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AN IMPLICITLY RESTARTED BIDIAGONAL LANCZOS METHOD FOR LARGE-SCALE SINGULAR VALUE PROBLEMS

XIAOHUI WANG, HONGYUAN ZHA
1999 Recent Advances in Numerical Methods and Applications II  
As we know for eigenproblem, implicit restart is prefered 2 A New Implicitly Restarted Algorithm to explicit one, the same thing is true for singular value problem.  ...  After we get the bidiagonal Lanczos reduction B k of A, we can use the singular values of B k to approximate those of A.  ... 
doi:10.1142/9789814291071_0016 fatcat:gk7k4h77wrbrjldlnizosrr5ku

Thick-restarted joint Lanczos bidiagonalization for the GSVD [article]

Fernando Alvarruiz, Carmen Campos, Jose E. Roman
2022 arXiv   pre-print
The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov  ...  We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance.  ...  Funding for open access charge: CRUE-Universitat Politècnica de València.  ... 
arXiv:2206.03768v1 fatcat:kyqxlpeceray5gvgxoink37c6m

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

James Baglama, Lothar Reichel
2005 SIAM Journal on Scientific Computing  
New restarted Lanczos bidiagonalization methods for the computation of a few of the largest or smallest singular values of a large matrix are presented.  ...  Computed examples show the new methods to be competitive with available schemes.  ...  We would like thank Stratis Gallopoulos for a copy of the paper [17] , Michiel Hochstenbach and Efi Kokiopoulou for providing copies of their codes, and Costas Bekas, Michiel Hochstenbach, and Efi Kokiopoulou  ... 
doi:10.1137/04060593x fatcat:kq3ap3ptvnbp7bvx7qcn6dyyki

Page 444 of Mathematical Reviews Vol. , Issue 96a [page]

1996 Mathematical Reviews  
Here, a two-way chasing scheme is introduced for the bidiagonalization step in downdating the singular-value decomposition.  ...  scheme for downdating the — singular-value decomposition.  ... 

An implicitly restarted block Lanczos bidiagonalization method using Leja shifts

James Baglama, Lothar Reichel
2012 BIT Numerical Mathematics  
In this paper, we propose an implicitly restarted block Lanczos bidiagonalization (IRBLB) method for computing a few extreme or interior singular values and associated right and left singular vectors of  ...  The method neither requires factorization of A nor of a matrix that contains A. This makes the method well suited for very large matrices.  ...  Acknowledgment We would like to thank the referees for carefully reading the paper and for comments that improved the presentation. Research in part supported by NSF grant DMS-1115385.  ... 
doi:10.1007/s10543-012-0409-x fatcat:jna6pko5yjbwhd6wtsx7ywlegy

Computation- and space-efficient implementation of SSA

Anton Korobeynikov
2010 Statistics and its Interface  
We outline several state-of-the-art algorithms including the Lanczos-based truncated Singular Value Decomposition (SVD) which can be modified to exploit the structure of the trajectory matrix.  ...  It is shown that the use of the general-purpose "blackbox" routines which can be found in packages like LAPACK leads to a huge waste of time since the Hankel structure of the trajectory matrix is not taken  ...  Golyandina and anonymous referees for thoughtful suggestions and comments that led to the significantly improved presentation of the paper. Received 30 November 2009  ... 
doi:10.4310/sii.2010.v3.n3.a9 fatcat:34nojpwakjgqvbrzc7suagfi7e

Parallel Bidiagonalization of a Dense Matrix

Carlos Campos, David Guerrero, Vicente Hernández, Rui Ralha
2007 SIAM Journal on Matrix Analysis and Applications  
A new stable method for the reduction of rectangular dense matrices to bidiagonal form has been proposed recently.  ...  In this paper we present such a modification. A block organization of the algorithm to use level 3 BLAS routines seems difficult and, at least for the moment, it relies upon level 2 BLAS routines.  ...  The values σ 1 ≥ · · · ≥ σ n ≥ 0 are called the singular values of A.  ... 
doi:10.1137/05062809x fatcat:mdfybt6nd5aptbo2lrkfm5ja3y

Computation- and Space-Efficient Implementation of SSA [article]

Anton Korobeynikov
2010 arXiv   pre-print
We outline several state-of-the-art algorithms (for example, Lanczos-based truncated SVD) which can be modified to exploit the structure of the trajectory matrix.  ...  The computational complexity of different steps of the basic SSA is discussed.  ...  Golyandina and anonymous reviewer for thoughtful suggestions and comments that let to significantly improved presentation of the article.  ... 
arXiv:0911.4498v2 fatcat:nqzs365qobhhnheios2ma3nzpe

