Takagi factorization or symmetric singular value decomposition is a special form of SVD applicable to symmetric complex matrices. The computation takes advantage of symmetry to reduce computation and storage requirements. The Jacobi method with chess tournament ordering was used to perform the computation in parallel on a GPU using the CUDA programming model. We were able to achieve speedups of...

Recent progress in the serial one-sided Jacobi method is the consequence of two main ideas. The first one is that of preconditioning of an original matrix by one (two) QR (and LQ) decomposition(s) with column pivoting. Drmač and Veselić [1] have shown (experimentally and, to some degree, also theoretically) that such a preconditioning leads to a significant concentration of an off-diagonal matr...

Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree n. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are sa...

Lyapunov exponents can be estimated by accurately computing the singular values of long products of matrices, with perhaps 1000 or more factor matrices. These products have extremely large ratios between the largest and smallest eigenvalues. A variant of Rutishauser’s Cholesky LR algorithm for computing eigenvalues of symmetric matrices is used to obtain a new algorithm for computing the singul...

One way, how to speed up the computation of the singular value decomposition of a given matrix A ∈ Cm×n, m ≥ n, by the parallel two-sided block-Jacobi method, consists of applying some pre-processing steps that would concentrate the Frobenius norm near the diagonal. Such a concentration should hopefully lead to fewer outer parallel iteration steps needed for the convergence of the entire algori...

The increasing popularity of Singular Value Decomposition Algorithms, used in real time signal processing, demands a rapid development of their fast and reliable implementations. This paper shows several modiications to the Jacobi-like parallel algorithm for Singular Value Decomposition (SVD) and their impact on the algorithm's performance. In particular, the optimisations for the parallel memo...

Recently the new field of parallel imaging accusation has the potential to revolutionize the field of fast MR imaging. The SPACE RIP technique is one of the parallel imaging methods which uses multiple receiver coil and utilizes the sensitivity profile information from a number of receiver coils in order to reduce the acquisition time. The image reconstruction problem of SPACE RIP is a computat...