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PRECONDITIONED PARALLEL BLOCK–JACOBI SVD ALGORITHM

GABRIEL OKŠA, MARIÁN VAJTERŠIC
2006 Parallel Processing Letters  
iteration steps in the parallel block-Jacobi SVD algorithm, whereby the details depend on the condition number and on the shape of spectrum, including the multiplicity of singular values.  ...  However, the gain in speed, as measured by the total parallel execution time, depends decisively on how efficient is the implementation of the distributed QR and LQ factorizations on a given parallel architecture  ...  In Section 2 we briefly introduce the parallel two-sided block-Jacobi SVD algorithm with the dynamic ordering.  ... 
doi:10.1142/s0129626406002708 fatcat:7mhzkmrjqvhjvii5wje7oh4kva

The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale

Jack Dongarra, Mark Gates, Azzam Haidar, Jakub Kurzak, Piotr Luszczek, Stanimire Tomov, Ichitaro Yamazaki
2018 SIAM Review  
They have likewise had many developments, including parallel and block versions and preconditioning to improve convergence.  ...  There are two main branches of dense SVD methods: bidiagonalization and Jacobi.  ...  We thank Martin Be\v cka, Gabriel Ok\v sa, and Mari\' an Va-jter\v sic for use of their block Jacobi code; Osni Marques for assistance with the MRRR code; and the anonymous reviewers for feedback to improve  ... 
doi:10.1137/17m1117732 fatcat:yj7hqon24rbuddmk5fuvcx4z5a

Page 4052 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
, preconditioned Jacobi-Davidson for large, sparse symmetric matrices.  ...  Preconditioners as- sessed include incomplete Cholesky, block Jacobi, the factorized approximate inverse (AINV), and the factorized sparse approxi- mate inverse (FSAI).  ... 

Page 3908 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews  
This is done by supplementing the QR updating with a Jacobi-type SVD procedure, where apparently only a few SVD steps after each QR update suffice in order to restore an acceptable approximation for the  ...  SVD.  ... 

Batched QR and SVD Algorithms on GPUs with Applications in Hierarchical Matrix Compression [article]

Wajih Halim Boukaram, George Turkiyyah, Hatem Ltaief, David E. Keyes
2017 arXiv   pre-print
The one-sided Jacobi algorithm is used for its simplicity and inherent parallelism as a building block for the SVD of low rank blocks using randomized methods.  ...  We implement multiple kernels based on the level of the GPU memory hierarchy in which the matrices can reside and show substantial speedups against streamed cuSOLVER SVDs.  ...  The main benefit of the block Jacobi algorithm is its high degree of parallelism; however, since we implement a batched routine for independent operations, we will use the serial block Jacobi algorithm  ... 
arXiv:1707.05141v1 fatcat:2zteuq5hsje4xk6in2qhon5eju

Efficient pre-processing in the parallel block-Jacobi SVD algorithm

Gabriel Okša, Marián Vajteršic
2006 Parallel Computing  
One way, how to speed up the computation of the singular value decomposition of a given matrix A ∈ C m×n , m ≥ n, by the parallel two-sided block-Jacobi method, consists of applying some pre-processing  ...  Such a concentration should hopefully lead to fewer outer parallel iteration steps needed for the convergence of the entire algorithm.  ...  In Section 2 we briefly introduce the parallel two-sided block-Jacobi SVD algorithm with the dynamic ordering.  ... 
doi:10.1016/j.parco.2005.06.006 fatcat:o26vvhky5rcu3kkrp4klc2ozgi

PRIMME_SVDS: A High-Performance Preconditioned SVD Solver for Accurate Large-Scale Computations [article]

Lingfei Wu, Eloy Romero, Andreas Stathopoulos
2017 arXiv   pre-print
PRIMME SVDS fills a gap in production level software for computing the partial SVD, especially with preconditioning.  ...  The numerical experiments demonstrate its superior performance compared to other state-of-the-art software and its good parallel performance under strong and weak scaling.  ...  For the rest, the preconditioner is based on block Jacobi on A T A with block size limited to 600.  ... 
arXiv:1607.01404v2 fatcat:ueyajok7f5d65ayac2vorojxui

Page 5644 of Mathematical Reviews Vol. , Issue 99h [page]

1999 Mathematical Reviews  
Berry, Information filtering using the Riemannian SVD (R-SVD) (386-395).  ...  Gilbert, Combinatorial preconditioning for sparse linear systems (1-4); Eric A. Schweitz and Dharma P.  ... 

