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Evaluating Applicability of Four Recursive Algorithms for Computation of the fully normalized associated legendre functions

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JOURNAL OF APPLIED GEODESY 2016年第4期10卷 267-275页

作者： Lei, Weiwei Li, Kai Henan Polytech Univ Sch Geodesy & Land informat Engn High Tech Zone Century Ave 2001 Jiaozuo 454000 Henan Peoples R China Chinese Acad Sci Shanghai Astron Observ Ctr Astrogeodynam Res Shanghai 200030 Peoples R China

There are four recursive algorithms used in the computation of the fully normalized associated legendre functions ( FNALFs): the standard forward column algorithm, the standard forward row algorithm, the recursive algorithm between every other degree, and the Belikov algorithm. These algorithms were evaluated in terms of their first relative numerical accuracy, second relative numerical accuracy, and computation speed and efficiency. The results show that when the degree n reaches 3000, both the recursive algorithm between every other degree and the Belikov algorithm are applicable for vertical bar cos theta vertical bar is an element of[0, 1], with the latter better second relative numerical accuracy than the former at a slower computation speed. In terms of vertical bar cos theta vertical bar is an element of[0, 1], the standard forward column algorithm, the recursive algorithm between every other degree, and the Belikov algorithm are applicable within degree n of 1900, and the standard forward column algorithm has the highest computation speed. The standard forward column algorithm is applicable for vertical bar cos theta vertical bar is an element of[0, 1] within degree n of 1900. This algorithm's range of applicability decreases as the degree increases beyond 1900;however, it remains applicable within a minute range when vertical bar cos theta vertical bar is approximately equal to 1. The standard forward row algorithm has the smallest range of applicability: it is only applicable within degree n of 100 for vertical bar cos theta vertical bar is an element of[0, 1], and its range of applicability decreases rapidly when the degree is greater than 100. The results of this research are expected to be useful to researchers in choosing the best algorithms for use in the computation of the FNALFs.

关键词： fully normalized associated legendre functions Recursive Algorithm Relative Numerical Accuracy Computation Speed Range of Applicability

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Algorithms for Computing Ultra-High-Degree Disturbing Potential Elements

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Chinese Journal of Geophysics 2013年第6期47卷 1150-1157页

作者： Fu-Qing PENG Zhe-Ren XIA

The fast development of modern spatial gravimetric technology makes it possible and more necessary to establish the ultra-high-degree global potential model. The numerical characters of the ultra-high-degree fully normalized associated legendre functions (ALFs) and their 1 st , 2 nd derivatives of the disturbing potential spherical harmonic expansions have been studied from the standard forward column recursions. Then, the abovementioned standard forward column methods are modified to avoid overflow. Finally, some practical algorithms for evaluating the ultra-high-degree disturbing potential elements by personal computer are presented.

关键词： Ultra-high-degree disturbing potential element fully normalized associated legendre functions Horner's scheme

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