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On Parallel Implementation of the Onesided Jacobi Algorithm for Singular Value Decompositions
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A Parallel Ring Ordering Algorithm for Efficient Onesided Jacobi SVD Computations
"... 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 ..."
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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
Singular Value Decomposition and Its Application to AutoRegressive Parametric Spectral Estimation
, 1991
"... During recent years much interest has been given to the application of Singular Value Decomposition in association with extendedorder and overdetermined evaluation in the finite parametric spectral estimation domain. Such approaches have been shown to perform superior to other methods for disconti ..."
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During recent years much interest has been given to the application of Singular Value Decomposition in association with extendedorder 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 widebanded AR processes. It is found that the socalled 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...
On JacobiLike Algorithms for Computing the Ordinary Singular Value Decomposition
, 1991
"... The increasing interest for using the OSVD in the realtime DSP domain necessitates an efficient computation of the OSVD. Special interest has been given to Jacobilike algorithms which also is the case in this paper. After a description of the basic orthogonal transformations, algorithms for comp ..."
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The increasing interest for using the OSVD in the realtime DSP domain necessitates an efficient computation of the OSVD. Special interest has been given to Jacobilike 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 Jacobilike algorithms enabling concurrent computation are described and compared. It is found that the wellknown cyclicbyrow scheme is the most suited for realtime DSP applications and it is shown that this scheme allows for concurrent implementations. Finally, some 6 Jacobilike 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...
Recent Developments in Dense Numerical Linear Algebra
, 1997
"... 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 comp ..."
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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 highperformance computers.
A Parallel Ring Ordering Algorithm for E cient Onesided Jacobi SVD Computations
"... In this paper we give evidence to show that in onesided 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 ..."
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In this paper we give evidence to show that in onesided 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 onesided 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 onesided 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
Generalization of the . . . the OneSided Block Jacobi SVD Algorithm: II. Implementation
, 2008
"... We have designed, implemented and tested (by simulation on a serial computer) the new dynamic ordering for the parallel onesided blockJacobi 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 ..."
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We have designed, implemented and tested (by simulation on a serial computer) the new dynamic ordering for the parallel onesided blockJacobi 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 JordanWielandt matrix with zero diagonal blocks, 21block X T Y and 12block Y T X; the order of the JordanWielandt 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 matrixvector 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 onesided 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.
Parallel OneSided . . . Algorithm: I. Analysis and Design
, 2007
"... 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 onesided Jacobi method (coupled with some ord ..."
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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 onesided 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 onesided 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 divideandconquer 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 sinecosine decomposition of certain matrix blocks. The possible parallelization strategy for the onesided blockJacobi algorithm is also discussed.