Experiments on orthogonalization by biorthogonal representations of orthogonal projectors

Miroslav Andrle, Laura Rebollo-Neira*

*Corresponding author for this work

Research output: Contribution to journalLetter

Abstract

A number of experiments are performed with the aim of enhancing a particular feature arising when biorthogonal sequences are used for the purpose of orthogonalization. It is shown that an orthogonalization process executed by biorthogonal sequences and followed by a re-orthogonalization step admits four numerically different realizations. The four possibilities are originated by the fact that, although an orthogonal projector is by definition a self-adjoint operator, due to numerical errors in finite precision arithmetic the biorthogonal representation does not fulfil such a property. In the experiments presented here one of the realizations is shown clearly numerically superior to the remaining three.

Original languageEnglish
Pages (from-to)545-551
Number of pages7
JournalJournal of Computational and Applied Mathematics
Volume205
Issue number1
DOIs
Publication statusPublished - 1 Aug 2007

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Orthogonal Projector
Orthogonalization
Experiment
Experiments
Self-adjoint Operator

Keywords

  • Biorthogonal basis
  • Gram-Schmidt orthogonalization
  • Orthogonal projections
  • Re-orthogonalization

Cite this

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abstract = "A number of experiments are performed with the aim of enhancing a particular feature arising when biorthogonal sequences are used for the purpose of orthogonalization. It is shown that an orthogonalization process executed by biorthogonal sequences and followed by a re-orthogonalization step admits four numerically different realizations. The four possibilities are originated by the fact that, although an orthogonal projector is by definition a self-adjoint operator, due to numerical errors in finite precision arithmetic the biorthogonal representation does not fulfil such a property. In the experiments presented here one of the realizations is shown clearly numerically superior to the remaining three.",
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Experiments on orthogonalization by biorthogonal representations of orthogonal projectors. / Andrle, Miroslav; Rebollo-Neira, Laura.

In: Journal of Computational and Applied Mathematics, Vol. 205, No. 1, 01.08.2007, p. 545-551.

Research output: Contribution to journalLetter

TY - JOUR

T1 - Experiments on orthogonalization by biorthogonal representations of orthogonal projectors

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AU - Rebollo-Neira, Laura

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AB - A number of experiments are performed with the aim of enhancing a particular feature arising when biorthogonal sequences are used for the purpose of orthogonalization. It is shown that an orthogonalization process executed by biorthogonal sequences and followed by a re-orthogonalization step admits four numerically different realizations. The four possibilities are originated by the fact that, although an orthogonal projector is by definition a self-adjoint operator, due to numerical errors in finite precision arithmetic the biorthogonal representation does not fulfil such a property. In the experiments presented here one of the realizations is shown clearly numerically superior to the remaining three.

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KW - Gram-Schmidt orthogonalization

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