Iterative algorithms for Gram-Schmidt orthogonalization

Hoffmann, W.

doi:10.1007/bf02241222

Cited by 111 publications

(88 citation statements)

References 4 publications

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“…The Moore-Penrose inverse is a powerful tool in computing polar decomposition, the areas of electrical networks, control theory, filtering, estimation theory and pattern recognition (see, e.g., [3,[13][14][15]). However, if one wants to solve the least-squares problem it is usually recommended to apply methods which do not compute A † directly but use numerically stable factorization of A instead.…”

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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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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“…A minor disadvantage of using the packaged routine is that the memory demands in that particular case are high and this places restrictions on the highest time-scale for decorrelation (encoded in the variable M here); this is currently under investigation. Some variants of the algorithm as described in [16] …”

Section: Discussionmentioning

confidence: 99%

“…One can now regard the estimation of as the problem of projecting onto the subspace of Euclidean space which is the orthogonal complement of the space spanned by the , (denoted here by ). In order to do this, it is necessary to establish an orthogonal basis on , (in fact, it might as well be orthonormal), denoted where Such a basis can be computed by GramSchmidt orthogonalisation as discussed in [16], (The Matlab function orth is used throughout here). In terms of the orthogonal basis, one has,…”

Section: The Orthogonalised Reverse Path Algorithmmentioning

confidence: 99%

On the orthogonalised reverse path method for nonlinear system identification

Muhamad¹,

Sims²,

Worden³

2012

Journal of Sound and Vibration

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Abstract. The problem of obtaining the underlying linear dynamic compliance matrix in the presence of nonlinearities in a general Multi-Degree-of-Freedom (MDOF) system can be solved using the Conditioned Reverse Path (CRP) method introduced by Richards and Singh (1998 Journal of Sound and Vibration, 213(4): p. 673-708). The CRP method also provides a means of identifying the coefficients of any nonlinear terms which can be specified a priori in the candidate equations of motion. Although the CRP has proved extremely useful in the context of nonlinear system identification, it has a number of small issues associated with it. One of these issues is the fact that the nonlinear coefficients are actually returned in the form of spectra which need to be averaged over frequency in order to generate parameter estimates. The parameter spectra are typically polluted by artefacts from the identification of the underlying linear system which manifest themselves at the resonance and anti-resonance frequencies. A further problem is associated with the fact that the parameter estimates are extracted in a recursive fashion which leads to an accumulation of errors. The first minor objective of this paper is to suggest ways to alleviate these problems without major modification to the algorithm. The results are demonstrated on numerically-simulated responses from MDOF systems. In the second part of the paper, a more radical suggestion is made, to replace the conditioned spectral analysis (which is the basis of the CRP method) with an alternative time domain decorrelation method. The suggested approach -the Orthogonalised Reverse Path (ORP) method -is illustrated here using data from simulated Single-Degree-of-Freedom (SDOF) and MDOF systems.

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“…Unfortunately, the classical Gram-Schmidt method is unstable with respect to rounding errors, so this method is rarely used. On the other hand, Hoffmann [11] gives experimental evidence indicating that a two-fold application of the classical Gram-Schmidt method is stable. A third method which has been suggested is the parallel implementation of Householder transformations, introduced by Walker [18].…”

Section: Orthogonalization Methodsmentioning

confidence: 99%

A parallel implementation of the block preconditioned GCR method

Vuik

Frank

1999

High-Performance Computing and Networking

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Abstract. The parallel implementation of GCR is addressed, with particular focus on communication costs associated with orthogonalization processes. This consideration brings up questions concerning the use of Householder reflections with GCR. To precondition the GCR method a block Gauss-Jacobi method is used. Approximate solvers are used to obtain a solution of the diagonal blocks. Experiments on a cluster of HP workstations and on a Cray T3E are given.

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Iterative algorithms for Gram-Schmidt orthogonalization

Cited by 111 publications

References 4 publications

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

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

On the orthogonalised reverse path method for nonlinear system identification

A parallel implementation of the block preconditioned GCR method

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