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c  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c  .                                                             .
c  .                  copyright (c) 1998 by UCAR                 .
c  .                                                             .
c  .       University Corporation for Atmospheric Research       .
c  .                                                             .
c  .                      all rights reserved                    .
c  .                                                             .
c  .                                                             .
c  .                         SPHEREPACK                       .
c  .                                                             .
c  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
c
c
c This file contains documentation for subroutine alfk.  The subroutine
c is in the file alf.f
c
c subroutine alfk (n,m,cp)
c
c dimension of           real cp(n/2 + 1)
c arguments
c
c purpose                routine alfk computes single precision fourier
c                        coefficients in the trigonometric series
c                        representation of the normalized associated
c                        legendre function pbar(n,m,theta) for use by
c                        routines lfp and lfpt in calculating single
c                        precision pbar(n,m,theta).
c
c                        first define the normalized associated
c                        legendre functions
c
c                        pbar(m,n,theta) = sqrt((2*n+1)*factorial(n-m)
c                        /(2*factorial(n+m)))*sin(theta)**m/(2**n*
c                        factorial(n)) times the (n+m)th derivative of
c                        (x**2-1)**n with respect to x=cos(theta)
c
c                        where theta is colatitude.
c
c                        then subroutine alfk computes the coefficients
c                        cp(k) in the following trigonometric
c                        expansion of pbar(m,n,theta).
c
c                        1) for n even and m even, pbar(m,n,theta) =
c                           .5*cp(1) plus the sum from k=1 to k=n/2
c                           of cp(k+1)*cos(2*k*th)
c
c                        2) for n even and m odd, pbar(m,n,theta) =
c                           the sum from k=1 to k=n/2 of
c                           cp(k)*sin(2*k*th)
c
c                        3) for n odd and m even, pbar(m,n,theta) =
c                           the sum from k=1 to k=(n+1)/2 of
c                           cp(k)*cos((2*k-1)*th)
c
c                        4) for n odd and m odd,  pbar(m,n,theta) =
c                           the sum from k=1 to k=(n+1)/2 of
c                           cp(k)*sin((2*k-1)*th)
c
c
c usage                  call alfk(n,m,cp)
c
c arguments
c
c on input               n
c                          nonnegative integer specifying the degree of
c                          pbar(n,m,theta)
c
c                        m
c                          is the order of pbar(n,m,theta). m can be
c                          any integer however cp is computed such that
c                          pbar(n,m,theta) = 0 if abs(m) is greater
c                          than n and pbar(n,m,theta) = (-1)**m*
c                          pbar(n,-m,theta) for negative m.
c
c on output              cp
c                          single precision array of length (n/2)+1
c                          which contains the fourier coefficients in
c                          the trigonometric series representation of
c                          pbar(n,m,theta)
c
c
c special conditions     none
c
c precision              single
c
c algorithm              the highest order coefficient is determined in
c                        closed form and the remainig coefficients are
c                        determined as the solution of a backward
c                        recurrence relation.
c
c accuracy               comparison between routines alfk and double
c                        precision dalfk on the cray1 indicates
c                        greater accuracy for smaller values
c                        of input parameter n.  agreement to 14
c                        places was obtained for n=10 and to 13
c                        places for n=100.
c