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Numerische Mathematik

, Volume 52, Issue 5, pp 499–509 | Cite as

On Clenshaws's method and a generalisation to faber series

  • S. W. Ellacott
  • E. B. Saff
Article

Summary

The method of expanding the solution of linear ordinary differential equations in power or laurent series is classical, and is usually associated with the name of Frobenius. In the early days of electronic computation, it was appreciated that expansions in Chebyshev series are often of more practical use, and the necessary techniques developed by Clenshaw. (This is usually carried out in order to approximate special functions defined by ordinary differential equations, rather than as a technique for actually solving such equations in general, for which finite difference methods are to be preferred.) In this paper we show that by means of a conformal map the problem of expansion in Chebyshev series can in fact be reduced to that of expansion in a Laurent series, yielding a method which is usually computationally simpler. Moreoveer, the method can be generalised to the case of Faber expansions on regions of the complex plane. A non-trivial example is explored in order to illustrate the method, and we also use the technique to generalise an identity relating Chebyshev polynomials to the Faber case.

Subject Classifications

AMS (MOS): 34A25 30E10, 33–04, 33A40, 65D20, 65E05, 65L07 CR: G1.2 

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. (9th Dover printing based on the U.S. Government Printing Office edition). Dover. New York (1972)Google Scholar
  2. 2.
    Coleman, J.P.: Polynomial Approximations in the Complex Plane. J. Comp. Appl. Math.18, 193–211 (1987)Google Scholar
  3. 3.
    Coleman, J.P., Smith, R.A.: The Faber Polynomials for Circular Sectors. Math. Comput.99 (179), 231–241 (1987)Google Scholar
  4. 4.
    Ellacott, S.W.: computation of Faber Series with Application to Numerical Polynomial Approximation in the Complex Plane. Math. Comput.40, 575–587 (1983)Google Scholar
  5. 5.
    Ellacott, S.W.: On the Faber Tranksform and Efficient Numerical Rational Approximation. SIAM J. Numer. Anal.20, 989–1000 (1983)Google Scholar
  6. 6.
    Ellacott, S.W., Gutknecht, M.H.: The Polynomial Caratheodory-Féjér Approximation Method for Jordan Regions. IMA J. Numer. Anal.3, 207–220 (1983)Google Scholar
  7. 7.
    Ellacott, S.W., Saff, E.B.: Computing with the Faber Transform. In: Rational Approximation and Interpolation (P. Graves-Morris, E.B. Saff, R.S. Varga, eds.), pp. 412–418. Springer Lecture Notes in Mathematics no. 1105, Heidelberg, Berlin, New York: Springer 1984Google Scholar
  8. 8.
    Fox, L.: Numerical Solution of Ordinary and Partial Differential Equations. Oxford: Pergamon Press 1962Google Scholar
  9. 9.
    Gaier, D.: Vorlesungen ueber Approximation im Komplexen. Basel: Birkhaeuser 1980Google Scholar
  10. 10.
    Kemp, P.: An Analysis of Chebyshev Series Methods for the Solution of Ordinary Differential Equations, with Applications Involving Interval Analysis. Ph.D. Dissertation, University of Cambridge, 1971Google Scholar
  11. 11.
    Luke, Y.: The Special Functions and their Applications (2 volumes). London: Academic Press 1969Google Scholar
  12. 12.
    Markushevich, A.I.: Theory of Functions of a Complex Variable, translated by R.A. Silverman. New York: Chelsea 1977Google Scholar

Copyright information

© Springe-Verlag 1988

Authors and Affiliations

  • S. W. Ellacott
    • 1
  • E. B. Saff
    • 2
  1. 1.Information Technology Research InstituteBrighton PolytechnicBrightonUK
  2. 2.Institute for Constructive Mathematics. Department of MathematicsUniversity of South FloridaTampaUSA

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