An algorithm and a stability theory for downdating the ULV decomposition

Barlow, Jesse L.; Yoon, Peter A.; Zha, Hongyuan

doi:10.1007/bf01740542

Cited by 22 publications

(15 citation statements)

References 24 publications

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“…In step 5 of our algorithm we eliminate the extra row by Givens rotations. Let Q = Q k Q k+1 : : : Q n be the product of rotations that eliminates last n k elements of the vector (F (1) G (2) ) T e n k as in Figure 4. Then de…ne…”

Section: Algorithmmentioning

confidence: 99%

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An Alternative Algorithm for a Sliding Window ULV Decomposition

Erbay

Barlow

2005

Computing

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The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be modi…ed much faster than the SVD.When modi…ying the ULVD, the accurate computation of the subspaces is required in certain time varying applications in signal processing. In this paper we present an updating algorithm which is suitable for large scaled matrices of low rank and as e¤ective as alternatives. The algorithm runs in O(n 2 ) time.

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Section: Algorithmmentioning

confidence: 99%

“…The most familiar formulation is due to Stewart [14]. A slightly di¤erent formulation is given below by Barlow, Yoon and Zha [2].…”

Section: Introductionmentioning

confidence: 99%

An Alternative Algorithm for a Sliding Window ULV Decomposition

Erbay

Barlow

2005

Computing

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“…Unfortunately, this so-called Linpack procedure is numerically inferior when q 2 is close to one, in which case it is safer to use algorithms based on the corrected semi-normal equation (CSNE) approach [7]. Two versions of this approach have been developed: the first by Bojanczyk and Lebak [8], and the second by Park and Eldén [45] and further improved by Barlow et al [3]. It is outside the scope of this work to present the details of these sophisticated downdating algorithms; instead we refer to the papers for details.…”

Section: Downdatingmentioning

confidence: 99%

“…These algorithms, in turn, are adapted from the downdating algorithms presented in the unpublished report [36]. They are not as sophisticated as the algorithms in [3,45], but more research is necessary to extend the latter algorithms to the ULLV decomposition. When A is downdated, then the matrix U A is first augmented with an additional column u 2 that is orthonormal to the columns of U A in such a way that the norm of the first row of (U A , u 2 ) is one.…”

Section: Downdating Algorithmsmentioning

confidence: 99%

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Fierro¹,

Hansen²,

Hansen³

1999

Numerical Algorithms

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We describe a Matlab 5.2 package for computing and modifying certain rank-revealing decompositions that have found widespread use in signal processing and other applications. The package focuses on algorithms for URV and ULV decompositions, collectively known as UTV decompositions. We include algorithms for the ULLV decomposition, which generalizes the ULV decomposition to a pair of matrices. For completeness a few algorithms for computation of the RRQR decomposition are also included. The software in this package can be used as is, or can be considered as templates for specialized implementations on signal processors and similar dedicated hardware platforms.

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“…Here Φ(n) is a modestly growing function of n, say, √ n. This formulation is a variant of the one in [24,8,4]. We refer to k as the -pseudorank of X, corresponding to a definition used by Lawson and Hanson [16,p.…”

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confidence: 99%

An Alternative Algorithm for the Refinement of ULV Decompositions

Barlow

Erbay

Slapniucar

2005

SIAM J. Matrix Anal. & Appl.

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Abstract. The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). It is useful in applications of the SVD such as principal components where we are interested in approximating a matrix by one of lower rank. It can be updated and downdated much more quickly than an SVD. In many instances, the ULVD must be refined to improve the approximation it gives for the important right singular subspaces or to improve the matrix approximation. Present algorithms to perform this refinement require O(mn) operations if the rank of the matrix is k, where k is very close to 0 or n, but these algorithms require O(mn 2 ) operations otherwise. Presented here is an alternative refinement algorithm that requires O(mn) operations no matter what the rank is. Our tests show that this new refinement algorithm produces similar improvement in matrix approximation and in the subspaces. We also propose slight improvements on the error bounds on subspaces and singular values computed by the ULVD.

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An algorithm and a stability theory for downdating the ULV decomposition

Cited by 22 publications

References 24 publications

An Alternative Algorithm for a Sliding Window ULV Decomposition

An Alternative Algorithm for a Sliding Window ULV Decomposition

Untitled

An Alternative Algorithm for the Refinement of ULV Decompositions

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