2006-01-16 11:39:11 -05:00
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SUBROUTINE FOUR2a (DATA,N,NDIM,ISIGN,IFORM)
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C Cooley-Tukey fast Fourier transform in USASI basic Fortran.
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C multi-dimensional transform, each dimension a power of two,
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C complex or real data.
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C TRANSFORM(K1,K2,...) = SUM(DATA(J1,J2,...)*EXP(ISIGN*2*PI*SQRT(-1)
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C *((J1-1)*(K1-1)/N(1)+(J2-1)*(K2-1)/N(2)+...))), summed for all
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C J1 and K1 from 1 to N(1), J2 and K2 from 1 TO N(2),
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C etc, for all NDIM subscripts. NDIM must be positive and
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C each N(IDIM) must be a power of two. ISIGN is +1 or -1.
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C Let NTOT = N(1)*N(2)*...*N(NDIM). Then a -1 transform
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C followed by a +1 one (or vice versa) returns NTOT
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C times the original data.
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C IFORM = 1, 0 or -1, as data is
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C complex, real, or the first half of a complex array. Transform
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C values are returned in array DATA. They are complex, real, or
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C the first half of a complex array, as IFORM = 1, -1 or 0.
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C The transform of a real array (IFORM = 0) dimensioned N(1) by N(2)
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C by ... will be returned in the same array, now considered to
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C be complex of dimensions N(1)/2+1 by N(2) by .... Note that if
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C IFORM = 0 or -1, N(1) must be even, and enough room must be
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C reserved. The missing values may be obtained by complex conjuga-
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C tion.
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C The reverse transformation of a half complex array dimensioned
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C N(1)/2+1 by N(2) by ..., is accomplished by setting IFORM
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C to -1. In the N array, N(1) must be the true N(1), not N(1)/2+1.
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C The transform will be real and returned to the input array.
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C Running time is proportional to NTOT*LOG2(NTOT), rather than
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C the naive NTOT**2. Furthermore, less error is built up.
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C Written by Norman Brenner of MIT Lincoln Laboratory, January 1969.
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C See IEEE Audio Transactions (June 1967), Special issue on FFT.
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parameter(NMAX=2048*1024)
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DIMENSION DATA(NMAX), N(1)
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NTOT=1
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DO 10 IDIM=1,NDIM
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10 NTOT=NTOT*N(IDIM)
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IF (IFORM) 70,20,20
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20 NREM=NTOT
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DO 60 IDIM=1,NDIM
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NREM=NREM/N(IDIM)
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NPREV=NTOT/(N(IDIM)*NREM)
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NCURR=N(IDIM)
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IF (IDIM-1+IFORM) 30,30,40
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30 NCURR=NCURR/2
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40 CALL BITRV (DATA,NPREV,NCURR,NREM)
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CALL COOL2 (DATA,NPREV,NCURR,NREM,ISIGN)
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IF (IDIM-1+IFORM) 50,50,60
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50 CALL FIXRL (DATA,N(1),NREM,ISIGN,IFORM)
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NTOT=(NTOT/N(1))*(N(1)/2+1)
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60 CONTINUE
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RETURN
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70 NTOT=(NTOT/N(1))*(N(1)/2+1)
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NREM=1
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DO 100 JDIM=1,NDIM
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IDIM=NDIM+1-JDIM
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NCURR=N(IDIM)
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IF (IDIM-1) 80,80,90
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80 NCURR=NCURR/2
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CALL FIXRL (DATA,N(1),NREM,ISIGN,IFORM)
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NTOT=NTOT/(N(1)/2+1)*N(1)
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90 NPREV=NTOT/(N(IDIM)*NREM)
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CALL BITRV (DATA,NPREV,NCURR,NREM)
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CALL COOL2 (DATA,NPREV,NCURR,NREM,ISIGN)
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100 NREM=NREM*N(IDIM)
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RETURN
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END
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SUBROUTINE BITRV (DATA,NPREV,N,NREM)
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C SHUFFLE THE DATA BY BIT REVERSAL.
