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The Effects of Loss of Orthogonality on Large Scale Numerical Computations

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Computational Science and Its Applications – ICCSA 2018 (ICCSA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10962))

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Abstract

Many useful large sparse matrix algorithms are based on orthogonality, but for efficiency this orthogonality is often obtained via short term recurrences. This can lead to both loss of orthogonality and loss of linear independence of computed vectors, yet with well designed algorithms high accuracy can still be obtained. Here we discuss a nice theoretical indicator of loss of orthogonality and linear independence and show how it leads to a related higher dimensional orthogonality that can be used to analyze and prove the effectiveness of such algorithms. We illustrate advantages and shortcomings of such algorithms with Cornelius Lanczos’ Hermitian matrix tridiagonalization process. The paper is reasonably expository, keeping simple by avoiding some detailed analyses.

The author’s work was supported by NSERC of Canada grant OGP0009236.

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Author information

Authors and Affiliations

  1. McGill University, Montreal, Quebec, H3A 0E9, Canada

    Christopher C. Paige

Authors
  1. Christopher C. Paige
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Correspondence to Christopher C. Paige .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. Saint Petersburg State University, Saint Petersburg, Russia

    Elena Stankova

  5. Polytechnic University of Bari, Bari, Italy

    Carmelo M. Torre

  6. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  7. Monash University, Clayton, Victoria, Australia

    David Taniar

  8. Kyushu Sangyo University, Fukuoka shi, Fukuoka, Japan

    Bernady O. Apduhan

  9. Politecnico di Bari, Bari, Italy

    Eufemia Tarantino

  10. Myongji University, Yongin, Korea (Republic of)

    Yeonseung Ryu

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Paige, C.C. (2018). The Effects of Loss of Orthogonality on Large Scale Numerical Computations. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2018. ICCSA 2018. Lecture Notes in Computer Science(), vol 10962. Springer, Cham. https://doi.org/10.1007/978-3-319-95168-3_29

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  • DOI: https://doi.org/10.1007/978-3-319-95168-3_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95167-6

  • Online ISBN: 978-3-319-95168-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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