# rfftf1.f

          SUBROUTINE RFFTF1 (N, C, CH, WA, IFAC)
    C***BEGIN PROLOGUE  RFFTF1
    C***PURPOSE  Compute the forward transform of a real, periodic sequence.
    C***LIBRARY   SLATEC (FFTPACK)
    C***CATEGORY  J1A1
    C***TYPE      SINGLE PRECISION (RFFTF1-S, CFFTF1-C)
    C***KEYWORDS  FFTPACK, FOURIER TRANSFORM
    C***AUTHOR  Swarztrauber, P. N., (NCAR)
    C***DESCRIPTION
    C
    C   Subroutine RFFTF1 computes the Fourier coefficients of a real
    C   periodic sequence (Fourier analysis).  The transform is defined
    C   below at output parameter C.
    C
    C   The arrays WA and IFAC which are used by subroutine RFFTB1 must be
    C   initialized by calling subroutine RFFTI1.
    C
    C   Input Arguments
    C
    C   N       the length of the array R to be transformed.  The method
    C           is most efficient when N is a product of small primes.
    C           N may change so long as different work arrays are provided.
    C
    C   C       a real array of length N which contains the sequence
    C           to be transformed.
    C
    C   CH      a real work array of length at least N.
    C
    C   WA      a real work array which must be dimensioned at least N.
    C
    C   IFAC    an integer work array which must be dimensioned at least 15.
    C
    C           The WA and IFAC arrays must be initialized by calling
    C           subroutine RFFTI1, and different WA and IFAC arrays must be
    C           used for each different value of N.  This initialization
    C           does not have to be repeated so long as N remains unchanged.
    C           Thus subsequent transforms can be obtained faster than the
    C           first.  The same WA and IFAC arrays can be used by RFFTF1
    C           and RFFTB1.
    C
    C   Output Argument
    C
    C   C       C(1) = the sum from I=1 to I=N of R(I)
    C
    C           If N is even set L = N/2; if N is odd set L = (N+1)/2
    C
    C             then for K = 2,...,L
    C
    C                C(2*K-2) = the sum from I = 1 to I = N of
    C
    C                     C(I)*COS((K-1)*(I-1)*2*PI/N)
    C
    C                C(2*K-1) = the sum from I = 1 to I = N of
    C
    C                    -C(I)*SIN((K-1)*(I-1)*2*PI/N)
    C
    C           If N is even
    C
    C                C(N) = the sum from I = 1 to I = N of
    C
    C                     (-1)**(I-1)*C(I)
    C
    C   Notes:  This transform is unnormalized since a call of RFFTF1
    C           followed by a call of RFFTB1 will multiply the input
    C           sequence by N.
    C
    C           WA and IFAC contain initialization calculations which must
    C           not be destroyed between calls of subroutine RFFTF1 or
    C           RFFTB1.
    C
    C***REFERENCES  P. N. Swarztrauber, Vectorizing the FFTs, in Parallel
    C                 Computations (G. Rodrigue, ed.), Academic Press,
    C                 1982, pp. 51-83.
    C***ROUTINES CALLED  RADF2, RADF3, RADF4, RADF5, RADFG
    C***REVISION HISTORY  (YYMMDD)
    C   790601  DATE WRITTEN
    C   830401  Modified to use SLATEC library source file format.
    C   860115  Modified by Ron Boisvert to adhere to Fortran 77 by
    C           changing dummy array size declarations (1) to (*).
    C   881128  Modified by Dick Valent to meet prologue standards.
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   900131  Routine changed from subsidiary to user-callable.  (WRB)
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  RFFTF1