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Backward Error Analysis for Perturbation Methods

  • Robert M. CorlessEmail author
  • Nicolas Fillion
Conference paper
Part of the Fields Institute Communications book series (FIC, volume 82)

Abstract

We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the validity of the solution, however obtained and by whichever method. This includes a nontrivial safeguard against slips, blunders, or bugs in the original computation. We also demonstrate its utility in deciding when to truncate an asymptotic series, improving on the well-known rule of thumb indicating truncation just prior to the smallest term. We also give an example of elimination of spurious secular terms even when genuine secularity is present in the equation. We give short expositions of several well-known perturbation methods together with computer implementations (as scripts that can be modified). We also give a generic backward error based method that is equivalent to iteration (but we believe useful as an organizational viewpoint) for regular perturbation.

Notes

Acknowledgements

We would like to thank Pei Yu, Robert H.C. Moir, and Julia Jankowski for their various contributions to this paper. We are also indebted to NSERC, Western University, and Galima Hassan for the key logistic support they provided.

References

  1. 1.
    Avrachenkov KE, Filar JA, Howlett PG (2013) Analytic perturbation theory and its applications. SIAM, PhiladelphiaCrossRefGoogle Scholar
  2. 2.
    Bellman RE (1972) Perturbation techniques in mathematics, physics, and engineering. Dover Publications, MineolazbMATHGoogle Scholar
  3. 3.
    Bender CM, Orszag SA (1978) Advanced mathematical methods for scientists and engineers: Asymptotic methods and perturbation theory, vol 1. Springer, New YorkzbMATHGoogle Scholar
  4. 4.
    Boas ML (1996) Mathematical methods in the physical sciences. Wiley, New YorkGoogle Scholar
  5. 5.
    Boyd JP (2014) Solving transcendental equations. SIAM, PhiladelphiaCrossRefGoogle Scholar
  6. 6.
    Corless RM (1993) What is a solution of an ODE? ACM SIGSAM Bull 27(4):15–19CrossRefGoogle Scholar
  7. 7.
    Corless RM, Corliss GF (1992) Rationale for guaranteed ODE defect control. In: Atanassova L, Herzberger J (eds) Computer arithmetic and enclosure methods, pp 3–12. North-Holland, AmsterdamGoogle Scholar
  8. 8.
    Corless RM, Fillion N (2013) A graduate introduction to numerical methods, from the viewpoint of backward error analysis. Springer, New York, 868 pp.CrossRefGoogle Scholar
  9. 9.
    Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5(1):329–359MathSciNetCrossRefGoogle Scholar
  10. 10.
    De Bruijn NG (1981) Asymptotic methods in analysis, vol 4. Dover Publications, MineolazbMATHGoogle Scholar
  11. 11.
    Deuflhard P, Hohmann A (2003) Numerical analysis in modern scientific computing: an introduction, vol 43. Springer, New YorkzbMATHGoogle Scholar
  12. 12.
    Enright WH (1989a) A new error-control for initial value solvers. Appl Math Comput 31:288–301MathSciNetzbMATHGoogle Scholar
  13. 13.
    Enright WH (1989b) Analysis of error control strategies for continuous Runge-Kutta methods. SIAM J Numer Anal 26(3):588–599MathSciNetCrossRefGoogle Scholar
  14. 14.
    Geddes KO, Czapor SR, Labahn G (1992) Algorithms for computer algebra. Kluwer Academic, BostonCrossRefGoogle Scholar
  15. 15.
    Grcar JF (2011) John von Neumann’s analysis of Gaussian elimination and the origins of modern numerical analysis. SIAM Rev 53(4):607–682MathSciNetCrossRefGoogle Scholar
  16. 16.
    Higham NJ (1996) Accuracy and stability of numerical algorithms, 2nd edn. SIAM, PhiladelphiazbMATHGoogle Scholar
  17. 17.
    Holmes MH (1995) Introduction to perturbation methods. Springer, New YorkCrossRefGoogle Scholar
  18. 18.
    Kirkinis E (2012) The renormalization group: a perturbation method for the graduate curriculum. SIAM Rev 54(2):374–388MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lanczos C (1988) Applied analysis. Dover Publications, MineolazbMATHGoogle Scholar
  20. 20.
    Lawden DF (2013) Elliptic functions and applications, vol 80. Springer Science & Business Media, HeidelbergzbMATHGoogle Scholar
  21. 21.
    Morrison JA (1966) Comparison of the modified method of averaging and the two variable expansion procedure. SIAM Rev 8(1):66–85MathSciNetCrossRefGoogle Scholar
  22. 22.
    Nayfeh AH (2011) Introduction to perturbation techniques. Wiley, New YorkzbMATHGoogle Scholar
  23. 23.
    Olver FWJ, Lozier DW, Boisvert RF, Clark CW (2010) NIST handbook of mathematical functions. Cambridge University Press, CambridgezbMATHGoogle Scholar
  24. 24.
    O’Malley RE (2014) Historical developments in singular pertubations. Springer, New YorkzbMATHGoogle Scholar
  25. 25.
    O’Malley RE, Kirkinis E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud Appl Math 124(4):383–410MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rand R, Armbruster D (2012) Perturbation methods, bifurcation theory and computer algebra, vol 65. Springer Science & Business Media, New YorkzbMATHGoogle Scholar
  27. 27.
    Roberts AJ (2014) Model emergent dynamics in complex systems. In: SIAMGoogle Scholar
  28. 28.
    Salvy B, Shackell J (2010) Measured limits and multiseries. J Lond Math Soc 82(3):747–762MathSciNetCrossRefGoogle Scholar
  29. 29.
    Van Dyke M (1964) Perturbation methods in fluid mechanics. Academic, New YorkzbMATHGoogle Scholar
  30. 30.
    Wilkinson JH (1963) Rounding errors in algebraic processes. Prentice-Hall series in automatic computation. Prentice-Hall, Englewood CliffsGoogle Scholar
  31. 31.
    Wilkinson JH (1965) The algebraic eigenvalue problem. Oxford University Press, New YorkzbMATHGoogle Scholar
  32. 32.
    Wilkinson JH (1971) Modern error analysis. SIAM Rev 13(4):548–568MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wilkinson JH (1984) The perfidious polynomial, vol 24. Mathematical Association of America, WashingtonzbMATHGoogle Scholar
  34. 34.
    Zhang Y, Corless RM (2014) High-accuracy series solution for two-dimensional convection in a horizontal concentric cylinder. SIAM J Appl Math 74(3):599–619MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The Rotman Institute of Philosophy, The Ontario Research Center for Computer Algebra and The School of Mathematical and Statistical SciencesThe University of Western OntarioLondonCanada
  2. 2.Department of PhilosophySimon Fraser University PhilosophyBurnabyCanada

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