# ezfftb.f

          SUBROUTINE EZFFTB (N, R, AZERO, A, B, WSAVE)
    C***BEGIN PROLOGUE  EZFFTB
    C***PURPOSE  A simplified real, periodic, backward fast Fourier
    C            transform.
    C***LIBRARY   SLATEC (FFTPACK)
    C***CATEGORY  J1A1
    C***TYPE      SINGLE PRECISION (EZFFTB-S)
    C***KEYWORDS  FFTPACK, FOURIER TRANSFORM
    C***AUTHOR  Swarztrauber, P. N., (NCAR)
    C***DESCRIPTION
    C
    C  Subroutine EZFFTB computes a real periodic sequence from its
    C  Fourier coefficients (Fourier synthesis).  The transform is
    C  defined below at Output Parameter R.  EZFFTB is a simplified
    C  but slower version of RFFTB.
    C
    C  Input Parameters
    C
    C  N       the length of the output array R.  The method is most
    C          efficient when N is the product of small primes.
    C
    C  AZERO   the constant Fourier coefficient
    C
    C  A,B     arrays which contain the remaining Fourier coefficients.
    C          These arrays are not destroyed.
    C
    C          The length of these arrays depends on whether N is even or
    C          odd.
    C
    C          If N is even, N/2    locations are required.
    C          If N is odd, (N-1)/2 locations are required
    C
    C  WSAVE   a work array which must be dimensioned at least 3*N+15
    C          in the program that calls EZFFTB.  The WSAVE array must be
    C          initialized by calling subroutine EZFFTI(N,WSAVE), and a
    C          different WSAVE array must be used for each different
    C          value of N.  This initialization does not have to be
    C          repeated so long as N remains unchanged.  Thus subsequent
    C          transforms can be obtained faster than the first.
    C          The same WSAVE array can be used by EZFFTF and EZFFTB.
    C
    C  Output Parameters
    C
    C  R       if N is even, define KMAX=N/2
    C          if N is odd,  define KMAX=(N-1)/2
    C
    C          Then for I=1,...,N
    C
    C               R(I)=AZERO plus the sum from K=1 to K=KMAX of
    C
    C               A(K)*COS(K*(I-1)*2*PI/N)+B(K)*SIN(K*(I-1)*2*PI/N)
    C
    C  ********************* Complex Notation **************************
    C
    C          For J=1,...,N
    C
    C          R(J) equals the sum from K=-KMAX to K=KMAX of
    C
    C               C(K)*EXP(I*K*(J-1)*2*PI/N)
    C
    C          where
    C
    C               C(K) = .5*CMPLX(A(K),-B(K))   for K=1,...,KMAX
    C
    C               C(-K) = CONJG(C(K))
    C
    C               C(0) = AZERO
    C
    C                    and I=SQRT(-1)
    C
    C  *************** Amplitude - Phase Notation ***********************
    C
    C          For I=1,...,N
    C
    C          R(I) equals AZERO plus the sum from K=1 to K=KMAX of
    C
    C               ALPHA(K)*COS(K*(I-1)*2*PI/N+BETA(K))
    C
    C          where
    C
    C               ALPHA(K) = SQRT(A(K)*A(K)+B(K)*B(K))
    C
    C               COS(BETA(K))=A(K)/ALPHA(K)
    C
    C               SIN(BETA(K))=-B(K)/ALPHA(K)
    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  RFFTB
    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   861211  REVISION DATE from Version 3.2
    C   881128  Modified by Dick Valent to meet prologue standards.
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  EZFFTB