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C ******************************************************************* |
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C ** THIS FORTRAN CODE IS INTENDED TO ILLUSTRATE POINTS MADE IN ** |
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C ** THE TEXT. TO OUR KNOWLEDGE IT WORKS CORRECTLY. HOWEVER IT IS ** |
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C ** THE RESPONSIBILITY OF THE USER TO TEST IT, IF IT IS USED IN A ** |
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C ** RESEARCH APPLICATION. ** |
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C ******************************************************************* |
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C ******************************************************************* |
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** FICHE F.37. ROUTINES TO CALCULATE FOURIER TRANSFORMS. ** |
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C ** THREE SEPARATE ROUTINES FOR DIFFERENT APPLICATIONS. ** |
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C ******************************************************************* |
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SUBROUTINE FILONC ( DT, DOM, NMAX, C, CHAT ) |
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C ******************************************************************* |
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C ** CALCULATES THE FOURIER COSINE TRANSFORM BY FILON'S METHOD ** |
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C ** ** |
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C ** A CORRELATION FUNCTION, C(T), IN THE TIME DOMAIN, IS ** |
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C ** TRANSFORMED TO A SPECTRUM CHAT(OMEGA) IN THE FREQUENCY DOMAIN.** |
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C ** ** |
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C ** REFERENCE: ** |
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C ** ** |
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C ** FILON, PROC ROY SOC EDIN, 49 38, 1928. ** |
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C ** ** |
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C ** PRINCIPAL VARIABLES: ** |
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C ** ** |
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C ** REAL C(NMAX) THE CORRELATION FUNCTION. ** |
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C ** REAL CHAT(NMAX) THE 1-D COSINE TRANSFORM. ** |
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C ** REAL DT TIME INTERVAL BETWEEN POINTS IN C. ** |
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C ** REAL DOM FREQUENCY INTERVAL FOR CHAT. ** |
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C ** INTEGER NMAX NO. OF INTERVALS ON THE TIME AXIS ** |
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C ** REAL OMEGA THE FREQUENCY ** |
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C ** REAL TMAX MAXIMUM TIME IN CORRL. FUNCTION ** |
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C ** REAL ALPHA, BETA, GAMMA FILON PARAMETERS ** |
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C ** INTEGER TAU TIME INDEX ** |
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C ** INTEGER NU FREQUENCY INDEX ** |
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C ** ** |
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C ** USAGE: ** |
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C ** ** |
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C ** THE ROUTINE REQUIRES THAT THE NUMBER OF INTERVALS, NMAX, IS ** |
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C ** EVEN AND CHECKS FOR THIS CONDITION. THE FIRST VALUE OF C(T) ** |
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C ** IS AT T=0. THE MAXIMUM TIME FOR THE CORRELATION FUNCTION IS ** |
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C ** TMAX=DT*NMAX. FOR AN ACCURATE TRANSFORM C(TMAX)=0. ** |
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C ******************************************************************* |
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INTEGER NMAX |
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REAL DT, DOM, C(0:NMAX), CHAT(0:NMAX) |
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REAL TMAX, OMEGA, THETA, SINTH, COSTH, CE, CO |
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REAL SINSQ, COSSQ, THSQ, THCUB, ALPHA, BETA, GAMMA |
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INTEGER TAU, NU |
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C ******************************************************************* |
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C ** CHECKS NMAX IS EVEN ** |
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IF ( MOD ( NMAX, 2 ) .NE. 0 ) THEN |
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STOP ' NMAX SHOULD BE EVEN ' |
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ENDIF |
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TMAX = REAL ( NMAX ) * DT |
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C ** LOOP OVER OMEGA ** |
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DO 30 NU = 0, NMAX |
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OMEGA = REAL ( NU ) * DOM |
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THETA = OMEGA * DT |
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C ** CALCULATE THE FILON PARAMETERS ** |
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SINTH = SIN ( THETA ) |
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COSTH = COS ( THETA ) |
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SINSQ = SINTH * SINTH |
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COSSQ = COSTH * COSTH |
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THSQ = THETA * THETA |
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THCUB = THSQ * THETA |
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IF ( THETA. EQ. 0.0 ) THEN |
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ALPHA = 0.0 |
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BETA = 2.0 / 3.0 |
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GAMMA = 4.0 / 3.0 |
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ELSE |
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ALPHA = ( 1.0 / THCUB ) |
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: * ( THSQ + THETA * SINTH * COSTH - 2.0 * SINSQ ) |
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BETA = ( 2.0 / THCUB ) |