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C DIMENSION DATA(NPREV,N,NREM)
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C COMPLEX DATA
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C EXCHANGE DATA(J1,J4REV,J5) WITH DATA(J1,J4,J5) FOR ALL J1 FROM 1
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C TO NPREV, ALL J4 FROM 1 TO N (WHICH MUST BE A POWER OF TWO), AND
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C ALL J5 FROM 1 TO NREM. J4REV-1 IS THE BIT REVERSAL OF J4-1. E.G.
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C SUPPOSE N = 32. THEN FOR J4-1 = 10011, J4REV-1 = 11001, ETC.
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parameter(NMAX=2048*1024)
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DIMENSION DATA(NMAX)
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IP0=2
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IP1=IP0*NPREV
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IP4=IP1*N
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IP5=IP4*NREM
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I4REV=1
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C I4REV = 1+(J4REV-1)*IP1
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DO 60 I4=1,IP4,IP1
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C I4 = 1+(J4-1)*IP1
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IF (I4-I4REV) 10,30,30
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10 I1MAX=I4+IP1-IP0
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DO 20 I1=I4,I1MAX,IP0
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C I1 = 1+(J1-1)*IP0+(J4-1)*IP1
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DO 20 I5=I1,IP5,IP4
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C I5 = 1+(J1-1)*IP0+(J4-1)*IP1+(J5-1)*IP4
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I5REV=I4REV+I5-I4
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C I5REV = 1+(J1-1)*IP0+(J4REV-1)*IP1+(J5-1)*IP4
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TEMPR=DATA(I5)
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TEMPI=DATA(I5+1)
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DATA(I5)=DATA(I5REV)
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DATA(I5+1)=DATA(I5REV+1)
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DATA(I5REV)=TEMPR
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20 DATA(I5REV+1)=TEMPI
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C ADD ONE WITH DOWNWARD CARRY TO THE HIGH ORDER BIT OF J4REV-1.
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30 IP2=IP4/2
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40 IF (I4REV-IP2) 60,60,50
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50 I4REV=I4REV-IP2
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IP2=IP2/2
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IF (IP2-IP1) 60,40,40
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60 I4REV=I4REV+IP2
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RETURN
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END
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SUBROUTINE COOL2 (DATA,NPREV,N,NREM,ISIGN)
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C DISCRETE FOURIER TRANSFORM OF LENGTH N. IN-PLACE COOLEY-TUKEY
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C ALGORITHM, BIT-REVERSED TO NORMAL ORDER, SANDE-TUKEY PHASE SHIFTS.
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C DIMENSION DATA(NPREV,N,NREM)
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C COMPLEX DATA
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C DATA(J1,K4,J5) = SUM(DATA(J1,J4,J5)*EXP(ISIGN*2*PI*I*(J4-1)*
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C (K4-1)/N)), SUMMED OVER J4 = 1 TO N FOR ALL J1 FROM 1 TO NPREV,
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C K4 FROM 1 TO N AND J5 FROM 1 TO NREM. N MUST BE A POWER OF TWO.
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C METHOD--LET IPREV TAKE THE VALUES 1, 2 OR 4, 4 OR 8, ..., N/16,
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C N/4, N. THE CHOICE BETWEEN 2 OR 4, ETC., DEPENDS ON WHETHER N IS
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C A POWER OF FOUR. DEFINE IFACT = 2 OR 4, THE NEXT FACTOR THAT
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C IPREV MUST TAKE, AND IREM = N/(IFACT*IPREV). THEN--
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C DIMENSION DATA(NPREV,IPREV,IFACT,IREM,NREM)
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C COMPLEX DATA
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C DATA(J1,J2,K3,J4,J5) = SUM(DATA(J1,J2,J3,J4,J5)*EXP(ISIGN*2*PI*I*
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C (K3-1)*((J3-1)/IFACT+(J2-1)/(IFACT*IPREV)))), SUMMED OVER J3 = 1
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C TO IFACT FOR ALL J1 FROM 1 TO NPREV, J2 FROM 1 TO IPREV, K3 FROM
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C 1 TO IFACT, J4 FROM 1 TO IREM AND J5 FROM 1 TO NREM. THIS IS
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C A PHASE-SHIFTED DISCRETE FOURIER TRANSFORM OF LENGTH IFACT.
