Accurate downdating of a modified Gram-Schmidt QR decomposition

Yoo, Kyeongah; Park, H.

doi:10.1007/bf01740553

Cited by 15 publications

(22 citation statements)

References 14 publications

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“…It turns out there are efficient, stable algorithms to perform this downdating (see [8,21]). For later reference when we come to operation counts, note that these algorithms removeB 12 a row at a time with a cost of O(n 2 ) for each row.…”

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confidence: 99%

An Efficient Algorithm for Computing the Generalized Null Space Decomposition

Guglielmi

Overton

Stewart

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. The generalized null space decomposition (GNSD) is a unitary reduction of a general matrix A of order n to a block upper triangular form that reveals the structure of the Jordan blocks of A corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya. It was extended first by Ruhe and then by Golub and Wilkinson, who based the reduction on the singular value decomposition. Unfortunately, if A has large Jordan blocks, the complexity of these algorithms can approach the order of n 4 . This paper presents an alternative algorithm, based on repeated updates of a QR decomposition of A, that is guaranteed to be of order n 3 . Numerical experiments confirm the stability of this algorithm, which turns out to produce essentially the same form as that of Golub and Wilkinson. The effect of errors in A on the ability to recover the original structure is investigated empirically. Several applications are discussed, including the computation of the Drazin inverse.

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confidence: 99%

An Efficient Algorithm for Computing the Generalized Null Space Decomposition

Guglielmi

Overton

Stewart

2015

SIAM J. Matrix Anal. & Appl.

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“…In this work, we modify the algorithms suggested by Daniel et al [8] and by Yoo and Park [23]. These modifications are based upon a framework whose development is the central subject of this paper.…”

Section: Introductionmentioning

confidence: 97%

“…However, whenÛ departs from left orthogonality, the vector d in (1.14) solves (1.12), whereas d =Û T e 1 does not, and that can make a dramatic difference. We arrived at the choice of d in (1.14) from analyzing a procedure of Yoo and Park [23] which produces an approximation of this value. We show that by altering the procedures in [8,23], we can compute better approximations of d in (1.14), thereby producing a more robust downdating procedure.…”

Section: Introductionmentioning

confidence: 99%

“…We arrived at the choice of d in (1.14) from analyzing a procedure of Yoo and Park [23] which produces an approximation of this value. We show that by altering the procedures in [8,23], we can compute better approximations of d in (1.14), thereby producing a more robust downdating procedure. We also show that the procedure is [23] is two modified Gram-Schmidt steps where the second step goes through the columns of U in reverse order.…”

Section: Introductionmentioning

confidence: 99%

“…Thus our discussion below concerns the computation of u 0 and d. A short history of these procedures is given in the introduction to [23] and in [2, pp. 137-140].…”

Section: Introductionmentioning

confidence: 99%

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Improved Gram–Schmidt Type Downdating Methods

2005

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The problem of deleting a row from a Q-R factorization (called downdating) using Gram-Schmidt orthogonalization is intimately connected to using classical iterative methods to solve a least squares problem with the orthogonal factor as the coefficient matrix. Past approaches to downdating have focused upon accurate computation of the residual of that least squares problem, then finding a unit vector in the direction of the residual that becomes a new column for the orthogonal factor. It is also important to compute the solution vector of the related least squares problem accurately, as that vector must be used in the downdating process to maintain good backward error in the new factorization. Using this observation, new algorithms are proposed.One of the new algorithms proposed is a modification of one due to Yoo and Park [BIT, 36:161-181, 1996]. That algorithm is shown to be a Gram-Schmidt procedure.Also presented are new results that bound the loss of orthogonality after downdating. An error analysis shows that the proposed algorithms' behavior in floating point arithmetic is close to their behavior in exact arithmetic. Experiments show that the changes proposed in this paper can have a dramatic impact upon the accuracy of the downdated Q-R decomposition. (2000): 65F20, 65F25. AMS subject classification

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Modifiable low‐rank approximation to a matrix

Barlow

Erbay

2009

Numerical Linear Algebra App

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SUMMARYA truncated ULV decomposition (TULVD) of an m ×n matrix X of rank k is a decomposition of the form X =U LV T + E, where U and V are left orthogonal matrices, L is a k ×k non-singular lower triangular matrix, and E is an error matrix. Only U ,V , L, and E F are stored, but E is not stored. We propose algorithms for updating and downdating the TULVD. To construct these modification algorithms, we also use a refinement algorithm based upon that in (SIAM J. Matrix Anal. Appl. 2005; 27(1):198-211) that reduces E F , detects rank degeneracy, corrects it, and sharpens the approximation.

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Accurate downdating of a modified Gram-Schmidt QR decomposition

Cited by 15 publications

References 14 publications

An Efficient Algorithm for Computing the Generalized Null Space Decomposition

An Efficient Algorithm for Computing the Generalized Null Space Decomposition

Improved Gram–Schmidt Type Downdating Methods

Modifiable low‐rank approximation to a matrix

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