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


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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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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