Computing the Bidiagonal SVD Using Multiple Relatively Robust Representations

Willems, Paul R.; Lang, Bruno; ̈mel, Christof Vo

doi:10.1137/050628301

Cited by 24 publications

(18 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The thesis [8] gives also a comparison of different numerically stable algorithms for computing eigenvalues and eigenvectors of Jacobi matrices; see also the survey and comments in [38,Sect. 2], and the recent work [9,10,30,58]. We can conclude that the computation and perturbation theory of quadrature nodes and weights from the recurrence coefficients is well understood.…”

Section: Sensitivity and Computing Of The Quadrature Nodes And Weightsmentioning

confidence: 90%

On sensitivity of Gauss–Christoffel quadrature

2007

View full text Add to dashboard Cite

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss-Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. 123 148 D. P. O'Leary et al.several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss-Christoffel quadrature estimates. We then discuss conditions under which the Gauss-Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems.

show abstract

Section: Sensitivity and Computing Of The Quadrature Nodes And Weightsmentioning

confidence: 90%

On sensitivity of Gauss–Christoffel quadrature

2007

View full text Add to dashboard Cite

show abstract

“…Step (1) involves Householder reflections and step (2) can either use QR iteration in Golub-Kahan-Reinsch (GKR) algorithm (81,82), divide-and-conquer method (35,83) or Multiple Relatively Robust Representations (MRRR) (84). An alternative SVD algorithm, combining steps (1) and (2), is to use Jacobi rotations and convergence criteria (85).…”

Section: Influence Of Algorithm Under Matlabmentioning

confidence: 99%

Denoising applied to spectroscopies – Part II: Decreasing computation time

Laurent

Gilles

Woelffel

et al. 2019

Applied Spectroscopy Reviews

View full text Add to dashboard Cite

Spectroscopies are of fundamental importance but can suffer from low sensitivity.Singular Value Decomposition (SVD) is a highly interesting mathematical tool, which can be conjugated with low-rank approximation to denoise spectra and increase sensitivity. SVD is also involved in data mining with Principal Component Analysis (PCA). In this paper, we focussed on the optimisation of SVD duration, which is a time-consuming computation. Both Intel processors (CPU) and Nvidia graphic cards (GPU) were benchmarked. A 100 times gain was achieved when combining divide and conquer algorithm, Intel Math Kernel Library (MKL), SSE3 (Streaming SIMD Extensions) hardware instructions and single precision. In such case, the CPU can outperform the GPU driven by CUDA technology. These results give a strong background to optimise SVD computation at the user scale.

show abstract

“…Thus, combining the implicit QR and the twisted factorization methods, we can compute the singular values and the left singular vectors of T in O(n 2 ) flops. There are O(n 2 ) methods for bidiagonal SVD [9,18]. Since we consider the SSVD of the complex symmetric tridiagonal T , we adapt the twisted method for symmetric tridiagonal eigendecomposition [5,6].…”

Section: Twisted Factorizationmentioning

confidence: 99%

A fast symmetric SVD algorithm for square Hankel matrices

Qiao

2008

Linear Algebra and its Applications

View full text Add to dashboard Cite

This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author's institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit:

show abstract

Computing the Bidiagonal SVD Using Multiple Relatively Robust Representations

Cited by 24 publications

References 10 publications

On sensitivity of Gauss–Christoffel quadrature

On sensitivity of Gauss–Christoffel quadrature

Denoising applied to spectroscopies – Part II: Decreasing computation time

A fast symmetric SVD algorithm for square Hankel matrices

Contact Info

Product

Resources

About