引用本文: 杨正辉, 魏子卿, 马健. 第二类连带勒让德函数及其一阶、二阶导数的递推计算方法[J]. 武汉大学学报 ( 信息科学版), 2020, 45(2): 213-218.
YANG Zhenghui, WEI Ziqing, MA Jian. Recursive Calculation Method for the Second Kind of Associated Legendre Functions and Its First and Second Derivatives[J]. Geomatics and Information Science of Wuhan University, 2020, 45(2): 213-218.
 Citation: YANG Zhenghui, WEI Ziqing, MA Jian. Recursive Calculation Method for the Second Kind of Associated Legendre Functions and Its First and Second Derivatives[J]. Geomatics and Information Science of Wuhan University, 2020, 45(2): 213-218.

Recursive Calculation Method for the Second Kind of Associated Legendre Functions and Its First and Second Derivatives

• 摘要: 推导出了地球重力场位模型椭球谐级数表达式中的第二类连带勒让德函数及其一阶、二阶导数的修正Jekeli递推计算方法，并与传统的Jekeli递推计算方法的结果进行比较。结果表明，在递推计算收敛精度相同的条件下，修正Jekeli递推计算比传统Jekeli递推计算需要的收敛项数减少一半，最大约为30项；随着阶次nm的增大以及收敛项k的增加，修正Jekeli递推计算的第二类连带勒让德微分方程的精度一直保持在1×10-6左右；不同高度下，阶数n与高度h之间的关系与球近似下（R/rn+1的阶数n与高度h之间的关系相似。

Abstract: This paper derives the ellipsoid harmonic series of earth's gravitational field about the second kind of associated Legendre functions and its first and second derivatives recursive calculation method, and it compares their results with those of traditional Jekeli recursive method. The result shows that the numbers with maxima of about 30 is required terms of the revised Jekeli's recurrences is about half numbers compared with the traditional recurrences, when the recursive computation has the same convergence accuracy. As degree n, order m and convergent k increase, the revised Jekeli's recurrences is all fulfilled to 1×10-6. Therefore, we also can see that the relationship of the height h and degree n in the ellipsoid approximation, as similar with the approximation of spherical harmonics's (R/r)n+1 in different heights.

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