76 / 2023-03-31 23:52:12

Applicability of standard forward column/row recurrence formulas for ALFs

Associated Legendre functions(ALFs),fully normalized Associated Legendre functions (fnALFs),standard forward column/row recurrence formulas,,the isolated singular factor method,the extended-range arithmetic

The Associative Legendre functions are a set of orthogonal basis functions. The so-called recurrence formula is usually used to calculate the fully normalized associated Legendre functions. This paper gives the applicability and universality of the standard forward column/row recurrence formula based on the method of the isolated singular factor method and the extended-range arithmetic. It is pointed out that the method of isolating singular factor is a special normalization method. It can significantly improve the applicability of the standard forward row recurrence formula to a certain extent, but it is invalid for standard forward column recurrence formula. In fact, the extended-range arithmetic is to expand the double precision number field to 4 precision number field, which can significantly improve the applicability of the column/row recurrence formula. However, this requires considerable computational efforts, 40-80 times slower than double precision computation, and also requires double storage space. Therefore, the extended-range arithmetic is not advisable in practical application.