New extension for Associated Legendre Functions (ALFs)

[Patrick Alken]

Hello all,

  Most current implementations of associated Legendre functions
(ALFs), including the current GSL version fail due to underflow
above degree and order (L,M) 1900.  A 2002 paper by Holmes and
Featherstone shows that with a very simple scaling modification
this limit can be extended to 2700 in double precision. I have
recently needed to compute high degree ALFs so I have implemented
their algorithm and made a new GSL extension if others are interested.

There are several other advantages in this extension over the
current GSL implementation:

1) I have included support for several normalization conventions:
   - Schmidt semi-normalized ALFs
   - Spherical harmonic normalized ALFs (GSL already has this)
   - Fully normalized ALFs
   - Unnormalized ALFs (GSL has this)

2) The user can choose whether to compute the Condon-Shortley
   phase of (-1)^m

3) The GSL array versions currently only compute ALFs for a fixed
   order m and multiple degrees l. This extension now efficiently
   computes all ALFs for a given maximum degree lmax for all
   (l,m) up to lmax.

4) The routines in this extension use a workspace which precomputes
   various factors in the recurrence relations which should lead
   to a (minor) speedup in the calculations.

This extension includes a complete test suite and documentation.
The docs contain a plot which shows how the current GSL
implementation fails for L,M = 2700.

I have put a link on the main GSL webpage for the extension.

Patrick Alken