Advertisement

Numerical Algorithms

, Volume 80, Issue 1, pp 235–251 | Cite as

The rational iteration method by Georges Lemaître

  • Hervé Le FerrandEmail author
Original Paper
  • 40 Downloads

Abstract

We explain in this article how the famous Belgian astronomer and physicist Georges Lemaître, a specialist in the theory of relativity, rediscovered, during the Second World War from Gauss’ work, Aitken’s Δ2 process. He called his discovery the rational iteration method and used it to solve an ordinary differential equation. After giving some historical elements on Aitken’s Δ2 process and Steffensen’s method, we examine in detail the genesis of the rational iteration, and then its application by Lemaître to Picard’s iterates to solve the differential equation \(y^{\prime }=dy/dx= 2y^{2}(y-x)\).

Keywords

Lemaître Aitken Steffensen Δ2 process Fixed point Ordinary differential equation Picard iterative process 

Mathematics Subject Classification (2010)

65B05 65H05 65L05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aitken, A.C.: On Bernoulli’s Numerical Solution of Algebraic Equations. Proc. Roy. Soc. Edinburgh 46, 289–305 (1927)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bacaër, N.: A Short History of Mathematical Population Dynamics. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bauer F.L.: My years with Rutishauser, Monday, February 18, 2002—LATSIS Symposium 2002—ETH ZürichGoogle Scholar
  4. 4.
    Bernoulli, D.: Observationes de seriebus quae formantur ex additione vel subtractione quacunque terminorum se mutuo consequentium. Commentarii Academiae Scientiarum Imperialis Petropolitanae 3, 85–100 (1728)Google Scholar
  5. 5.
    Brezinski, C.: History of Continued Fractions and Padé Approximants. Springer Series in Computational Mathematics, vol. 12. Springer, Berlin (1991)Google Scholar
  6. 6.
    Brezinski, C.: Extrapolation algorithms and Padé approximations: a historical survey. Appl. Numer. Math. 20, 299–318 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brezinski, C., Redivo-Zaglia, M.: Extrapolation Methods, Theory and Practice. Studies in Computational mathematics, vol. 2. North-Holland, Amsterdam (1991)zbMATHGoogle Scholar
  8. 8.
    Brezinski, C., Redivo-Zaglia, M.: The genesis and early developments of Aitken’s process, Shanks’ transformation, and related fixed point methods, this issueGoogle Scholar
  9. 9.
    Chabert, J.-L.: A History of Algorithms. From the Pebble to the Microchip. Springer, Berlin (1999)CrossRefGoogle Scholar
  10. 10.
    Davis, P.J.: Interpolation and Approximation. Dover, New York (1975)zbMATHGoogle Scholar
  11. 11.
    Demidovitch, B., Maron, I.: Eléments de Calcul Numérique. Éditions MIR, Moscou, (translated from Russian) (1979)Google Scholar
  12. 12.
    Euler, L.: De usu serierum recurrentium in radicibus aequationum indagandis, Introductio in analysin infinitorum, 1748. Introduction à l’Analyse Infinitésimale, Paris, 1796, livre 1, chap XVIIGoogle Scholar
  13. 13.
    Gauss, C.F.: Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. F. Perthes and J.H. Besser, Hamburg (1809)zbMATHGoogle Scholar
  14. 14.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley, New York (1988)zbMATHGoogle Scholar
  15. 15.
    Hermans, M., Stoffel, J.-F. (eds.): Le père Henri Bosmans sj (1852–1928). Acad. Belgique, Bull. Cl. Sci. Tome XXI (2010)Google Scholar
  16. 16.
    Hubbard, J., West, B.: Differential Equations. A Dynamical Systems Approach, Part I. Springer, New York (1991)Google Scholar
  17. 17.
    Jones, C.B., Lloyd, J.L. (eds.): Dependable and Historing Computing. Springer, Berlin (2011)Google Scholar
  18. 18.
    Lambert, D.: Un Atome d’Univers. La Vie et l’Oeuvre de Georges Lemaître. Editions Lessius, Belgium (2000)Google Scholar
  19. 19.
    Lambert, D.: The Atom of the Universe, The Life and Work of Georges Lemaître. Copernicus Center Press, Poland (2015)Google Scholar
  20. 20.
    Lemaître, G.: L’itération rationnelle. Acad. Belgique, Bull. Cl. Sci. (5) 28, 347–354 (1942)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lemaître, G.: Intégration d’une équation différentielle par itération rationnelle. Acad. Belgique, Bull. Cl. Sci. (5) 28, 815–825 (1942)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Luminet, J.-P.: L’Invention du Big Bang, Points. Seuil, Paris (2014)Google Scholar
  23. 23.
    Ogborn, M.E.: Johan Frederik Steffensen, 1873–1961. J. R. Stat. Soc. Ser. A (General) 125(4), 672–673 (1962)Google Scholar
  24. 24.
    Osada, N.: The early history of convergence acceleration methods. Numer. Algorithms 60, 205–221 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Picard, E.: Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J. Math. Pures Appl., 4e série, tome 6, 145–210 (1890)zbMATHGoogle Scholar
  26. 26.
    Runge, C., König, H.: Vorlesungen über numerisches Rechnen. Springer, Berlin (1924)CrossRefzbMATHGoogle Scholar
  27. 27.
    Rutishauser, H.: Der Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Phys. 5, 233–251 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Smith, D.E.: History of Mathematics, vol. I. Dover Publications, New York (1958)Google Scholar
  29. 29.
    Steffensen, J.F.: Remark on iteration. Skand. Aktuarietidskr. 16, 64–72 (1933)zbMATHGoogle Scholar
  30. 30.
    Whittaker, E.T., Robinson, G.: The Calculus of Observations. A Treatise on Numerical Mathematics. Blackie and Son, London (1924)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijon cedexFrance

Personalised recommendations