| 计算子午线弧长与底点纬度的常微分方程数值解法 |
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| 引用本文： | 杨丽坤,雷伟伟.计算子午线弧长与底点纬度的常微分方程数值解法[J].测绘科学技术学报,2017(6):560-563. |
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| 作者姓名： | 杨丽坤 雷伟伟 |
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| 作者单位： | 1. 郑州工业贸易学校,河南郑州,450007;2. 河南理工大学测绘与国土信息工程学院,河南焦作,454000 |
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| 基金项目： | 国家自然科学基金项目(41272373)，国家测绘地理信息局测绘基础研究基金项目(15-01-05) |
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| 摘 要： | 计算子午线弧长与底点纬度本质上是解算标准的一阶常微分方程。为了研究利用常微分方程数值解法进行子午线弧长与底点纬度计算的可行性与可靠性,选取大地纬度自0°起以步长1″依次增大至90°,共计324 001个样本数据,分别基于求解常微分方程的Euler算法、改进的Euler算法以及二阶、三阶、四阶Runge-Kutta算法对其进行了数值计算。并与传统算法结果进行比较,从数值算法结果的精度、运算速度、自洽程度等方面对数值算法质量进行评价。计算结果表明:利用常微分方程数值解法求解子午线弧长与底点纬度的方法,能够得到与传统算法精度一致的结果;且数值算法运算速度大约是传统算法的2倍,其中四阶Runge-Kutta算法的精度与自洽程度最高。这表明,常微分方程数值解法比传统算法更适用于子午线弧长和底点纬度的大数据计算。
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| 关 键 词： | 常微分方程 Euler算法 Runge-Kutta算法 算法精度 算法速度 自洽程度 |
Calculation of Meridian Arc Length and Latitude of Pedal Based on the Numerical Solution of Ordinary Differential Equations |
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| Abstract: | The calculation of meridian arc length and latitude of pedal is to resolve the standard first order ordinary differential equation essentially.In order to study the feasibility and reliability of calculating the meridian arc length and latitude of pedal based on the numerical solution of ordinary differential equation,it is selected the geodetic latitude from 0° to 90° with the step of 1″,in total of 324 001 sample data,the numerical calculation of this equation is carried out based on Euler algorithm,improved Euler algorithm and the two order,three order,and four order Runge-Kutta algorithm.Then comparing with the traditional algorithm,the quality of these algorithms are evaluated through the accuracy of the calculation results,computation speed and self-consistency degree.The results show that the method of calculating the meridian arc length and latitude of pedal by using the numerical solution of ordinary differential equation can obtain the result that is consistent with the precision of traditional algorithm,and the speed of numerical algorithm is about 2 times of that of the traditional algorithm and the four order Runge-Kutta algorithm has the highest accuracy and self-consistency.This indicates that the numerical solution of ordinary differential equation is more suitable for the big data calculation of meridian arc and latitude of pedal. |
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| Keywords: | ordinary differential equation Euler algorithm Runge-Kutta algorithm algorithm accuracy algorithm speed self-consistency |
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