Estimating the largest singular values of large sparse matrices via modified moments

1991 ◽  
Vol 1 (3) ◽  
pp. 353-373 ◽  
Author(s):  
Michael Berry ◽  
Gene Golub
Keyword(s):  
Sparse Matrices ◽  
Singular Values ◽  
Modified Moments ◽  
Large Sparse Matrices

Approximating dominant singular triplets of large sparse matrices via modified moments

1996 ◽  
Vol 13 (1) ◽  
pp. 123-152 ◽  
Author(s):  
Sowmini Varadhan ◽  
Michael W. Berry ◽  
Gene H. Golub
Keyword(s):  
Sparse Matrices ◽  
Modified Moments ◽  
Large Sparse Matrices

Large-Scale Sparse Singular Value Computations

1992 ◽  
Vol 6 (1) ◽  
pp. 13-49 ◽  
Author(s):  
Michael W. Berry
Keyword(s):  
Information Retrieval ◽  
Large Scale ◽  
Sparse Matrices ◽  
Singular Values ◽  
Singular Value ◽  
Practical Applications ◽  
Singular Vectors ◽  
Subspace Iteration ◽  
Value Decomposition ◽  
Large Sparse Matrices

We present four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture. We emphasize Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left- and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography. The target architectures for our implementations are the CRAY-2S/4–128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices. This research may help advance the development of future out-of-core sparse SVD methods, which can be used, for example, to handle extremely large sparse matrices 0 × (106) rows or columns associated with extremely large databases in query-based information-retrieval applications.

An Efficient Parallel Algorithm for Extreme Eigenvalues of Sparse Nonsymmetric Matrices

1992 ◽  
Vol 6 (1) ◽  
pp. 98-111 ◽  
Author(s):  
S. K. Kim ◽  
A. T. Chrortopoulos
Keyword(s):  
Shared Memory ◽  
Message Passing ◽  
Sparse Matrices ◽  
Data Locality ◽  
Main Memory ◽  
Global Memory ◽  
Global Communication ◽  
Step Method ◽  
Arnoldi Algorithm ◽  
Large Sparse Matrices

Main memory accesses for shared-memory systems or global communications (synchronizations) in message passing systems decrease the computation speed. In this paper, the standard Arnoldi algorithm for approximating a small number of eigenvalues, with largest (or smallest) real parts for nonsymmetric large sparse matrices, is restructured so that only one synchronization point is required; that is, one global communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory machine per each iteration is required. We also introduce an s-step Arnoldi method for finding a few eigenvalues of nonsymmetric large sparse matrices. This method generates reduction matrices that are similar to those generated by the standard method. One iteration of the s-step Arnoldi algorithm corresponds to s iterations of the standard Arnoldi algorithm. The s-step method has improved data locality, minimized global communication, and superior parallel properties. These algorithms are implemented on a 64-node NCUBE/7 Hypercube and a CRAY-2, and performance results are presented.

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