C
C
C             * * * * * * * * * * * * * * * * * * * * * *
C             *                                         *
C             * SUBROUTINE VRFFTF(M,N,R,RT,MDIMR,WSAVE) *
C             *                                         *
C             * * * * * * * * * * * * * * * * * * * * * *
C
C
C     SUBROUTINE VRFFTF COMPUTES THE FOURIER COEFFICIENTS (FORWARD
C     TRANSFORM) OF A NUMBER OF REAL PERIODIC SEQUENCES.  SPECIFICALLY,
C     FOR EACH SEQUENCE THE SUBROUTINE CALCULATES THE INDEPENDENT
C     FOURIER COEFFICIENTS DESCRIBED ABOVE.
C
C
C     INPUT PARAMETERS
C
C     M       THE NUMBER OF TRANSFORMS TO BE COMPUTED.
C
C     N       THE LENGTH OF THE TRANSFORMS.  THE METHOD IS MOST
C             EFFICIENT WHEN N IS A PRODUCT OF SMALL PRIMES, HOWEVER N
C             MAY BE ANY POSITIVE INTEGER.
C
C     R       A REAL TWO-DIMENSIONAL ARRAY OF SIZE MDIMR X N THAT
C             CONTAINS THE SEQUENCES TO BE TRANSFORMED.  THE I-TH
C             SEQUENCE TO BE TRANSFORMED, X(I,J), J=0,1,...,N-1, IS
C             STORED AS
C
C               R(I,J) = X(I,J-1) , J=1, 2, . . . , N.
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 AND
C             VRFFTB.
C
C
C     OUTPUT PARAMETERS
C
C     R       CONTAINS THE FOURIER COEFFICIENTS F(K) DESCRIBED ABOVE
C             FOR EACH OF THE M INPUT SEQUENCES.  SPECIFICALLY, ROW I
C             OF R, R(I,J), J=1,2,..,N, CONTAINS THE INDEPENDENT
C             FOURIER COEFFICIENTS F(I,K), FOR THE I-TH INPUT SEQUENCE
C             STORED AS
C
C             R(I,1) = REAL( F(I,0) ),
C                    = SQRT(1/N)*SUM(J=0,N-1)[ X(I,J) ],
C
C             R(I,2*K) = REAL( F(I,K) )
C                      = SQRT(1/N)*SUM(J=0,N-1)[X(I,J)*COS(2J*K*PI/N)]
C
C             R(I,2*K+1) = IMAG( F(I,K) )
C                        =-SQRT(1/N)*SUM(J=0,N-1)[X(I,J)*SIN(2J*K*PI/N)]
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                     = SQRT(1/N)*SUM(J=0,N-1)[ (-1)**J*X(I,J) ].
C
C     WSAVE   CONTAINS RESULTS THAT MUST NOT BE DESTROYED BETWEEN
C             CALLS TO VRFFTF OR VRFFTB.
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