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Parallel Numerical Linear Algebra
, 1993
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illust ..."
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Cited by 542 (26 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, and the singular value decomposition. We consider dense, band and sparse matrices.
An SVD Updating Algorithm for Subspace Tracking
, 1992
"... . In this paper, we extend the well known QRupdating scheme to a similar but more versatile and generally applicable scheme for updating the singular value decomposition (SVD). This is done by supplementing the QRupdating with a Jacobitype SVD procedure, where apparently only a few SVD steps afte ..."
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Cited by 31 (9 self)
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. In this paper, we extend the well known QRupdating scheme to a similar but more versatile and generally applicable scheme for updating the singular value decomposition (SVD). This is done by supplementing the QRupdating with a Jacobitype SVD procedure, where apparently only a few SVD steps after each QRupdate suffice in order to restore an acceptable approximation for the SVD. This then results in a reduced computational cost, comparable to the cost for merely QRupdating. We examine the usefulness of such an approximate updating scheme when applied to subspace tracking. It is shown how an O(n 2 ) SVD updating algorithm can restore an acceptable approximation at every stage, with a fairly small tracking error of approximately the time variation in O(n) time steps. Finally, an error analysis is performed, proving that the algorithm is stable, when supplemented with a Jacobitype reorthogonalization procedure, which can easily be incorporated into the updating scheme. Key wor...
Computing the Generalized Singular Value Decomposition
 SIAM J. Sci. Comput
, 1991
"... We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The first is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second ..."
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Cited by 19 (1 self)
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We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The first is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new 2 \Theta 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm. We present proofs of stability and high accuracy of the 2 \Theta 2 GSVD algorithm, and demonstrate it using examples on which all previous algorithms fail. 1 Introduction The purpose of this paper is to describe a variation of Paige's algorithm [28] for computing the following generalized singular value decomposition (GSVD) introduced by Van Loan [33], and Paige and Saunders [25]. This is also called the quotient singular value decomposition (QSVD) in [8]. Theorem 1.1 Let A 2 IR m\Thetan and B 2 IR p\Thetan have rank(A T ; B T ) = n. 1 Then there are orthogonal matrices U , V and Q su...
On the Parallel Implementation of Jacobi and Kogbetliantz Algorithms
"... Modified Jacobi and Kogbetliantz algorithms are derived by combining methods for modifying the orthogonal rotations. These methods are characterized by the use of approximate orthogonal rotations and the factorization of these rotations. The presented new approximations exhibit better properties and ..."
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Cited by 14 (8 self)
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Modified Jacobi and Kogbetliantz algorithms are derived by combining methods for modifying the orthogonal rotations. These methods are characterized by the use of approximate orthogonal rotations and the factorization of these rotations. The presented new approximations exhibit better properties and require less computational cost than known approximations. Suitable approximations are applied together with factorized rotation schemes in order to gain square root free or square root and division free algorithms. The resulting approximate and factorized rotation schemes are highly suited for parallel implementations. The convergence of the algorithms is analyzed and an application in signal processing is discussed.
The CSD, GSVD, their Applications and Computations
 University of Minnesota
, 1992
"... Since the CS decomposition (CSD) and the generalized singular value decomposition (GSVD) emerged as the generalization of the singular value decomposition about fifteen years ago, they have been proved to be very useful tools in numerical linear algebra. In this paper, we review the theoretical and ..."
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Cited by 10 (0 self)
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Since the CS decomposition (CSD) and the generalized singular value decomposition (GSVD) emerged as the generalization of the singular value decomposition about fifteen years ago, they have been proved to be very useful tools in numerical linear algebra. In this paper, we review the theoretical and numerical development of the decompositions, discuss some of their applications and present some new results and observations. We also point out some open problems. A Fortran 77 code has been written that computes the CSD and the GSVD. Keywords: singular value decomposition, CS decomposition, generalized singular value decomposition. Subject Classifications: AMS(MOS): 65F30; CR:G1.3 1 Introduction The singular value decomposition (SVD) of a matrix is one of the most important tools in numerical linear algebra. It has been widely used in scientific computing. Recently, Stewart [52] gave an excellent survey on the early history of the SVD back to the contributions of E. Beltrami and C. Jord...
The QR decomposition and the singular value decomposition in the symmetrized maxplus algebra
, 1998
"... ..."
Monitoring the Stage of Diagonalization in JacobiType Methods
 In Int. Conf. on Acoust., Speech and Signal Processing
, 1994
"... ..."
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 comput ..."
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Cited by 2 (2 self)
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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...