*DECK RFFTB1 SUBROUTINE RFFTB1 (N, C, CH, WA, IFAC) C***BEGIN PROLOGUE RFFTB1 C***PURPOSE Compute the backward fast Fourier transform of a real C coefficient array. C***LIBRARY SLATEC (FFTPACK) C***CATEGORY J1A1 C***TYPE SINGLE PRECISION (RFFTB1-S, CFFTB1-C) C***KEYWORDS FFTPACK, FOURIER TRANSFORM C***AUTHOR Swarztrauber, P. N., (NCAR) C***DESCRIPTION C C Subroutine RFFTB1 computes the real periodic sequence from its C Fourier coefficients (Fourier synthesis). The transform is defined C below at output parameter C. C C The arrays WA and IFAC which are used by subroutine RFFTB1 must be C initialized by calling subroutine RFFTI1. C C Input Arguments C C N the length of the array R to be transformed. The method C is most efficient when N is a product of small primes. C N may change so long as different work arrays are provided. C C C a real array of length N which contains the sequence C to be transformed. C C CH a real work array of length at least N. C C WA a real work array which must be dimensioned at least N. C C IFAC an integer work array which must be dimensioned at least 15. C C The WA and IFAC arrays must be initialized by calling C subroutine RFFTI1, and different WA and IFAC arrays must be C used for each different value of N. This initialization C does not have to be repeated so long as N remains unchanged. C Thus subsequent transforms can be obtained faster than the C first. The same WA and IFAC arrays can be used by RFFTF1 C and RFFTB1. C C Output Argument C C C For N even and for I = 1,...,N C C C(I) = C(1)+(-1)**(I-1)*C(N) C C plus the sum from K=2 to K=N/2 of C C 2.*C(2*K-2)*COS((K-1)*(I-1)*2*PI/N) C C -2.*C(2*K-1)*SIN((K-1)*(I-1)*2*PI/N) C C For N odd and for I = 1,...,N C C C(I) = C(1) plus the sum from K=2 to K=(N+1)/2 of C C 2.*C(2*K-2)*COS((K-1)*(I-1)*2*PI/N) C C -2.*C(2*K-1)*SIN((K-1)*(I-1)*2*PI/N) C C Notes: This transform is unnormalized since a call of RFFTF1 C followed by a call of RFFTB1 will multiply the input C sequence by N. C C WA and IFAC contain initialization calculations which must C not be destroyed between calls of subroutine RFFTF1 or C RFFTB1. 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 RADB2, RADB3, RADB4, RADB5, RADBG 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 881128 Modified by Dick Valent to meet prologue standards. C 891214 Prologue converted to Version 4.0 format. (BAB) C 900131 Routine changed from subsidiary to user-callable. (WRB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE RFFTB1 DIMENSION CH(*), C(*), WA(*), IFAC(*) C***FIRST EXECUTABLE STATEMENT RFFTB1 NF = IFAC(2) NA = 0 L1 = 1 IW = 1 DO 116 K1=1,NF IP = IFAC(K1+2) L2 = IP*L1 IDO = N/L2 IDL1 = IDO*L1 IF (IP .NE. 4) GO TO 103 IX2 = IW+IDO IX3 = IX2+IDO IF (NA .NE. 0) GO TO 101 CALL RADB4 (IDO,L1,C,CH,WA(IW),WA(IX2),WA(IX3)) GO TO 102 101 CALL RADB4 (IDO,L1,CH,C,WA(IW),WA(IX2),WA(IX3)) 102 NA = 1-NA GO TO 115 103 IF (IP .NE. 2) GO TO 106 IF (NA .NE. 0) GO TO 104 CALL RADB2 (IDO,L1,C,CH,WA(IW)) GO TO 105 104 CALL RADB2 (IDO,L1,CH,C,WA(IW)) 105 NA = 1-NA GO TO 115 106 IF (IP .NE. 3) GO TO 109 IX2 = IW+IDO IF (NA .NE. 0) GO TO 107 CALL RADB3 (IDO,L1,C,CH,WA(IW),WA(IX2)) GO TO 108 107 CALL RADB3 (IDO,L1,CH,C,WA(IW),WA(IX2)) 108 NA = 1-NA GO TO 115 109 IF (IP .NE. 5) GO TO 112 IX2 = IW+IDO IX3 = IX2+IDO IX4 = IX3+IDO IF (NA .NE. 0) GO TO 110 CALL RADB5 (IDO,L1,C,CH,WA(IW),WA(IX2),WA(IX3),WA(IX4)) GO TO 111 110 CALL RADB5 (IDO,L1,CH,C,WA(IW),WA(IX2),WA(IX3),WA(IX4)) 111 NA = 1-NA GO TO 115 112 IF (NA .NE. 0) GO TO 113 CALL RADBG (IDO,IP,L1,IDL1,C,C,C,CH,CH,WA(IW)) GO TO 114 113 CALL RADBG (IDO,IP,L1,IDL1,CH,CH,CH,C,C,WA(IW)) 114 IF (IDO .EQ. 1) NA = 1-NA 115 L1 = L2 IW = IW+(IP-1)*IDO 116 CONTINUE IF (NA .EQ. 0) RETURN DO 117 I=1,N C(I) = CH(I) 117 CONTINUE RETURN END