A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions
MetadataShow full item record
Originally published in Journal of Geodesy 2002 76(5) pp.279-299.
The original article is available at springerlink.com.
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.
Showing items related by title, author, creator and subject.
Tuthill, John D. (1995)In antenna array processing it is generally required to enhance the reception or detection of a signal from a particular direction while suppressing noise and interference signals from other directions. An optimisation ...
Nadarajah, Nandakumaran; Kirubarajan, T.; Lang, T.; McDonald, M.; Punithakumar, K. (2011)In general, for multitarget problems where the number of targets and their states are time varying, the optimal Bayesian multitarget tracking is computationally demanding. The Probability Hypothesis Density (PHD) filter, ...
Teunissen, Peter; Khodabandeh, A. (2013)In this contribution, we extend ‘Kalman-filter’ theory by introducing a new BLUE–BLUP recursion of the partitioned measurement and dynamic models. Instead of working with known state-vector means, we relax the model and ...