地球物理学报 ›› 2021, Vol. 36 ›› Issue (5): 1900–1904.doi: 10.6038/pg2021EE0409

• 固体地球物理及空间物理学(大气、行星、地球动力学、重磁电及地震学、地热学) • 上一篇    下一篇

缔合勒让德函数的精度评定方法研究

张红兵1, 喻铮铮2, 张捍卫2,*   

  1. 1.河南工业和信息化职业学院,焦作 454000;
    2.河南理工大学,焦作 454000
  • 收稿日期:2020-12-18 修回日期:2021-05-21 发布日期:2021-11-11
  • 通讯作者: * 张捍卫,男,1967年生,博士,教授,博士生导师,研究方向为大地测量.E-mail: zhanwei800@163.com
  • 作者简介:张红兵,男,1971年生,硕士,副教授,研究方向为矿业工程.E-mail: zhhb_1971@126.com
  • 基金资助:
    国家自然科学基金(42074002,41931075)资助.

Study on accuracy evaluation method of the associated Legendre functions

ZHANG HongBing1, YU ZhengZheng2, ZHANG HanWei2,*   

  1. 1. Henan College of Industry and Information Technology, Jiaozuo 454000, China;
    2. Henan Polytechnic University, Jiaozuo 454000, China
  • Received:2020-12-18 Revised:2021-05-21 Published:2021-11-11

摘要: 勒让德方程是物理学和其他技术领域常常遇到的一类常微分方程.当试图在球坐标中求解三维拉普拉斯方程(或相关的其他偏微分方程)时,问题便会归结为勒让德方程的求解,解的形式是一系列勒让德多项式(函数).完全规格化缔合勒让德函数(fnALFs)的数值计算和精度评定,是地球重力场以及相关领域中的重要研究课题之一.本文利用缔合勒让德函数(ALFs)的三角函数级数展开式,研究了其定积分性质.基于球函数加法定理,给出了fnALFs定积分和fnALFs导数数值的精度评定公式.另外,利用连续函数在特定区间上的平均值概念,给出了计算fnALFs数值的精度评定方法.最后,利用引力场模型,给出了各种精度评定方法中相对误差数值确定的理论依据.

关键词: 完全规格化缔合勒让德函数(fnALFs), fnALFs的定积分和导数, 精度评定方法

Abstract: Legendre equation is a kind of ordinary differential equation often encountered in physics and other technical fields. When trying to solve the three dimensional Laplace equation (or other related partial differential equation) in spherical coordinates, the problem boils down to solving the Legendre equation. Its solution’form is a series of Legendre polynomials. The numerical calculation and accuracy evaluation of fully normalized Associated Legendre Functions (fnALFs) are important research topics in the Earth’s gravity field and related fields. In this paper, the properties of definite integral of Associated Legendre Functions (ALFs) are studied by means of its trigonometric function series expansion. Based on the spherical function addition theorem, the accuracy evaluation formulas of fnALFs definite integral and fnALFs derivative are given. In addition, the accuracy evaluation method of fnALFs is given by using the average value concept of continuous function in a specific interval. Finally, using the gravitational field model, the theoretical basis for determining the relative error in various accuracy evaluation methods is given.

Key words: Fully normalized Associated Legendre Functions (fnALFs), Definite integral and derivative of fnALFs, Accuracy evaluation method

中图分类号: 

  • P223
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