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C FACTORING N BY FOURS SAVES ABOUT TWENTY FIVE PERCENT OVER FACTOR-
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C ING BY TWOS. DATA MUST BE BIT-REVERSED INITIALLY.
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C IT IS NOT NECESSARY TO REWRITE THIS SUBROUTINE INTO COMPLEX
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C NOTATION SO LONG AS THE FORTRAN COMPILER USED STORES REAL AND
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C IMAGINARY PARTS IN ADJACENT STORAGE LOCATIONS. IT MUST ALSO
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C STORE ARRAYS WITH THE FIRST SUBSCRIPT INCREASING FASTEST.
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parameter(NMAX=2048*1024)
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DIMENSION DATA(NMAX)
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real*8 twopi,wstpr,wstpi,wr,wi,w2r,w2i,w3r,w3i,wtempr
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TWOPI=6.2831853072*FLOAT(ISIGN)
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IP0=2
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IP1=IP0*NPREV
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IP4=IP1*N
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IP5=IP4*NREM
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IP2=IP1
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C IP2=IP1*IPROD
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NPART=N
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10 IF (NPART-2) 60,30,20
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20 NPART=NPART/4
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GO TO 10
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C DO A FOURIER TRANSFORM OF LENGTH TWO
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30 IF (IP2-IP4) 40,160,160
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40 IP3=IP2*2
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C IP3=IP2*IFACT
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DO 50 I1=1,IP1,IP0
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C I1 = 1+(J1-1)*IP0
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DO 50 I5=I1,IP5,IP3
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C I5 = 1+(J1-1)*IP0+(J4-1)*IP3+(J5-1)*IP4
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I3A=I5
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I3B=I3A+IP2
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C I3 = 1+(J1-1)*IP0+(J2-1)*IP1+(J3-1)*IP2+(J4-1)*IP3+(J5-1)*IP4
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TEMPR=DATA(I3B)
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TEMPI=DATA(I3B+1)
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DATA(I3B)=DATA(I3A)-TEMPR
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DATA(I3B+1)=DATA(I3A+1)-TEMPI
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DATA(I3A)=DATA(I3A)+TEMPR
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50 DATA(I3A+1)=DATA(I3A+1)+TEMPI
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IP2=IP3
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C DO A FOURIER TRANSFORM OF LENGTH FOUR (FROM BIT REVERSED ORDER)
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60 IF (IP2-IP4) 70,160,160
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70 IP3=IP2*4
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C IP3=IP2*IFACT
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C COMPUTE TWOPI THRU WR AND WI IN DOUBLE PRECISION, IF AVAILABLE.
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THETA=TWOPI/FLOAT(IP3/IP1)
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SINTH=SIN(THETA/2)
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WSTPR=-2*SINTH*SINTH
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WSTPI=SIN(THETA)
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WR=1.
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WI=0.
