A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions
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 f...
| Main Authors: | , |
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| Format: | Journal Article |
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Springer-Verlag
2002
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| Online Access: | http://hdl.handle.net/20.500.11937/22940 |
| _version_ | 1848751013345886208 |
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| author | Holmes, S. Featherstone, Will |
| author_facet | Holmes, S. Featherstone, Will |
| author_sort | Holmes, S. |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T07:45:58Z |
| format | Journal Article |
| id | curtin-20.500.11937-22940 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T07:45:58Z |
| publishDate | 2002 |
| publisher | Springer-Verlag |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-229402017-09-13T13:57:02Z A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions Holmes, S. Featherstone, Will Spherical Harmonic Expansions - Fully Normalised Associated Legendre Functions - Clenshaw Summation - Recursion - Horner's Scheme 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. 2002 Journal Article http://hdl.handle.net/20.500.11937/22940 10.1007/s00190-002-0216-2 Springer-Verlag fulltext |
| spellingShingle | Spherical Harmonic Expansions - Fully Normalised Associated Legendre Functions - Clenshaw Summation - Recursion - Horner's Scheme Holmes, S. Featherstone, Will A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions |
| title | A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions |
| title_full | A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions |
| title_fullStr | A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions |
| title_full_unstemmed | A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions |
| title_short | A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions |
| title_sort | unified approach to the clenshaw summation and the recursive computation of very high degree and order normalised associated legendre functions |
| topic | Spherical Harmonic Expansions - Fully Normalised Associated Legendre Functions - Clenshaw Summation - Recursion - Horner's Scheme |
| url | http://hdl.handle.net/20.500.11937/22940 |