BEGIN PROLOGUE VRFFTB DATE WRITTEN R

      SUBROUTINE VRFFTB(M,N,R,RT,MDIMR,WSAVE)

C***BEGIN PROLOGUE  VRFFTB

C***DATE WRITTEN   850801   (YYMMDD)

C***REVISION DATE  900509   (YYMMDD)

C***CATEGORY NO.  J1A1

C***KEYWORDS  FAST FOURIER TRANSFORM, REAL PERIODIC TRANSFORM, 

C             FOURIER SYNTHESIS, BACKWARD TRANSFORM, MULTIPLE SEQUENCES

C***AUTHOR  SWEET, R.A. (NIST) AND LINDGREN, L.L. (NIST)

C***PURPOSE  Backward real periodic transform, M sequences.

C***DESCRIPTION

C

C  Subroutine VRFFTB computes the synthesis (backward transform) of a

C  number of real periodic sequences from their Fourier coefficients. 

C  Specifically, for each set of independent Fourier coefficients

C  F(K), the corresponding real periodic sequence is computed. 

C

C  The array WSAVE which is used by subroutine VRFFTB must be

C  initialized by calling subroutine VRFFTI(N,WSAVE).

C

C

C  Input Parameters

C

C  M       the number of sets of coefficients.

C

C  N       the length of the sequences of coefficients to be 

C          transformed.  The method is most efficient when N is a

C          product of small primes, however n may be any positive 

C          integer.

C

C  R       areal two-dimensional array of size MDIMX x N containing the

C          coefficients to be transformed.  Each set of coefficients

C          F(K), K\0,1,..,N-1, is stored as a ROW of R.  Specifically,

C          the I-th set of independent Fourier coefficients is stored

C

C                R(I,1) = REAL( F(I,0) ),

C

C                R(I,2*K) = REAL( F(I,K) )

C

C                R(I,2*K+1) = IMAG( F(I,K) )

C

C                   for K = 1, 2, . . . , M-1,

C

C                and, when N is even,

C

C                R(I,N) = REAL( F(I,N/2) ).

C

C  RT      a real two-dimensional work array of size MDIMX x N.

C

C  MDIMR   the row (or first) dimension of the arrays R and RT exactly 

C          as they appear in the calling program.  This parameter is 

C          used to specify the variable dimension of these arrays.

C

C  WSAVE   a real one-dimensional work array which must be dimensioned

C          at least N+15.  The WSAVE array must be initialized by 

C          calling subroutine VRFFTI.  A different WSAVE array must be

C          used for each different value of N.  This initialization does

C          not have to be repeated so long as N remains unchanged.  The

C          same WSAVE array may be used by VRFFTB and VRFFTB.

C

C  Output Parameters

C

C  R       contains M real periodic sequences corresponding to the given

C          coefficients.  Specifically, the I-th row of R contains the 

C          real periodic sequence corresponding to the I-th set of

C          independent Fourier coefficients F(I,K) stored as

C

C               R(I,J) = X(I,J-1) ,   J = 1, 2, . . . , N, where

C

C               X(I,J) = SQRT(1/N)* F(I,0) + (-1)**J*F(I,N/2)

C                        + 2*SUM(K=1,M)[ REAL(F(I,2K))*COS(2K*J*PI/N)

C                        - IMAG(F(I,2K+1))*SIN(2K*J*PI/N) ]  ,

C

C                 when N is even, and

C

C               X(I,J) = SQRT(1/N)* F(I,0) +

C                        2*SUM(K=1,M)[ REAL(F(I,2K))*COS(2K*J*PI/N)

C                        - IMAG(F(I,2K+1))*SIN(2K*J*PI/N) ]  ,

C

C                 when N is odd.

C

C  WSAVE   contains results which must not be destroyed between calls

C          to VRFFTF or VRFFTB.

C

C  -----------------------------------------------------------------

C

C  NOTE  -  A call of VRFFTF followed immediately by a call of

C           of VRFFTB will return the original sequences R.  Thus,

C           VRFFTB is the correctly normalized inverse of VRFFTF.

C

C  -----------------------------------------------------------------

C

C  VRFFTB is a straightforward extension of the subprogram RFFTB to

C  handle M simultaneous sequences.  RFFTB was originally developed

C  by P. N. Swarztrauber of NCAR.

C

C

C              * * * * * * * * * * * * * * * * * * * * *

C              *                                       *

C              *         PROGRAM SPECIFICATIONS        *

C              *                                       *

C              * * * * * * * * * * * * * * * * * * * * *

C

C

C     DIMENSION OF    R(MDIMR,N), RT(MDIMR,N), WSAVE(N+15)

C     ARGUMENTS

C

C     LATEST          AUGUST 1, 1985

C     REVISION

C

C     SUBPROGRAMS     VRFFTI, VRFTI1, VRFFTF, VRFTF1, VRADF2, VRADF3,

C     REQUIRED        VRADF4, VRADF5, VRADFG, VRFFTB, VRFTB1, VRADB2,

C                     VRADB3, VRADB4, VRADB5, VRADBG, PIMACH

C

C     SPECIAL         NONE

C     CONDITIONS

C

C     COMMON          NONE

C     BLOCKS

C

C     I/O             NONE

C

C     PRECISION       SINGLE

C

C     SPECIALIST      ROLAND SWEET

C

C     LANGUAGE        FORTRAN

C

C     HISTORY         WRITTEN BY LINDA LINDGREN AND ROLAND SWEET AT THE

C                     NATIONAL BUREAU OF STANDARDS (BOULDER).

C

C     ALGORITHM       A REAL VARIANT OF THE STOCKHAM AUTOSORT VERSION

C                     OF THE COOLEY-TUKEY FAST FOURIER TRANSFORM.

C

C     PORTABILITY     AMERICAN NATIONAL STANDARDS INSTITUTE FORTRAN 77.

C                     THE ONLY MACHINE DEPENDENT CONSTANT IS LOCATED IN

C                     THE FUNCTION PIMACH.

C

C     REQUIRED        COS,SIN

C     RESIDENT

C     ROUTINES

C

C

C***REFERENCES  P. N. Swarztrauber, Vectorizing the FFTs, in Parallel

C               Computations, (G. Rodrigue, ed.), Academic Press, 1982,

C               pp. 51-83.

C***ROUTINES CALLED  VRFTB1

C***END PROLOGUE  VRFFTB

C

C     VRFFTPK, VERSION 1, AUGUST 1985

C

      DIMENSION     R(MDIMR,N),RT(MDIMR,N),WSAVE(N+15)

      IF (N .EQ. 1) RETURN

      CALL VRFTB1 (M,N,R,RT,MDIMR,WSAVE(1),WSAVE(N+1))

      RETURN

      END

.