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DO 150 I2=1,IP2,IP1
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C I2 = 1+(J2-1)*IP1
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IF (I2-1) 90,90,80
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80 W2R=WR*WR-WI*WI
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W2I=2*WR*WI
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W3R=W2R*WR-W2I*WI
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W3I=W2R*WI+W2I*WR
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90 I1MAX=I2+IP1-IP0
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DO 140 I1=I2,I1MAX,IP0
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C I1 = 1+(J1-1)*IP0+(J2-1)*IP1
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DO 140 I5=I1,IP5,IP3
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C I5 = 1+(J1-1)*IP0+(J2-1)*IP1+(J4-1)*IP3+(J5-1)*IP4
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I3A=I5
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I3B=I3A+IP2
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I3C=I3B+IP2
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I3D=I3C+IP2
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C I3 = 1+(J1-1)*IP0+(J2-1)*IP1+(J3-1)*IP2+(J4-1)*IP3+(J5-1)*IP4
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IF (I2-1) 110,110,100
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C APPLY THE PHASE SHIFT FACTORS
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100 TEMPR=DATA(I3B)
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DATA(I3B)=W2R*DATA(I3B)-W2I*DATA(I3B+1)
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DATA(I3B+1)=W2R*DATA(I3B+1)+W2I*TEMPR
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TEMPR=DATA(I3C)
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DATA(I3C)=WR*DATA(I3C)-WI*DATA(I3C+1)
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DATA(I3C+1)=WR*DATA(I3C+1)+WI*TEMPR
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TEMPR=DATA(I3D)
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DATA(I3D)=W3R*DATA(I3D)-W3I*DATA(I3D+1)
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DATA(I3D+1)=W3R*DATA(I3D+1)+W3I*TEMPR
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110 T0R=DATA(I3A)+DATA(I3B)
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T0I=DATA(I3A+1)+DATA(I3B+1)
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T1R=DATA(I3A)-DATA(I3B)
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T1I=DATA(I3A+1)-DATA(I3B+1)
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T2R=DATA(I3C)+DATA(I3D)
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T2I=DATA(I3C+1)+DATA(I3D+1)
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T3R=DATA(I3C)-DATA(I3D)
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T3I=DATA(I3C+1)-DATA(I3D+1)
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DATA(I3A)=T0R+T2R
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DATA(I3A+1)=T0I+T2I
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DATA(I3C)=T0R-T2R
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DATA(I3C+1)=T0I-T2I
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IF (ISIGN) 120,120,130
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120 T3R=-T3R
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T3I=-T3I
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130 DATA(I3B)=T1R-T3I
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DATA(I3B+1)=T1I+T3R
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DATA(I3D)=T1R+T3I
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140 DATA(I3D+1)=T1I-T3R
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WTEMPR=WR
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WR=WSTPR*WTEMPR-WSTPI*WI+WTEMPR
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150 WI=WSTPR*WI+WSTPI*WTEMPR+WI
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IP2=IP3
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GO TO 60
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160 RETURN
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END
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SUBROUTINE FIXRL (DATA,N,NREM,ISIGN,IFORM)
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C FOR IFORM = 0, CONVERT THE TRANSFORM OF A DOUBLED-UP REAL ARRAY,
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C CONSIDERED COMPLEX, INTO ITS TRUE TRANSFORM. SUPPLY ONLY THE
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C FIRST HALF OF THE COMPLEX TRANSFORM, AS THE SECOND HALF HAS
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C CONJUGATE SYMMETRY. FOR IFORM = -1, CONVERT THE FIRST HALF
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C OF THE TRUE TRANSFORM INTO THE TRANSFORM OF A DOUBLED-UP REAL
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C ARRAY. N MUST BE EVEN.
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C USING COMPLEX NOTATION AND SUBSCRIPTS STARTING AT ZERO, THE
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C TRANSFORMATION IS--
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C DIMENSION DATA(N,NREM)
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C ZSTP = EXP(ISIGN*2*PI*I/N)
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C DO 10 I2=0,NREM-1
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C DATA(0,I2) = CONJ(DATA(0,I2))*(1+I)
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C DO 10 I1=1,N/4
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C Z = (1+(2*IFORM+1)*I*ZSTP**I1)/2
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C I1CNJ = N/2-I1
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C DIF = DATA(I1,I2)-CONJ(DATA(I1CNJ,I2))
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C TEMP = Z*DIF
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C DATA(I1,I2) = (DATA(I1,I2)-TEMP)*(1-IFORM)
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C 10 DATA(I1CNJ,I2) = (DATA(I1CNJ,I2)+CONJ(TEMP))*(1-IFORM)
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C IF I1=I1CNJ, THE CALCULATION FOR THAT VALUE COLLAPSES INTO
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C A SIMPLE CONJUGATION OF DATA(I1,I2).
