Rounding error analysis of the classical Gram-Schmidt orthogonalization process

Giraud, Luc; Langou, Julien; Rozložńık, Miroslav; Eshof, Jasper van den

doi:10.1007/s00211-005-0615-4

Cited by 114 publications

(122 citation statements)

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“…This well-known phenomenon of "twice is enough" in numerical Gram-Schmidt orthogonalization, has been deeply studied and explained in [21]. Observe that we don't really need orthogonality, but rather a reasonable conditioning of the Vandermonde matrix in the transformed basis, namely V (a; ϕ) = V (a; p)T s = V s .…”

Section: Weakly Admissible Meshes (Wams)mentioning

confidence: 99%

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Bos¹,

Marchi²,

Sommariva³

et al. 2010

SIAM J. Numer. Anal.

117

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We discuss and compare two greedy algorithms, that compute discrete versions of Fekete-like points for multivariate compact sets by basic tools of numerical linear algebra. The first gives the so-called "Approximate Fekete Points" by QR factorization with column pivoting of Vandermonde-like matrices. The second computes Discrete Leja Points by LU factorization with row pivoting. Moreover, we study the asymptotic distribution of such points when they are extracted from Weakly Admissible Meshes.2000 AMS subject classification: 41A10, 41A63, 65D05.

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Section: Weakly Admissible Meshes (Wams)mentioning

confidence: 99%

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Bos¹,

Marchi²,

Sommariva³

et al. 2010

SIAM J. Numer. Anal.

117

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show abstract

“…This algorithm was developed and analyzed by Abdelmalek in [1] and its detailed error analysis was given by Giraud, Langou, Rozložnik, and Van Den Eshof in [9].…”

Section: Algorithm IV (Qr Cgs2 )mentioning

confidence: 99%

Numerical aspects of computing the Moore-Penrose inverse of full column rank matrices

Smoktunowicz

Wróbel

2011

Bit Numer Math

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This paper presents a comparison of certain direct algorithms for computing the Moore-Penrose inverse, for matrices of full column rank, from the point of view of numerical stability. It is proved that the algorithm using Householder QR decomposition, implemented in floating point arithmetic, is forward stable but only conditionally mixed forward-backward stable. A similar result holds also for the Classical Gram-Schmidt algorithm with reorthogonalization (CGS2). This algorithm was developed and analyzed by Abdelmalek (BIT, 11 (4) We incorporate the recent result of Byers and Xu (SIAM J. Matrix Anal. Appl. 30:822-843, 2008) to show that the algorithm based on the Golub-Kahan bidiagonalization is mixed forward-backward stable under natural assumptions.

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“…See Refs. [67,68] for discussions about the conditioning and numerical stability of different orthonormalization procedures.…”

Section: Appendix A: the Reduced Basis Methodsmentioning

confidence: 99%

Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models

et al. 2014

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We propose a solution to the problem of quickly and accurately predicting gravitational waveforms within any given physical model. The method is relevant for both real-time applications and more traditional scenarios where the generation of waveforms using standard methods can be prohibitively expensive. Our approach is based on three offline steps resulting in an accurate reduced order model in both parameter and physical dimensions that can be used as a surrogate for the true or fiducial waveform family. First, a set of m parameter values is determined using a greedy algorithm from which a reduced basis representation is constructed. Second, these m parameters induce the selection of m time values for interpolating a waveform time series using an empirical interpolant that is built for the fiducial waveform family. Third, a fit in the parameter dimension is performed for the waveform's value at each of these m times. The cost of predicting L waveform time samples for a generic parameter choice is of order OðmL þ mc fit Þ online operations, where c fit denotes the fitting function operation count and, typically, m ≪ L. The result is a compact, computationally efficient, and accurate surrogate model that retains the original physics of the fiducial waveform family while also being fast to evaluate. We generate accurate surrogate models for effective-one-body waveforms of nonspinning binary black hole coalescences with durations as long as 10 5 M, mass ratios from 1 to 10, and for multiple spherical harmonic modes. We find that these surrogates are more than 3 orders of magnitude faster to evaluate as compared to the cost of generating effective-one-body waveforms in standard ways. Surrogate model building for other waveform families and models follows the same steps and has the same low computational online scaling cost. For expensive numerical simulations of binary black hole coalescences, we thus anticipate extremely large speedups in generating new waveforms with a surrogate. As waveform generation is one of the dominant costs in parameter estimation algorithms and parameter space exploration, surrogate models offer a new and practical way to dramatically accelerate such studies without impacting accuracy.

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Rounding error analysis of the classical Gram-Schmidt orthogonalization process

Cited by 114 publications

References 20 publications

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Computing Multivariate Fekete and Leja Points by Numerical Linear Algebra

Numerical aspects of computing the Moore-Penrose inverse of full column rank matrices

Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models

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