第二类勒让德函数的重规格化及其优化

doi:10.11947/j.AGCS.2024.20230283

测绘学报 ›› 2024, Vol. 53 ›› Issue (7): 1298-1307.doi: 10.11947/j.AGCS.2024.20230283

第二类勒让德函数的重规格化及其优化

张捍卫¹^,²(), 杨永勤²(), 李晓玲², 张华²

^1.山东理工大学建筑工程与信息工程学院，山东　淄博　255000
^2.河南理工大学测绘与国土信息工程学院，河南　焦作　454003

收稿日期:2023-07-12 发布日期:2024-08-12
通讯作者: 杨永勤 E-mail:zhanwei800@163.com;212104010025@home.hpu.edu.cn
作者简介:张捍卫（1967—），男，博士，教授，博士生导师，主要从事大地测量学的教学与研究。E-mail：zhanwei800@163.com
基金资助:
国家自然科学基金(42074002)

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

Hanwei ZHANG¹^,²(), Yongqin YANG²(), Xiaoling LI², Hua ZHANG²

^1.School of Civil Engineering and Geomatics, Shandong University of Technology, Zibo 255000, China
^2.School of Surveying and Land Information Engineering, Henan Polytechnic University, Jiaozuo 454003, China

Received:2023-07-12 Published:2024-08-12
Contact: Yongqin YANG E-mail:zhanwei800@163.com;212104010025@home.hpu.edu.cn
About author:ZHANG Hanwei (1967—), male, PhD, professor, PhD supervisor, majors in teaching and research of geodesy. E-mail: zhanwei800@163.com
Supported by:
The National Natural Science Foundation of China(42074002)

摘要/Abstract

摘要：

椭球谐函数级数展开是地球重力场椭球谐建模的基础。然而，处理椭球谐函数级数的主要困难在于计算第二类勒让德函数。而Jekeli的重规格化方法简化了这一计算过程。本文在Jekeli重规格化的基础上，详细推导了两种基于高斯超几何函数变换的优化递归方法，同时利用这两种优化递归的方法计算了第二类勒让德函数，并将其展开到二阶导数。通过数值计算，证明了优化递归方法可以有效加快收敛，缩短计算时间，并且适用阶数更高，这使得椭球谐函数级数在实际应用中更加方便、可行。

关键词: 第二类勒让德函数, 缔合勒让德微分方程, 重规格化, 高斯超几何函数

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

中图分类号:

P223

张捍卫, 杨永勤, 李晓玲, 张华. 第二类勒让德函数的重规格化及其优化[J]. 测绘学报, 2024, 53(7): 1298-1307.

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.

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参考文献 25

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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

第二类勒让德函数的重规格化及其优化

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

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