Applicability of standard forward column/row recurrence equations for ALFs

ALFs标准按列/行递推公式的适用性

Fully normalized associated Legendre functions(fnALFs)are a set of orthogonal basis functions that are usually calculated by using the recurrence equation.This paper presented the applicability and universality of the standard forward column/row recurrence equation based on the isolated singular factor method and extended-range arithmetic.Isolating a singular factor is a special normalization method that can improve the universality of the standard forward row recurrence equation to a certain extent,its universality can up to degree hundreds.However,it is invalid for standard forward column recurrence equation.The extended-range arithmetic expands the double-precision number field to the quad-precision numberfield.The quad-precision numberfield can retain more significant digits in the operation process and express larger and smaller numbers.The extended-range arithmetic can significantly improve the applicability and universality of the standard forward column/row recurrence equations,its universality can up to degree several thousand.However,the quad-precision numberfield operation needs to occupy more storage space,which is why its operation speed is slow and undesirable in practical applications.In this paper,the X-number method is introduced in the standard forward row recurrence equation for thefirst time.With the use of the X-number method,fnALFs can be recursed to 4.2 billion degree by using standard forward column/row recurrence equations.

完全规格化的缔合勒让德函数(fnALFs)是一组正交的基函数。通常利用所谓的递推公式进行计算。本文基于分离奇异因子方法和扩充数域方法,给出了标准向前按列/行递推公式的适用性和普适性。分离奇异因子方法,在一定程度上,提升标准向前按行递推公式的普适性,其普适性可达到几百阶,但该方法对标准向前按列递推公式无效。扩充数域方法就是将双精度数域运算扩展到4精度数域。由于4精度数在运算过程中能够保留更多的有效数字,且能够表达更大和更小的数,因此扩充数域方法可显著提升标准向前按列/行递推公式的普适性和适用性,其普适性可达到几千阶。但是4精度运算需要占据更多存储空间,因此运算速度很慢,在实际应用中并不可取。本文首次将X-数方法引入到标准向前按行递推公式,基于X-数方法,利用标准向前按列/行递推公式都可将fnALFs递推至42亿阶。