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parameter(NMAX=2048*1024)
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DIMENSION DATA(NMAX)
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TWOPI=6.283185307*FLOAT(ISIGN)
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IP0=2
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IP1=IP0*(N/2)
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IP2=IP1*NREM
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IF (IFORM) 10,70,70
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C PACK THE REAL INPUT VALUES (TWO PER COLUMN)
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10 J1=IP1+1
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DATA(2)=DATA(J1)
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IF (NREM-1) 70,70,20
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20 J1=J1+IP0
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I2MIN=IP1+1
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DO 60 I2=I2MIN,IP2,IP1
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DATA(I2)=DATA(J1)
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J1=J1+IP0
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IF (N-2) 50,50,30
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30 I1MIN=I2+IP0
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I1MAX=I2+IP1-IP0
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DO 40 I1=I1MIN,I1MAX,IP0
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DATA(I1)=DATA(J1)
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DATA(I1+1)=DATA(J1+1)
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40 J1=J1+IP0
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50 DATA(I2+1)=DATA(J1)
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60 J1=J1+IP0
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70 DO 80 I2=1,IP2,IP1
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TEMPR=DATA(I2)
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DATA(I2)=DATA(I2)+DATA(I2+1)
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80 DATA(I2+1)=TEMPR-DATA(I2+1)
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IF (N-2) 200,200,90
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90 THETA=TWOPI/FLOAT(N)
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SINTH=SIN(THETA/2.)
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ZSTPR=-2.*SINTH*SINTH
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ZSTPI=SIN(THETA)
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ZR=(1.-ZSTPI)/2.
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ZI=(1.+ZSTPR)/2.
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IF (IFORM) 100,110,110
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100 ZR=1.-ZR
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ZI=-ZI
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110 I1MIN=IP0+1
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I1MAX=IP0*(N/4)+1
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DO 190 I1=I1MIN,I1MAX,IP0
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DO 180 I2=I1,IP2,IP1
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I2CNJ=IP0*(N/2+1)-2*I1+I2
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IF (I2-I2CNJ) 150,120,120
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120 IF (ISIGN*(2*IFORM+1)) 130,140,140
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130 DATA(I2+1)=-DATA(I2+1)
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140 IF (IFORM) 170,180,180
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150 DIFR=DATA(I2)-DATA(I2CNJ)
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DIFI=DATA(I2+1)+DATA(I2CNJ+1)
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TEMPR=DIFR*ZR-DIFI*ZI
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TEMPI=DIFR*ZI+DIFI*ZR
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DATA(I2)=DATA(I2)-TEMPR
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DATA(I2+1)=DATA(I2+1)-TEMPI
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DATA(I2CNJ)=DATA(I2CNJ)+TEMPR
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DATA(I2CNJ+1)=DATA(I2CNJ+1)-TEMPI
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IF (IFORM) 160,180,180
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160 DATA(I2CNJ)=DATA(I2CNJ)+DATA(I2CNJ)
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DATA(I2CNJ+1)=DATA(I2CNJ+1)+DATA(I2CNJ+1)
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170 DATA(I2)=DATA(I2)+DATA(I2)
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DATA(I2+1)=DATA(I2+1)+DATA(I2+1)
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180 CONTINUE
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TEMPR=ZR-.5
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ZR=ZSTPR*TEMPR-ZSTPI*ZI+ZR
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190 ZI=ZSTPR*ZI+ZSTPI*TEMPR+ZI
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C RECURSION SAVES TIME, AT A SLIGHT LOSS IN ACCURACY. IF AVAILABLE,
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C USE DOUBLE PRECISION TO COMPUTE ZR AND ZI.
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200 IF (IFORM) 270,210,210
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C UNPACK THE REAL TRANSFORM VALUES (TWO PER COLUMN)
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210 I2=IP2+1
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I1=I2
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J1=IP0*(N/2+1)*NREM+1
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GO TO 250
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220 DATA(J1)=DATA(I1)
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DATA(J1+1)=DATA(I1+1)
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I1=I1-IP0
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J1=J1-IP0
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230 IF (I2-I1) 220,240,240
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240 DATA(J1)=DATA(I1)
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DATA(J1+1)=0.
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250 I2=I2-IP1
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J1=J1-IP0
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DATA(J1)=DATA(I2+1)
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DATA(J1+1)=0.
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I1=I1-IP0
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J1=J1-IP0
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IF (I2-1) 260,260,230
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260 DATA(2)=0.
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270 RETURN
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END
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