计算子午线弧长的三类算法及其分析比较"

多项式展开算法是计算子午线弧长的传统方法,为了研究利用数值积分算法和常微分方程数值解法进行子午线弧长计算的可行性与可靠性,本文选取大地纬度自0°至90°的3组样本数据(间隔距离分别为1°、1'、1″),分别基于多项式展开数值积分算法和常微分方程数值解法,计算得到各组样本数据的子午线弧长,并通过算法计算结果精度和运算速度两个方面对数值算法的质量进行了评价。计算结果表明:数值积分算法和常微分方程数值解法均可以得到与多项式展开算法精度相同的结果;数值积分算法可通过减小步长以提高计算结果精度,但运算速度急剧降低;3阶、4阶的Runge-Kutta算法不仅运算结果精度高,而且运算速度也比传统算法快3倍多,表明了常微分方程数值解法更适用于子午线弧长的大数据计算。

The Three Kinds of Algorithm for Calculating Meridian Arc Length and Their Analysis and Comparison

The Polynomial expansion algorithm is the traditional method to calculate the meridian arc length.In order to study the feasibility and reliability of using numerical integration algorithm and numerical solution of ordinary differential equations for meridian arc length calculation,it was selected 3 sets of sample data within the geodetic latitude from 0°to 90°,whose intervals are 1°,1',1",respectively.Based on the polynomial expansion algorithms,numerical integral algorithms and numerical solution of ordinary differential equations,the corresponding meridian arc length results were calculated and then evaluated the quality of each numerical algorithm with regard to algorithm accuracy and computation speed.The results show that the numerical integration algorithm and numerical solution of ordinary differential equations can be obtained the same accuracy results of the polynomial expansion algorithm,and the numerical integral can increase calculation accuracy by decreasing the step length,but the computation speed drops dramatically;The 3 and 4 order Runge-Kutta algorithm not only have high precision but the computing speed is twice more than the traditional expanded algorithm,which showed that the numerical solution of the ordinary differential equation is more suitable for large data calculation of the meridian arc length.