Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization

Kokiopoulou, Effrosyni; Bekas, Costas; Gallopoulos, Efstratios

doi:10.1016/j.apnum.2003.11.011

Cited by 51 publications

(63 citation statements)

References 38 publications

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“…However, for a small value of m, the desired singular triplets of A may be approximated poorly by computed approximate singular triplets {σ j }. In order to circumvent this difficulty, several methods have been proposed that are based on the computation of partial Lanczos bidiagonalizations (1.3)-(1.4) with m small for a sequence of initial vectors p 1 ; see, e.g., [5,13,14,15,16,17,28]. These methods are commonly referred to as restarted partial Lanczos bidiagonalization methods.…”

mentioning

confidence: 99%

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

Baglama¹,

Reichel²

2005

SIAM J. Sci. Comput.

281

316

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mentioning

confidence: 99%

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

Baglama¹,

Reichel²

2005

SIAM J. Sci. Comput.

281

316

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“…In fact, the computation of s(z), even by means of advanced methods (e.g. those presented in [2,23,25,26]), is very expensive, especially when the smallest singular values are clustered. Moreover, we need to compute s(z) for many values of z.…”

Section: Matrix-based Methods and Hybridsmentioning

confidence: 99%

“…To appreciate these results, we note that using a parallel implementation of the classical GRID algorithm together with a state-of-the-art general purpose SVD solver [26] for the large (n = 40 000) sparse matrix of the previous example on a 50 × 50 mesh and an eight-node COW, the pseudospectrum took in excess of 4 h of runtime.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The environment also incorporates and lets one use, via appropriate selections in PPsGUI, many state-of-the-art algorithms for computing singular values. So, in addition to the transfer function framework, one can use MATLAB's own direct and iterative methods svd, svds as well as methods described in [2,23,25,26]. PPsGUI also allows the superposition of plots as well as three-dimensional plotting.…”

Section: Ppsguimentioning

confidence: 99%

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The design of a distributed MATLAB-based environment for computing pseudospectra

Bekas

Kokiopoulou

Gallopoulos

2005

Future Generation Computer Systems

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“…(2) While being mathematically nearly equivalent to Sorensen's strategy, Aggdef(2) achieves significant improvements in both efficiency and stability. To emphasize this point, we compare Aggdef(2) with another, more straightforward extension of Sorensen's deflation strategy for the bidiagonal case, described in [18]. The authors in [18] use both the left and right singular vectors of a target singular value to form two unitary matrices Q S and P S (determined by letting y be the singular vectors that determines the unitary matrix) such that diag(I n−k , P T S ) · B · diag(I n−k , Q S ) is bidiagonal except for the nonzero (n − k, n)th element, and its bottom diagonal is "isolated".…”

Section: Comparison With Aggdef(1)mentioning

confidence: 99%

dqds with Aggressive Early Deflation

Nakatsukasa¹,

Aishima²,

Yamazaki³

2012

SIAM J. Matrix Anal. & Appl.

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Abstract. The dqds algorithm computes all the singular values of an n-by-n bidiagonal matrix to high relative accuracy in O(n 2 ) cost. Its efficient implementation is now available as a LAPACK subroutine and is the preferred algorithm for this purpose. In this paper we incorporate into dqds a technique called aggressive early deflation, which has been applied successfully to the Hessenberg QR algorithm. Extensive numerical experiments show that aggressive early deflation often reduces the dqds runtime significantly. In addition, our theoretical analysis suggests that with aggressive early deflation, the performance of dqds is largely independent of the shift strategy. We confirm through experiments that the zero-shift version is often as fast as the shifted version. We give a detailed error analysis to prove that with our proposed deflation strategy, dqds computes all the singular values to high relative accuracy.Key words. aggressive early deflation, dqds, singular values, bidiagonal matrix AMS subject classifications. 65F15, 15A181. Introduction. The differential quotient difference with shifts (dqds) algorithm computes all the singular values of an n-by-n bidiagonal matrix to high relative accuracy in O(n 2 ) cost [11]. Its efficient implementation has been developed and is now available as a LAPACK subroutine DLASQ [30]. Because of its guaranteed relative accuracy and efficiency, dqds has now replaced the QR algorithm [7], which had been the default algorithm to compute the singular values of a bidiagonal matrix. The standard way of computing the singular values of a general matrix is to first apply suitable orthogonal transformations to reduce the matrix to bidiagonal form, then use dqds [6]. dqds is also a major computational kernel in the MRRR algorithm for computing orthogonal eigenvectors of a symmetric tridiagonal matrix [8,9,10] and the singular value decomposition of a bidiagonal matrix [35] in O(n 2 ) cost. The aggressive early deflation strategy, introduced in [5], is known to greatly improve the performance of the Hessenberg QR algorithm for computing the eigenvalues of a general square matrix by deflating converged eigenvalues long before a conventional deflation strategy does. The primary contribution of this paper is the proposal of two deflation strategies for dqds based on aggressive early deflation. The first strategy is a direct specialization of aggressive early deflation to dqds. The second strategy, which takes full advantage of the bidiagonal structure of the matrix, is computationally more efficient. We present a detailed mixed forward-backward stability analysis that proves the second strategy guarantees high relative accuracy of all the computed singular values. The results of extensive numerical experiments demonstrate that performing aggressive early deflation significantly reduces the solution time of dqds in many cases. We observed speedups of up to a factor 50, and in all our experiments the second strategy was at least as fast as DLASQ for any matrix larger than 3000.

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Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization

Cited by 51 publications

References 38 publications

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

Augmented Implicitly Restarted Lanczos Bidiagonalization Methods

The design of a distributed MATLAB-based environment for computing pseudospectra

dqds with Aggressive Early Deflation

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