C
C
C             * * * * * * * * * * * * * * * * * * * * * *
C             *                                         *
C             * SUBROUTINE VRFFTB(M,N,R,RT,MDIMR,WSAVE) *
C             *                                         *
C             * * * * * * * * * * * * * * * * * * * * * *
C
C
C     SUBROUTINE VRFFTB COMPUTES THE SYNTHESIS (BACKWARD TRANSFORM) OF
C     A NUMBER OF REAL PERODIC SEQUENCES FROM THEIR FOURIER
C     COEFFICIENTS.  SPECIFICALLY, FOR EACH SET OF INDEPENDENT FOURIER
C     COEFFICIENTS F(K) DESCRIBED ABOVE, THE CORRESPONDING REAL PERIODIC
C     SEQUENCE IS COMPUTED.
C
C
C     INPUT PARAMETERS
C
C     M       THE NUMBER OF SETS OF COEFFIENTS.
C
C     N       THE NUMBER OF COEFFIENTS TO BE TRANSFORMED.  THE METHOD
C             IS MOST EFFICIENT WHEN N IS A PRODUCT OF SMALL PRIMES,
C             HOWEVER N MAY BE ANY POSITIVE INTEGER.
C
C     R       A REAL TWO-DIMENSIONAL ARRAY OF SIZE MDIMR X N THAT
C             CONTAINS THE COEFFICIENTS TO BE TRANSFORMED.  EACH SET OF
C             COEFFICIENTS F(K), K=0,1,...,N-1, IS STORED AS A ROW OF R.
C             SPECIFICALLY, THE I-TH SET OF INDEPENDENT FOURIER
C             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 MDIMR X N.
C
C     MDIMR   THE ROW (OR FIRST) DIMENSION OF THE ARRAYS R AND RT AS
C             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 THAT MUST BE DIMENSIONED
C             AT LEAST N+15.  THE WSAVE ARRAY MUST BE INITIALIZED BY
C             CALLING SUBROUTINE VRFFTI.  A DIFFERENT WSAVE ARRAY MUST
C             BE USED FOR EACH DIFFERENT VALUE OF N. THIS INITIALIZATION
C             DOES NOT HAVE TO BE REPEATED SO LONG AS N REMAINS
C             UNCHANGED.  THE SAME WSAVE ARRAY CAN BE USED BY VRFFTF
C             AND VRFFTB.
C
C
C     OUTPUT PARAMETERS
C
C     R       CONTAINS THE M REAL PERIODIC SEQUENCES CORRESPONDING TO
C             THE GIVEN COEFFICIENTS.  SPECIFICALLY, THE I-TH ROW OF THE
C             MATRIX R CONTAINS THE REAL PERIODIC SEQUENCE CORRESPONDING
C             TO THE I-TH SET OF INDEPENDENT FOURIER COEFFICIENTS F(I,K)
C             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
C             CALLS TO VRFFTB OR VRFFTF.
C
C
C
C              * * * * * * * * * * * * * * * * * * * * *
C              *                                       *
C              *         PROGRAM SPECIFICATIONS        *
C              *                                       *
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