缔合Legendre函数二阶导数的快速稳定递推算法

根据初等代数的基本原理,推导了一种缔合Legendre函数二阶导数的快速稳定递推算法。数值测试结果表明,在阶次高达3 600时,该方法与其他几种现有方法的计算精度相当,但计算效率比其他方法提高了一倍以上,并且该方法没有奇异性,适用于快速精确地计算任意纬度的缔合Legendre函数二阶导数值

Fast and Stably Recursive Algorithm for Computing Second Derivative of Associated Legendre Functions

Based on the basic principles of elementary algebra,a fast and stably recursive algorithm for computing second derivative of associated Legendre's functions is derived.Numerical tests suggest that this new approach is exactly as precise as general ones.But the principal strength of the new approach is that it is much faster than general ones in computation speed(at least twice as fast).The method is non-singular(the relative accuracy can be achieved to 5×10-10 up to degree and order 3 600 at the poles) and simplicity of formulation and implementation(just need a few lines of code).The approach can compute the second derivative of associated Legendre functions of any latitude quickly and accurately,which is very important for gravity gradient data processing of GOCE satellite.