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Computing the Bidiagonal SVD Through an Associated Tridiagonal Eigenproblem

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High Performance Computing for Computational Science – VECPAR 2016 (VECPAR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10150))

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Abstract

In this paper, we present an algorithm for the singular value decomposition (SVD) of a bidiagonal matrix by means of the eigenpairs of an associated symmetric tridiagonal matrix. The algorithm is particularly suited for the computation of a subset of singular values and corresponding vectors. We focus on a sequential implementation of the algorithm, discuss special cases and other issues. We use a large set of bidiagonal matrices to assess the accuracy of the implementation and to identify potential shortcomings. We show that the algorithm can be up to three orders of magnitude faster than existing algorithms, which are limited to the computation of a full SVD.

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References

  1. Bai, Z., Demmel, J., Dongarra, J., Ruhe, A., van der Vorst, H. (eds.): Templates for the solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia (2000)

    MATH  Google Scholar 

  2. Marques, O., Demmel, J., Voemel, C., Parlett, B.N.: A testing infrastructure for symmetric tridiagonal eigensolvers. ACM TOMS 35, 1–13 (2008)

    Article  Google Scholar 

  3. Demmel, J., Marques, O., Voemel, C., Parlett, B.N.: Performance and accuracy of LAPACK’s symmetric tridiagonal eigensolvers. SIAM J. Sci. Comput. 30, 1508–1526 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dhillon, I.S., Parlett, B.N.: Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices. Linear Algebra Appl. 387, 1–28 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dhillon, I.S., Parlett, B.N., Voemel, C.: The design and implementation of the MRRR algorithm. ACM TOMS 32, 533–560 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Willems, P., Lang, B.: A framework for the MR\(^3\) algorithm: theory and implementation. SIAM J. Sci. Comput. 35, 740–766 (2013)

    Article  MathSciNet  Google Scholar 

  7. Demmel, J., Kahan, W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci. Stat. Comput. 11, 873–912 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Willems, P., Lang, B., Voemel, C.: Computing the bidiagonal SVD using multiple relatively robust representations. SIAM. J. Matrix Anal. Appl. 28, 907–926 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Willems, P.: On MR\(^3\)-type algorithms for the tridiagonal symmetric eigenproblem and the bidiagonal SVD, PhD dissertation, University of Wuppertal (2010)

    Google Scholar 

  10. Voemel, C.: ScaLAPACK’s MRRR algorithm. ACM TOMS 37, 1–35 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    Osni Marques

  2. Faculdade de Economia and CMUP, Universidade Do Porto, Porto, Portugal

    Paulo B. Vasconcelos

Authors
  1. Osni Marques
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  2. Paulo B. Vasconcelos
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Corresponding authors

Correspondence to Osni Marques or Paulo B. Vasconcelos .

Editor information

Editors and Affiliations

  1. University of Porto, Porto, Portugal

    Inês Dutra

  2. University of Porto, Porto, Portugal

    Rui Camacho

  3. University of Porto, Porto, Portugal

    Jorge Barbosa

  4. Lawrence Berkeley National Laboratory, Berkeley, California, USA

    Osni Marques

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Marques, O., Vasconcelos, P.B. (2017). Computing the Bidiagonal SVD Through an Associated Tridiagonal Eigenproblem. In: Dutra, I., Camacho, R., Barbosa, J., Marques, O. (eds) High Performance Computing for Computational Science – VECPAR 2016. VECPAR 2016. Lecture Notes in Computer Science(), vol 10150. Springer, Cham. https://doi.org/10.1007/978-3-319-61982-8_8

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  • DOI: https://doi.org/10.1007/978-3-319-61982-8_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-61981-1

  • Online ISBN: 978-3-319-61982-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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