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Nonlinear Numerical Methods and Rational Approximation II pp 291–301Cite as

On the Vector-Valued Padé Approximants and the Vector ε-Algorithm

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Part of the book series: Mathematics and Its Applications ((MAIA,volume 296))

Abstract

In a recent work, thanks to the use of Clifford algebras and designants, we have shown that Wynn’s vector ε -algorithm can be written as a ratio of two designants. The present work, according to these new results, has the aim of

  1. 1.

    Finding easily and differently the results of some authors in this scope, particularly those of Wynn, D.E. Roberts and P.R. Graves-Morris.

  2. 2.

    Computing explicitly the expressions of the vector-valued Padé approximants as a ratio of two designants, which generalizes the results given by the authors above, and establishing the connection between these approximants and the vector ε -algorithm.

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References

  1. A.C. Aitken, Determinants and Matrices, Oliver Boyd(Edinburgh),1965.

    Google Scholar 

  2. A. Artin, Geometric algebra, Interscience(NewYork)1966.

    Google Scholar 

  3. C. Brezinski, M. Redivo Zaglia, Extrapolation Methods. Theory and Practice, NorthHolland(Amesterdam),1991.

    Google Scholar 

  4. C. Brezinski, A general extrapolation algorithm, Numer.Math, 35(1980)175-187.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Brezinski, Some results in the theory of the vector ε-algorithm., Linear Algebra Appl., 8(1974)77-86.

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Brezinski, Computation of the eigenelements of a matrix by the a-algorithm, Linear Algebra Appl., 11(1975)7-20.

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Brezinski, Application de l’ε-algorithme à la résolution des systèmes non linéaires, C.R. Acad.Sci.Paris, 271A(1970)1174-1177.

    MathSciNet  Google Scholar 

  8. C. Brezinski,Padé type approximants and general orthogonal polynomials, Birkauser Verlag, Basel (1980).

    Google Scholar 

  9. R. Deheuvels,formes quadratiques et groupes classiques, Presses Universitaires de France,1981.

    Google Scholar 

  10. F.J. Dyson, Quaternion determinants,Helv.Phys.Acta,45(1972)289-302.

    Google Scholar 

  11. E. Gekeler, On the solution of systems of equations by the epsilon algorithm of Wynn,Math.Comput.,26(1972)427-436.

    Article  MathSciNet  MATH  Google Scholar 

  12. P.R. Graves-Morris, Vector-valued Rational Interpolants I, Numer.Math., 42(1983)331-348.

    Article  MathSciNet  MATH  Google Scholar 

  13. P.R. Graves-Morris, Vector-valued Rational Interpolants II,I.M.A.J.Num.Anal, 4(1984)209-224.

    Article  MathSciNet  MATH  Google Scholar 

  14. P.R. Graves-Morris and CD Jenkins, Vector-valued Rational Interpolants III,Constr.Approx, 2(1986)263-289.

    Article  MathSciNet  MATH  Google Scholar 

  15. P.R. Graves-Morris and DE Roberts,From Matrix to Vector Padé Approximants, J.Comp.Appld.Math., to appear.

    Google Scholar 

  16. T. Håvie, Generalized Neville type extrapolation schemes, BIT, 19(1979)204-213.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Heyting, Die theorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation,Math.Ann,98(1927)465-490.

    Article  MathSciNet  MATH  Google Scholar 

  18. G.N. Hile and P. Lounesto, Matrix representations of Clifford algebras,Linear Algebra Appl.,128(1990)51-63.

    Article  MathSciNet  MATH  Google Scholar 

  19. M.L. Mehta, Matrix theory. Selected Topics and Useful Results, Les éditions de Physique, Les Ulis(1989).

    Google Scholar 

  20. J.B. McLeod, A note on the e-algorithm, Computing, 7(1971)17-24.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Penrose, A generalised inverse for matrices, Pro.Cambridge Phil.Soc, 51(1955),406-413.

    Article  MathSciNet  MATH  Google Scholar 

  22. IR. Porteous, Topological Geometry,2nd ed,Cambridge University Press,1981.

    Book  MATH  Google Scholar 

  23. DE. Roberts, Clifford algebras and vector-valued rational forms I,Proc.Roy.Soc,A431(1990)285-300.

    Google Scholar 

  24. DE. Roberts, Clifford algebras and vector-valued rational forms II, to appear in the proceedings of the International Congres on Extrapolation and Rational Approximation, to be published in Numerical Algorithms.

    Google Scholar 

  25. A. Salam, Extrapolation: Extension et Nouveaux Résultats,Thesis Université des Sciences et Technologies de Lille,1993.

    Google Scholar 

  26. A. Salam, Non-commutative extrapolation algorithms, to appear.

    Google Scholar 

  27. J. Van Iseghem, Vector Padé approximants, 11th IMACS Worl Congres Numer. Math. and Appl. 1 ed. Vichnevetsky and Vignes (1986), North Holland, Amsterdam.

    Google Scholar 

  28. P. Wynn, On a device for computing the em(Sn) transformation,MTAC,10(1956)91-96.

    MathSciNet  MATH  Google Scholar 

  29. P. Wynn, Vector continued fractions,Linear Algebra Appl.,1(1968)357-395.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratoire d’analyse numérique et d’optimisation, Université des Sciences et Technologies de Lille, UFR IEEA-M3, 59655, Villeneuve d’Ascq-, France

    A. Salam

Authors
  1. A. Salam
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of Antwerp, Antwerp, Belgium

    Annie Cuyt

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© 1994 Springer Science+Business Media Dordrecht

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Salam, A. (1994). On the Vector-Valued Padé Approximants and the Vector ε-Algorithm. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_25

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  • DOI: https://doi.org/10.1007/978-94-011-0970-3_25

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4420-2

  • Online ISBN: 978-94-011-0970-3

  • eBook Packages: Springer Book Archive

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