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Abstract not found

1 Introduction Let A be a real m \Theta n matrix. Without loss of generality we assume that m * n. The singular value decomposition (SVD) of A is its factorisation into a product of three matrices

During recent years much interest has been given to the application of Singular Value Decomposition in association with extended--order and overdetermined evaluation in the finite parametric spectral estimation domain. Such approaches have been shown to perform superior to other methods for discontinuous frequency signal, e.g. the harmonic retrieval problem. In this report a similar approach is applied to wide--banded AR processes. It is found that the so--called extraneous poles of the lower rank solution spoil the spectral estimate. A new approach, the direction weighted total least squares solution, which enforces the extraneous poles to be located at the origin while maintaining the good properties of the aforementioned approaches, is therefore introduced. Computer simulation experiments clearly indicate that this approach is superior to existing overdetermined and extended-- or parsimonic order methods. The author is currently being supported by a grant from the Danish Technical...

The increasing interest for using the OSVD in the real--time DSP domain necessitates an efficient computation of the OSVD. Special interest has been given to Jacobi--like algorithms which also is the case in this paper. After a description of the basic orthogonal transformations, algorithms for computing the OSVD are classified and shortly described. Various rotation schemes for Jacobi--like algorithms enabling concurrent computation are described and compared. It is found that the well--known cyclic--by--row scheme is the most suited for real--time DSP applications and it is shown that this scheme allows for concurrent implementations. Finally, some 6 Jacobi--like algorithms, including a new one presented here, are described and compared in detail. The differences of the various algorithms can be summarized in four. (i) The assumed structure of the matrix. (ii) How the rotation formula is expressed. (iii) The applied rotation scheme. (iv) How the result is delivered. All four items ar...

We survey recent developments in dense numerical linear algebra, covering linear systems, least squares problems and eigenproblems. Topics considered include the design and analysis of block, partitioned and parallel algorithms, condition number estimation, componentwise error analysis, and the computation of practical error bounds. Frequent reference is made to LAPACK, the state of the art package of Fortran software designed to solve linear algebra problems efficiently and accurately on high-performance computers.

In this paper we give evidence to show that in one-sided Jacobi SVD computation the sorting of column norms in each sweep is very important. An e cient parallel ring Jacobi ordering for computing singular value decomposition is described. This ordering can generate n(n 1)=2 di erent index pairs and sort column norms at the same time. The one-sided Jacobi SVD algorithm using this parallel ordering converges in about the same number of sweeps as the sequential cyclic Jacobi algorithm. The issue of equivalence of orderings for one-sided Jacobi is also discussed. We show how an ordering which does not sort column norms into order may still perform e ciently as long as it can generate the same index pairs at the same step as one which does sorting. Some experimental results on a Fujitsu AP1000 are presented. 1

We have designed, implemented and tested (by simulation on a serial computer) the new dynamic ordering for the parallel one-sided block-Jacobi SVD algorithm. Our idea is based on the estimation of the cosines of principal angles between two block columns X and Y of the same width without explicitly forming the matrix product X T Y (or Y T X) and computing its SVD. Instead, we propose to use a fixed number 2q of iterations in the Lanczos algorithm applied to the symmetric 2x2 block Jordan-Wielandt matrix with zero diagonal blocks, 21-block X T Y and 12-block Y T X; the order of the Jordan-Wielandt matrix is the sum of the block column widths. However, the matrix blocks X T Y and Y T X are never formed explicitly; the needed matrix-vector multiplications are computed exchanging intermediate product vectors between two processors that host the block column X and Y. After computing 2q iterations, the Frobenius norm of an auxiliary tridiagonal matrix of order 2q estimates the square root of twice the sum of squares of q largest cosines (representing q smallest principal angles) between X and Y. In the parallel algorithm using p processors, these weights can be used for choosing p pairs of block columns, which are far from orthogonality with respect to those q smallest angles. We show how to implement this new parallel ordering in the distributed paradigm of parallel computing using the Message Passing Interface (MPI). First numerical results obtained by simulation show that the one-sided parallel dynamic ordering can lead to a substantial decrease of the number of parallel iteration steps needed for the convergence as compared to a cyclic ordering.

The computation of a singular value decomposition of an m × n matrix A is certainly one of the most often demanded tasks in various applications. There are many algorithms for computing the full or partial singular value decomposition. Among them, the one-sided Jacobi method (coupled with some orderings) is reputable for its ability to compute the singular values as well as left and right singular vectors with high relative accuracy. This is important, for example, in applications like quantum physics or chemistry, where the atomic and/or molecular energies of tiny values have to be computed very accurately (these energies are modeled as the eigenvalues of symmetric operators, thus they equal to singular values). Unfortunately, the Jacobi method belongs also to the slowest algorithms, and as such has been almost abandoned. Recently, some new ideas for accelerating the one-sided serial Jacobi algorithm were presented and implemented. Numerical experiments have shown that the modified Jacobi algorithm is as fast as the QR algorithm and slightly slower than the divide-and-conquer one. We describe in detail main ideas of an acceleration, namely, working with matrix blocks rather than elements, the preprocessing of an original matrix, the special initialization procedure, the new matrix recursion and the sine-cosine decomposition of certain matrix blocks. The possible parallelization strategy for the one-sided block-Jacobi algorithm is also discussed.