The serial Jacobi algorithm (either one-sided or two-sided) for the computation of a singular value decomposition (SVD) of a general matrix has excellent numerical properties and parallelization potential, but it is considered to be the slowest method for computing the SVD. Even its parallelization with some parallel cyclic (static) ordering of subproblems does not lead to much improvement when...

In an earlier paper, an approximate SVD updating scheme has been derived as an interlacing of a QR updating on the one hand and a Jacobi-type SVD procedure on the other hand, possibly supplemented with a certain re-orthogonalization scheme. In this paper, this updating algorithm is mapped onto a systolic array with O(n 2 ) parallelism for O(n 2 ) complexity, resulting in an O(n 0 ) throughput. ...

In this study, new highly parallel algorithm of two-sided Jacobi 8-D transformation is suggested. It is oriented on VLSI-implementation of special processor array. This array is built using 8-D CORDIC algorithm for quaternion valued matrix singular value decomposition. Accuracy analysis and simulation results are added. Such array can be utilized to speed up the Jacobi method realization to com...

This technical report is devoted to the description of implementation details of the accelerated parallel one-sided block Jacobi SVD algorithm, whose analysis and design was described in [21]. We provide discuss a suitable data layout for a parallel implementation of the algorithm on a parallel computer with distributed memory. This discussion is complicated by the fact that different computati...

We discuss a new method for the iterative computation of a portion of the singular values and vectors of a large sparse matrix. Similar to the Jacobi-Davidson method for the eigenvalue problem, we compute in each step a correction by (approximately) solving a correction equation. It is shown that this JDSVD method can be seen as an accelerated (inexact) Newton scheme. We compare the method with...

We show experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the Rfactor, can lead to a substantial decrease of the number of outer parallel 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 value...

A new dynamic ordering is presented for the parallel one-sided block Jacobi SVD algorithm. Similarly to the two-sided variant, which has been analyzed and implemented in last 10 years, the dynamic ordering takes into account the actual status of a matrix—this time of its block columns with respect to their mutual orthogonality. Using p processors, in each parallel iteration step the p mostly in...

In this paper, we introduce a joint hyperbolic-orthogonal decomposition of two matrices which we call a Generalized Hyperbolic SVD, or GHSVD. This decomposition can be used for nding the eigenvalues and eigenvectors of a symmetric indeenite pencil X T X ? Y T Y where = diag(1). We also present an implicit Jacobi-like method for computing this GHSVD.

The practical and accurate computation of the singular value decomposition of a quaternion matrix is of importance in vector signal processing using quaternions. We present a Jacobi algorithm for computing such an SVD, and discuss its utility and accuracy. The algorithm is included in an open-source Matlab toolbox for quaternions where it serves as an accurate reference implementation.