Renormalization and its optimization of the Legendre function of the second kind

doi:10.11947/j.AGCS.2024.20230283

Abstract

Abstract:

The series expansion of ellipsoidal harmonic functions is the basis for ellipsoid harmonic modeling of the Earth's gravity field. However, the main difficulty in dealing with ellipsoidal harmonics series lies in the calculation of Legendre functions of the second kind. Jekeli's renormalization method simplifies this calculation process. Based on Jekeli's renormalization, this paper deduces two optimization recursive methods based on transformations of Gaussian hypergeometric functions are derived in details. At the same time, these two optimization recursive methods are used to calculate the second type of Legendre function, and expand it to the second derivative. Numerical calculations have proven that the optimization recursive method can effectively accelerate convergence, shorten calculation time, and is applicable to higher orders, which makes the ellipsoid harmonic function series more convenient and feasible in practical applications.

Key words: the Legendre function of the second kind, the associated Legendre differential equation, renormalization, Gaussian hypergeometric function

CLC Number:

P223

Hanwei ZHANG, Yongqin YANG, Xiaoling LI, Hua ZHANG. Renormalization and its optimization of the Legendre function of the second kind[J]. Acta Geodaetica et Cartographica Sinica, 2024, 53(7): 1298-1307.

Figures/Tables 9

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原始递归的\|LDE\|	n=1500	n=2000	n=2500	n=3000
最大值	1.90×10^-2	3.45	3.75×10³	4.20×10⁶
平均值	3.79×10^-6	3.10×10^-2	2.52	2.06×10³

阶数n	同时减小α、β项的优化递归的\|LDE\|		只减小β项的优化递归的\|LDE\|
阶数n	最大值	平均值	最大值	平均值
1500	1.86×10^-8	3.56×10^-10	1.12×10^-8	1.96×10^-10
3000	1.81×10^-5	4.16×10^-8	7.15×10^-7	1.56×10^-8
5000	8.95×10^-1	6.12×10^-4	6.10×10^-5	7.60×10^-7
10 000	3.38×10¹¹	4.95×10⁷	1.25	5.00×10^-2

阶数n	原始递归	同时减小α、β项的优化递归	只减小β项的优化递归
1500	15.247	7.324	9.100
2000	32.961	14.126	17.589
2500	57.399	23.733	29.791
5000	—	124.493	151.844
10 000	—	724.021	820.401