/S0895479898334605