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Fierro, Ricardo D.; Hansen, Per Christian; Hansen, Poul Henrik Kyvsgaard

doi:10.1023/a:1019112103049

Cited by 38 publications

(2 citation statements)

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“…For example, the QR decomposition with column pivoting is a popular and fast alternative to the SVD in this context. See, e.g., Fierro, Hansen, and Hansen [18] for an overview of the qualitative differences between different URV decompositions in floating-point arithmetic.…”

Section: Theorem 5 (Zha [28 Def 21]) the Regular Restricted Singular ...mentioning

confidence: 99%

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Towards a more robust algorithm for computing the restricted singular value decomposition

Zwaan¹

2020

Preprint

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A new algorithm to compute the restricted singular value decomposition of dense matrices is presented. Like Zha's method [29], the new algorithm uses an implicit Kogbetliantz iteration, but with four major innovations. The first innovation is a useful quasi-upper triangular generalized Schur form that just requires orthonormal transformations to compute. Depending on the application, this Schur form can be used instead of the full decomposition. The second innovation is a new preprocessing phase that requires fewer rank determinations than previous methods. The third innovation is a numerically stable RSVD algorithm for 2 × 2 upper-triangular matrices, which forms a key component of the implicit Kogbetliantz iteration. The fourth innovation is an alternative scaling for the restricted singular triplets that results in elegant formulas for their computation. Beyond these four innovations, the qualitative (numerical) characteristics of the algorithm are discussed extensively. Some numerical challenges in the (optional) postprocessing phase are considered too; though, their solutions require further research. Numerical tests and examples confirm the effectiveness of the method.

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Section: Theorem 5 (Zha [28 Def 21]) the Regular Restricted Singular ...mentioning

confidence: 99%

“…Here, we use a similar notation as in the preprocessing phase, with a similar numbering of the blocks. Define (7), where S and T are as in (18), and let ℓ) , and…”

Section: Postprocessing In Exact Arithmeticmentioning

confidence: 99%

Towards a more robust algorithm for computing the restricted singular value decomposition

Zwaan¹

2020

Preprint

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Adaptive Kernel Classifiers Via Matrix Decomposition Updating for Biological Data Analysis

Kim

Park

2008

Machine Learning in Bioinformatics

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Gram‐Schmidt orthogonalization: 100 years and more

Leon

Björck

Gander

2012

Numerical Linear Algebra App

121

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In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P. Gram in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthogonalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary, we present a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Its application for solving least squares problems and in Krylov subspace methods are also reviewed. Software and implementation aspects are also discussed.

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Untitled

Cited by 38 publications

References 50 publications

Towards a more robust algorithm for computing the restricted singular value decomposition

Towards a more robust algorithm for computing the restricted singular value decomposition

Adaptive Kernel Classifiers Via Matrix Decomposition Updating for Biological Data Analysis

Gram‐Schmidt orthogonalization: 100 years and more

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