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    A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions

    18932_119976.pdf (305.5Kb)
    Access Status
    Open access
    Authors
    Holmes, S.
    Featherstone, Will
    Date
    2002
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Holmes, S.A. and Featherstone, W.E.. 2002. A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. Journal of Geodesy 76 (5): 279-299.
    Source Title
    Journal of Geodesy
    DOI
    10.1007/s00190-002-0216-2
    Faculty
    Division of Resources and Environment
    Department of Spatial Sciences
    Remarks

    Originally published in Journal of Geodesy 2002 76(5) pp.279-299.

    Copyright Springer-Verlag

    The original article is available at springerlink.com.

    URI
    http://hdl.handle.net/20.500.11937/22940
    Collection
    • Curtin Research Publications
    Abstract

    Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude. This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums, complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method. However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis.

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