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: * ( THETA * ( 1.0 + COSSQ ) -2.0 * SINTH * COSTH ) |
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GAMMA = ( 4.0 / THCUB ) * ( SINTH - THETA * COSTH ) |
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ENDIF |
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C ** DO THE SUM OVER THE EVEN ORDINATES ** |
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CE = 0.0 |
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DO 10 TAU = 0, NMAX, 2 |
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CE = CE + C(TAU) * COS ( THETA * REAL ( TAU ) ) |
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10 CONTINUE |
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C ** SUBTRACT HALF THE FIRST AND LAST TERMS ** |
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CE = CE - 0.5 * ( C(0) + C(NMAX) * COS ( OMEGA * TMAX ) ) |
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C ** DO THE SUM OVER THE ODD ORDINATES ** |
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CO = 0.0 |
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DO 20 TAU = 1, NMAX - 1, 2 |
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CO = CO + C(TAU) * COS ( THETA * REAL ( TAU ) ) |
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20 CONTINUE |
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C ** FACTOR OF TWO FOR THE REAL COSINE TRANSFORM ** |
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CHAT(NU) = 2.0 * ( ALPHA * C(NMAX) * SIN ( OMEGA * TMAX ) |
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: + BETA * CE + GAMMA * CO ) * DT |
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30 CONTINUE |
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RETURN |
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END |
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SUBROUTINE LADO ( DT, NMAX, C, CHAT ) |
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C ******************************************************************* |
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C ** CALCULATES THE FOURIER COSINE TRANSFORM BY LADO'S METHOD ** |
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C ** ** |
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C ** A CORRELATION FUNCTION, C(T), IN THE TIME DOMAIN, IS ** |
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C ** TRANSFORMED TO A SPECTRUM CHAT(OMEGA) IN THE FREQUENCY DOMAIN.** |
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C ** ** |
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C ** REFERENCE: ** |
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C ** ** |
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C ** LADO, J COMPUT PHYS, 8 417, 1971. ** |
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C ** ** |
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C ** PRINCIPAL VARIABLES: ** |
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C ** ** |
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C ** REAL C(NMAX) THE CORRELATION FUNCTION. ** |
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C ** REAL CHAT(NMAX) THE 1-D COSINE TRANSFORM. ** |
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C ** REAL DT TIME INTERVAL BETWEEN POINTS IN C. ** |
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C ** REAL DOM FREQUENCY INTERVAL BETWEEN POINTS IN CHAT.** |
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C ** INTEGER NMAX NO. OF INTERVALS ON THE TIME AXIS ** |
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C ** ** |
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C ** USAGE: ** |
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C ** ** |
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C ** THE CORRELATION FUNCTION IS REQUIRED AT HALF INTEGER ** |
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C ** INTERVALS, I.E. C(T), T=(TAU-0.5)*DT FOR TAU=1 .. NMAX. ** |
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C ** THE COSINE TRANSFORM IS RETURNED AT HALF INTERVALS, I.E. ** |
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C ** CHAT(OMEGA), OMEGA=(NU-0.5)*DOM FOR NU = 1 .. NMAX. ** |
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C ******************************************************************* |
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INTEGER NMAX |
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REAL DT, C(NMAX), CHAT(NMAX) |
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INTEGER TAU, NU |
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REAL TAUH, NUH, NMAXH, PI, SUM |
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PARAMETER ( PI = 3.1415927 ) |
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C ******************************************************************* |
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NMAXH = REAL ( NMAX ) - 0.5 |
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C ** LOOP OVER OMEGA ** |
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DO 20 NU = 1, NMAX |
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NUH = REAL ( NU ) - 0.5 |
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SUM = 0.0 |
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C ** LOOP OVER T ** |
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DO 10 TAU = 1, NMAX |
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TAUH = REAL ( TAU ) - 0.5 |
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SUM = SUM + C(TAU) * COS ( TAUH * NUH * PI / NMAXH ) |
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10 CONTINUE |
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C ** FACTOR OF TWO FOR THE REAL COSINE TRANSFORM ** |
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CHAT(NU) = 2.0 * DT * SUM |
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20 CONTINUE |
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RETURN |
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END |
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SUBROUTINE FILONS ( DR, DK, NMAX, H, HHAT ) |
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C ******************************************************************* |
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C ** FOURIER SINE TRANSFORM BY FILON'S METHOD ** |
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C ** ** |
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C ** A SPATIAL CORRELATION FUNCTION, H(R), IS TRANSFORMED TO ** |
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C ** HHAT(K) IN RECIPROCAL SPACE. ** |
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C ** ** |