Special issue on parallel matrix algorithms and applications

Erricos John Kontoghiorghes, Ahmed Sameh, Denis Trystram
2002 Parallel Computing  
Papers presented at the workshop covered many aspects of parallel numerical linear algebra algorithms.  ...  , as well as parallel matrix algorithms that dominate performance in air pollution modelling codes.  ...  A method for the parallel computation of the singular value decomposition is proposed by Be c cka, Ok s sa and Vajter s sic in Dynamic Ordering for a Parallel Block-Jacobi SVD Algorithm.  ... 
doi:10.1016/s0167-8191(01)00133-8 fatcat:sslnrtnrwnckzpq32dn3g4x6bu

New Fast and Accurate Jacobi SVD Algorithm. II

Zlatko Drmač, Krešimir Veselić
2008 SIAM Journal on Matrix Analysis and Applications  
This paper presents new implementation of one-sided Jacobi SVD for triangular matrices and its use as the core routine in a new preconditioned Jacobi SVD algorithm, recently proposed by the authors.  ...  If used in the preconditioned Jacobi SVD algorithm, it delivers superior performance leading to the currently fastest method for computing SVD decomposition with high relative accuracy.  ...  Assembling the SVD of A is straightforward. In this report we unwrap the black-box and show how it performs in the framework of the new preconditioned Jacobi SVD algorithm [13] .  ... 
doi:10.1137/05063920x fatcat:iz5tyieoqjhc5onoiyhsjsvc5m

Page 5406 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
(B-PRL) On Kogbetliantz’s SVD algorithm in the presence of clusters. Linear Algebra Appl. 95 (1987), 135-160.  ...  J. (1-SUNYB-C) On one-sided Jacobi methods for parallel computation. SIAM J. Algebraic Discrete Methods 8 (1987), no. 4, 790-796.  ... 

New Fast and Accurate Jacobi SVD Algorithm. I

Zlatko Drmač, Krešimir Veselić
2008 SIAM Journal on Matrix Analysis and Applications  
This makes the bidiagonalization based SVD computation numerically inferior to the Jacobi SVD algorithm [13] .  ...  Our quest for a highly accurate and efficient SVD algorithm has led us to a new, superior variant of the Jacobi algorithm.  ...  Preconditioned Jacobi SVD algorithm. In case of m n, the QR factorization of A, A = Q R 0 , reduces the computational complexity of all classical SVD methods.  ... 
doi:10.1137/050639193 fatcat:l5nbjqgszneotjnx2lcdfvgx4e

A Global Convergence Proof for Cyclic Jacobi Methods with Block Rotations

Zlatko Drmač
2010 SIAM Journal on Matrix Analysis and Applications  
This solves a long standing open problem of convergence of block cyclic Jacobi methods.  ...  This paper introduces a globally convergent block (column-and row-) cyclic Jacobi method for diagonalization of Hermitian matrices and for computation of the singular value decomposition of general matrices  ...  Similar conclusion holds for the general case of the new preconditioned Jacobi SVD algorithm in [13] , [14] . (1) P T HP = LL * (Cholesky factorization with optional pivoting) (2) LW =ȖΣ one sided Jacobi  ... 
doi:10.1137/090748548 fatcat:xoajfbdoqzdjdojhstlx2oqsaa

Numerical Analysis 2000 Vol. III: Linear Algebra

2000 Journal of Computational and Applied Mathematics  
The behavior of algorithms under rounding errors was a great source of inspiration for the further development of perturbation theory.  ...  Simply stated: one had only to increase the order of the problem and to implement the well-known algorithms e ciently on modern computers.  ...  Tang consider reÿnement techniques for this way of preconditioning. This includes symbolic factorization algorithms, reorderings, and blocking techniques.  ... 
doi:10.1016/s0377-0427(00)00453-2 fatcat:6mkanjyjcbgzzj3ksbrg4dwxnm

Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications

Jun Zhang
2000 Computer Methods in Applied Mechanics and Engineering  
The preconditioned iterative methods consist of Krylov subspace accelerators and a powerful general purpose multilevel block ILU (BILUM) preconditioner.  ...  We conduct an experimental study on the behavior of several preconditioned iterative methods to solve nonsymmetric matrices arising from computational¯uid dynamics (CFD) applications.  ...  In all tables with numerical results,``'' is the size of the uniform blocks.``s'' and``p'' are parameters used in the double dropping strategy.``x'' is the SVD threshold value (see Algorithm 3.2).  ... 
doi:10.1016/s0045-7825(99)00345-x fatcat:4mww3kg6drgqlekfowegk5gr2u
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