FPGA-Based Co-processor for Singular Value Array Reconciliation Tomography

Jack Coyne, David Cyganski, R. James Duckworth
2008 2008 16th International Symposium on Field-Programmable Custom Computing Machines  
This thesis describes a co-processor system that has been designed to accelerate computations associated with Singular Value Array Reconciliation Tomography (SART), a method for locating a wide-band RF  ...  Compared to a Pentium 4 CPU running at 3 GHz, use of the co-processor system provides a speed-up of about 6 times for the current signal matrix size of 128-by-16.  ...  Therefore, singular value decomposition must be performed once for each location on the scan-grid.  ... 
doi:10.1109/fccm.2008.35 dblp:conf/fccm/CoyneCD08 fatcat:wem3ipr3zfaqjo4uilwoj47ch4

Singular Value Decomposition Approximation via Kronecker Summations for Imaging Applications [article]

Clarissa Garvey, Chang Meng, James G. Nagy
2018 arXiv   pre-print
In this paper we propose an approach to approximate a truncated singular value decomposition of a large structured matrix.  ...  By first decomposing the matrix into a sum of Kronecker products, our approach can be used to approximate a large number of singular values and vectors more efficiently than other well known schemes, such  ...  In this work, we explore an approach to compute approximations of the largest singular values and corresponding singular vectors of a large scale matrix K.  ... 
arXiv:1803.11525v2 fatcat:lvvv4j47sfeq3chmxcx66p2uiq

Bidiagonalization and R-Bidiagonalization: Parallel Tiled Algorithms, Critical Paths and Distributed-Memory Implementation

Mathieu Faverge, Julien Langou, Yves Robert, Jack Dongarra
2017 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS)  
of the initial matrix, then performing the band-bidiagonalization of the Rfactor.  ...  of the standard scalar bidiagonalization algorithm; and (ii) the R-bidiagonalization algorithm R-BIDIAG, which is a tiled version of the algorithm which consists in first performing the QR factorization  ...  In many algorithms, the bidiagonal form is a critical step to compute the singular value decomposition (SVD) of a matrix.  ... 
doi:10.1109/ipdps.2017.46 dblp:conf/ipps/FavergeLRD17 fatcat:sn2qcahlbbgiroxspkamkxzfyi

CPU-GPU hybrid bidiagonal reduction with soft error resilience

Yulu Jia, Piotr Luszczek, George Bosilca, Jack J. Dongarra
2013 Proceedings of the Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems - ScalA '13  
Soft errors manifest themselves as bit-flips that alter the user value, and numerical software is a category of software that is sensitive to such data changes.  ...  In this paper, we present a design of a bidiagonal reduction algorithm that is resilient to soft errors, and we also describe its implementation on hybrid CPU-GPU architectures.  ...  INTRODUCTION Bidiagonalization of a general M × N matrix A is prerequisite to computing the singular value decomposition (SVD) of A.  ... 
doi:10.1145/2530268.2530270 dblp:conf/sc/JiaLBD13 fatcat:fnzowqdodzhz5bl57mwkeujere

Bidiagonalization with Parallel Tiled Algorithms [article]

Mathieu Faverge and Julien Langou and Yves Robert and Jack Dongarra
2016 arXiv   pre-print
R-BiDiag, which is a tiled version of the algorithm which consists in first performing the QR factorization of the initial matrix, then performing the band-bidiagonalization of the R-factor.  ...  We provide experiments on a single multicore node, and on a few multicore nodes of a parallel distributed shared-memory system, to show the superiority of the new algorithms on a variety of matrix sizes  ...  We note that going to bidiagonal form is not a necessary step to compute the singular value decomposition of a matrix.  ... 
arXiv:1611.06892v1 fatcat:2sfb75c4xzhrtfzyzqdqlahyyy

Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization

E. Kokiopoulou, C. Bekas, E. Gallopoulos
2004 Applied Numerical Mathematics  
A matrix-free algorithm, IRLANB, for the efficient computation of the smallest singular triplets of large and possibly sparse matrices is described.  ...  Key characteristics of the approach are its use of Lanczos bidiagonalization, implicit restarting, and harmonic Ritz values.  ...  A first version of this work was developed in the context of the first author's diploma thesis [26] and presented during the International Workshop on Parallel Matrix Algorithms and Applications (PMAA  ... 
doi:10.1016/j.apnum.2003.11.011 fatcat:ikp5yyqs2vbxxomqd5uz4jhdqe

One-sided reduction to bidiagonal form

Rui Ralha
2003 Linear Algebra and its Applications  
In other papers we have shown that a method based upon this idea may become a serious competitor (in terms of speed) for computing the singular values of large matrices and also that it is well suited  ...  Nevertheless, we give examples of ill-conditioned matrices for which we have been able to produce a bidiagonal form whose singular values are much more accurate than the ones computed with the standard  ...  I acknowledge the support of the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI.  ... 
doi:10.1016/s0024-3795(01)00569-9 fatcat:35jvcaejqjejzdlqxmxk6wzgle
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