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Efficient Certification of Numeric Solutions to Eigenproblems

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Mathematical Aspects of Computer and Information Sciences (MACIS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10693))

Abstract

In this paper, we present an efficient algorithm for the certification of numeric solutions to eigenproblems. The algorithm relies on a mixture of ball arithmetic, a suitable Newton iteration, and clustering of eigenvalues that are close.

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References

  1. Alefeld, G., Herzberger, J.: Introduction to Interval Analysis. Academic Press, New York (1983)

    MATH  Google Scholar 

  2. Armentano, D., Beltrán, C., Bürgisser, P., Cucker, F., Shub, M.: A stable, polynomial-time algorithm for the eigenpair problem. Technical report, arXiv (2014). http://arxiv.org/abs/1505.03290

  3. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-0701-6

    Book  MATH  Google Scholar 

  4. Dongarra, J.J., Moler, C.B., Wilkinson, J.H.: Improving the accuracy of computed eigenvalues and eigenvectors. SIAM J. Numer. Anal. 20(1), 23–45 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of ISSAC 2014, Kobe, Japan, pp. 296–303, 23–25 July 2014

    Google Scholar 

  6. Golub, G.H., Van Loan, F.: Matrix Computations. JHU Press, Baltimore (1996)

    MATH  Google Scholar 

  7. Graillat, S., Trébuchet, P.: A new algorithm for computing certified numerical approximations of the roots of a zero-dimensional system. In: Proceedings of ISSAC 2009, pp. 167–174. ACM Press (2009)

    Google Scholar 

  8. Harvey, D.: Faster truncated integer multiplication (2017). https://arxiv.org/abs/1703.00640

  9. Harvey, D., van der Hoeven, J.: On the complexity of integer matrix multiplication. Technical report, HAL (2014, accepted for publication in JSC). http://hal.archives-ouvertes.fr/hal-01071191

  10. Harvey, D., van der Hoeven, J., Lecerf, G.: Even faster integer multiplication. J. Complex. 36, 1–30 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. van der Hoeven, J.: Ball arithmetic. Technical report, HAL (2009). http://hal.archives-ouvertes.fr/hal-00432152

  12. van der Hoeven, J., Lecerf, G., Mourrain, B., et al.: Mathemagix (2002). http://www.mathemagix.org

  13. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer, London (2001). https://doi.org/10.1007/978-1-4471-0249-6

    Book  MATH  Google Scholar 

  14. Johansson, F.: Arb: a C library for ball arithmetic. ACM Commun. Comput. Algebra 47(3/4), 166–169 (2014)

    Article  Google Scholar 

  15. Kulisch, U.W.: Computer Arithmetic and Validity: Theory, Implementation, and Applications. Studies in Mathematics, vol. 33. de Gruyter, Berlin (2008)

    Book  MATH  Google Scholar 

  16. Miyajima, S.: Fast enclosure for all eigenvalues in generalized eigenvalue problems. J. Comput. Appl. Math. 233(11), 2994–3004 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Moore, R.E.: Interval Analysis. Prentice Hall, Englewood Cliffs (1966)

    MATH  Google Scholar 

  18. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  19. Rump, S., Graillat, S.: Verified error bounds for multiple roots of systems of nonlinear equations. Numer. Algorithms 54, 359–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rump, S.M.: Guaranteed inclusions for the complex generalized eigenproblem. Computing 42(2), 225–238 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers, Dordrecht (1999). http://www.ti3.tu-harburg.de/rump/

    Chapter  Google Scholar 

  22. Rump, S.M.: Computational error bounds for multiple or nearly multiple eigenvalues. Linear Algebra Appl. 324(1–3), 209–226 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yamamoto, T.: Error bounds for computed eigenvalues and eigenvectors. Numerische Mathematik 34(2), 189–199 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’informatique, UMR 7161 CNRS, Campus de l’École polytechnique 1, rue Honoré d’Estienne d’Orves Bâtiment Alan Turing, CS35003, 91120, Palaiseau, France

    Joris van der Hoeven

  2. Inria Sophia Antipolis Méditerranée, AROMATH, 2004 route des Lucioles, 06902, Sophia Antipolis, France

    Bernard Mourrain

Authors
  1. Joris van der Hoeven
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  2. Bernard Mourrain
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Corresponding author

Correspondence to Joris van der Hoeven .

Editor information

Editors and Affiliations

  1. University of Paderborn, Paderborn, Germany

    Johannes Blömer

  2. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  3. Johannes Kepler University Linz, Linz, Austria

    Temur Kutsia

  4. SBA Research, Vienna, Austria

    Dimitris E. Simos

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van der Hoeven, J., Mourrain, B. (2017). Efficient Certification of Numeric Solutions to Eigenproblems. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_6

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  • DOI: https://doi.org/10.1007/978-3-319-72453-9_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72452-2

  • Online ISBN: 978-3-319-72453-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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