超高阶次Legendre函数的跨阶数递推算法

本文引入了Legendre函数的跨阶数递推算法,并利用该算法在双精度数范围内计算了按间隔为1°余纬从1°变化至89°对应的直到完整的20000阶次的归一化连带Legendre函数的值.为验证计算精度,通过多种途径对该算法的计算结果进行检验,结果表明:该算法算得的每个阶次连带Legendre函数的值至少具有10-10这样的绝对精度.此外还对该算法的计算用时进行了统计,结果为该算法的计算用时大约是Legendre函数计算中常用的按阶数递推算法用时的1.6倍.

A recursion arithmetic formula for Legendre functions of ultra-high degree and order on every other degree

The most popular arithmetic for Legendre functions in the study of the gravity field is the increasing degree recursion method (IDR). Although IDR has a simple expression in mathematics, it is not suitable to compute Legendre functions of ultra-high degree and order. For example, when degree reaches 1800, IDR cannot be used to compute Legendre functions because of under-flow in the double floating-point range. Hence, some modified versions for IDR are discussed. A typical modification is to extend the double floating-point range, and the result is that the run-time in computation increases rapidly too. In order to solve the problem in computing Legendre functions of ultra-high degree and order, a recursion arithmetic approach on every other degree for Legendre functions is presented in this paper. Our aim is to illustrate that this approach can be not only used to compute Legendre functions, but the run-time of computation is also saved.#br#The main method is computation, that is, the values of Legendre functions are computed based on the recursion formulas of Legendre functions on every other degree, and then the computation accuracies and the run-time are assessed.#br#The values of the fully normalized associated Legendre functions up to complete degree and order 20000 are computed from colatitude 1° to 89° with 1° interval in the double floating-point range. The computation accuracies are estimated according to the properties of Legendre functions. From our computation results, it can be acclaimed that the computation accuracies of Legendre functions up to complete degree and order 20000 can reach 10^-10 at least if the recursion arithmetic on every other degree is applied. The run-time for computation is listed. From the statistical results, the run-time of the arithmetic in this work is approximately 1.6 times that of IDR.#br#The computation of Legendre functions of ultra-high degree and order plays an important role in refining the gravity field. The recursion arithmetic on every other degree is suitable to compute Legendre functions of ultra-high degree and order. The arithmetic has two main advantages: it has enough accuracies and its run-time is also acceptable compared with IDR. In all, the recursion arithmetic on every other degree is an efficient approach to compute Legendre functions of ultra-high degree and order.