#
# = ALF: a collection of routines for computing associated Legendre polynomials (ALFs)
# ALF is an extension library for GSL to compute associated Legendre polynomials developed by Patrick Alken.
# Ruby/GSL includes interfaces to it if ALF is found during installlation.
#
# The class and method descriptions below are based on references from the document of ALF (alf-1.0/doc/alf.texi) by P.Alken.
#
# == {}[link:index.html"name="1] Module structure
# * GSL::ALF (module)
#   * GSL::ALF::Workspace (Class)
#
# == {}[link:index.html"name="2] Creating ALF workspace
# ---
# * GSL::ALF::Workspace.alloc(lmax)
# * GSL::ALF.alloc(lmax)
#
# Creates a workspace for computing associated Legendre polynomials (ALFs). The maximum ALF degree is specified by lmax. The size of this workspace is O(lmax).
#
# == {}[link:index.html"name="3] Methods
# ---
# * GSL::ALF::Workspace#params(csphase, cnorm, norm)
#
#   Sets various parameters for the subsequent computation of ALFs. If
#   <tt>csphase</tt> is set to a non-zero value, the Condon-Shortley phase of
#   (-1)^m will be applied to the associated Legendre functions. The
#   Condon-Shortley phase is included by default. If <tt>cnorm</tt> is set to
#   zero, the real normalization of the associated Legendre functions will be
#   used. The default is to use complex normalization. The norm parameter
#   defines the type of normalization which will be used. The possible values
#   are as follows.
#   * ALF::NORM_SCHMIDT: Schmidt semi-normalized associated Legendre polynomials S_l^m(x). (default)
#   * ALF::NORM_SPHARM: Associated Legendre polynomials Y_l^m(x) suitable for the calculation of spherical harmonics.
#   * ALF::NORM_ORTHO: Fully orthonormalized associated Legendre polynomials N_l^m(x).
#   * ALF::NORM_NONE:: Unnormalized associated Legendre polynomials P_l^m(x).
# ---
# * GSL::ALF::Workspace#Plm_array(x)
# * GSL::ALF::Workspace#Plm_array(lmax, x)
# * GSL::ALF::Workspace#Plm_array(x, result)
# * GSL::ALF::Workspace#Plm_array(lmax, x, result)
# * GSL::ALF::Workspace#Plm_array(x, result, deriv)
# * GSL::ALF::Workspace#Plm_array(lmax, x, result, deriv)
#
#   Compute all associated Legendre polynomials P_l^m(x) and optionally their
#   first derivatives dP_l^m(x)/dx for 0 <= l <= lmax, 0 <= m <= l. The value
#   of <tt>lmax</tt> cannot exceed the previously specified lmax parameter to
#   <tt>ALF.alloc</tt>, but may be less. If <tt>lmax</tt> is not given, the
#   parameter to <tt>ALF.alloc()</tt> is used. The results are stored in
#   <tt>result</tt>, an instance of <tt>GSL::Vector</tt>. Note that this vector
#   must have enough length to store all the values for the polynomial
#   P_l^m(x), and the length required can be known using
#   <tt>ALF::array_size(lmax)</tt>. If a vector is not given, a new vector is
#   created and returned. 
#
#   The indices of <tt>result</tt> (and <tt>deriv</tt> corresponding to the
#   associated Legendre function of degree <tt>l</tt> and order <tt>m</tt> can
#   be obtained by calling <tt>ALF::array_index(l, m)</tt>.
# ---
# * GSL::ALF::Workspace#Plm_deriv_array(x)
# * GSL::ALF::Workspace#Plm_deriv_array(lmax, x)
# * GSL::ALF::Workspace#Plm_deriv_array(x, result, deriv)
# * GSL::ALF::Workspace#Plm_deriv_array(lmax, x, result, deriv)
#
#   Compute all associated Legendre polynomials P_l^m(x) and their first
#   derivatives dP_l^m(x)/dx for 0 <= l <= lmax, 0 <= m <= l. 
#
# ---
# * GSL::ALF::array_size(lmax)
#
#   Returns the size of arrays needed for the array versions of P_l^m(x).
# ---
# * GSL::ALF::array_index(l, m)
#
#   Returns the array index of results of <tt>Plm_array()</tt> and
#   <tt>Plm_deriv_array()</tt> corresponding to P_l^m(x) and dP_l^m(x)/dx
#   respectively. The index is given by l(l + 1)/2 + m.
#
#
