Singular Value Decomposition Problem Research Articles

Matrix Methods for Estimating the Coherence Functions from Estimates of the Cross-Spectral Density Matrix

It is shown that the usual method for estimating the coherence functions (ordinary, partial, and multiple) for a general multiple-input! multiple-output problem can be expressed as a modified form of Cholesky decomposition of the cross-spectral density matrix of the input and output records. The results can be equivalently obtained using singular value decomposition (SVD) of the cross-spectral density matrix. Using SVD suggests a new form of fractional coherence. The formulation as a SVD problem also suggests a way to order the inputs when a natural physical order of the inputs is absent.

Large-Scale Sparse Singular Value Computations

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.

Computing the singular value decomposition on a distributed system of vector processors

Jacobi methods for computing the singular value decomposition (SVD) of a matrix are ideally suited for multiprocessor environments due to their inherent parallelism. In this paper we show how a block version of the two-sided Jacobi method can be used to compute the SVD efficiently on a distributed architecture. We compare two variants of this method that differ mainly in the degree to which they diagonalize a given subproblem. The first method is a true block generalization of the scalar scheme in that each off-diagonal block is completely annihilated. The second method is a scalar Jacobi algorithm reorganized in such a manner that it conforms to the block decomposition of the problem. Experiments on the LCAP system at IBM Kingston show that the block Jacobi algorithm performs very well on a distributed system, especially when the processors have vector processing hardware. As an example, we were able to achieve a sustained performance of 159 MFlops on a 960 × 720 SVD problem using eight processors. A suprising outcome of these experiments is that the determining factor for the performance of the algorithm on high-performance architectures is the subproblem solver, not the communication overhead of the algorithm.

Self-Equivalent Flows Associated Decomposition

A family of flows which are continuous analogues of the constant and variable shift $QR$ algorithms for the singular value decomposition problem is presented, and it is shown that certain of these flows interpolate the $QR$ algorithm exactly. Here attention is not restricted to bidiagonal matrices; arbitrary rectangular matrices are considered.