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C ** REFERENCE: ** |
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C ** ** |
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C ** FILON, PROC ROY SOC EDIN, 49 38, 1928. ** |
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C ** ** |
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C ** PRINCIPAL VARIABLES: ** |
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C ** ** |
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C ** REAL KVEC THE WAVENUMBER ** |
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C ** REAL RMAX MAXIMUM DIST IN CORREL. FUNCTION ** |
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C ** REAL ALPHA, BETA, GAMMA FILON PARAMETERS ** |
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C ** REAL H(NMAX) THE CORRELATION FUNCTION ** |
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C ** REAL HHAT(NMAX) THE 3-D TRANSFORM ** |
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C ** REAL DR INTERVAL BETWEEN POINTS IN H ** |
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C ** REAL DK INTERVAL BETWEEN POINTS IN HHAT ** |
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C ** INTEGER NMAX NO. OF INTERVALS ** |
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C ** ** |
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C ** USAGE: ** |
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C ** ** |
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C ** THE ROUTINE REQUIRES THAT THE NUMBER OF INTERVALS, NMAX, IS ** |
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C ** EVEN AND CHECKS FOR THIS CONDITION. THE FIRST VALUE OF H(R) ** |
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C ** IS AT R=0. THE MAXIMUM R FOR THE CORRELATION FUNCTION IS ** |
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C ** RMAX=DR*NMAX. FOR AN ACCURATE TRANSFORM H(RMAX)=0. ** |
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C ******************************************************************* |
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INTEGER NMAX |
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REAL DR, DK, H(0:NMAX), HHAT(0:NMAX) |
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REAL RMAX, K, THETA, SINTH, COSTH |
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REAL SINSQ, COSSQ, THSQ, THCUB, ALPHA, BETA, GAMMA |
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REAL SE, SO, FOURPI, R |
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INTEGER IR, IK |
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C ******************************************************************* |
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C ** CHECKS NMAX IS EVEN ** |
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IF ( MOD ( NMAX, 2 ) .NE. 0 ) THEN |
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STOP ' NMAX SHOULD BE EVEN ' |
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ENDIF |
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FOURPI = 16.0 * ATAN ( 1.0 ) |
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RMAX = REAL ( NMAX ) * DR |
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C ** LOOP OVER K ** |
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DO 30 IK = 0, NMAX |
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K = REAL ( IK ) * DK |
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THETA = K * DR |
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C ** CALCULATE THE FILON PARAMETERS ** |
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SINTH = SIN ( THETA ) |
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COSTH = COS ( THETA ) |
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SINSQ = SINTH * SINTH |
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COSSQ = COSTH * COSTH |
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THSQ = THETA * THETA |
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THCUB = THSQ * THETA |
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IF ( THETA. EQ. 0.0 ) THEN |
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ALPHA = 0.0 |
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BETA = 2.0 / 3.0 |
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GAMMA = 4.0 / 3.0 |
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ELSE |
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ALPHA = ( 1.0 / THCUB ) |
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: * ( THSQ + THETA * SINTH * COSTH - 2.0 * SINSQ ) |
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BETA = ( 2.0 / THCUB ) |
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: * ( THETA * ( 1.0 + COSSQ ) -2.0 * SINTH * COSTH ) |
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GAMMA = ( 4.0 / THCUB ) * ( SINTH - THETA * COSTH ) |
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ENDIF |
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C ** THE INTEGRAND IS H(R) * R FOR THE 3-D TRANSFORM ** |
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C ** DO THE SUM OVER THE EVEN ORDINATES ** |
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SE = 0.0 |
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DO 10 IR = 0, NMAX, 2 |
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R = REAL ( IR ) * DR |
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SE = SE + H(IR) * R * SIN ( K * R ) |
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10 CONTINUE |
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C ** SUBTRACT HALF THE FIRST AND LAST TERMS ** |
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C ** HERE THE FIRST TERM IS ZERO ** |
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SE = SE - 0.5 * ( H(NMAX) * RMAX * SIN ( K * RMAX ) ) |
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C ** DO THE SUM OVER THE ODD ORDINATES ** |
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SO = 0.0 |
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DO 20 IR = 1, NMAX - 1, 2 |
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R = REAL ( IR ) * DR |
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SO = SO + H(IR) * R * SIN ( K * R ) |
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20 CONTINUE |
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HHAT(IK) = ( - ALPHA * H(NMAX) * RMAX * COS ( K * RMAX) |
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: + BETA * SE + GAMMA * SO ) * DR |
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C ** INCLUDE NORMALISING FACTOR ** |
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HHAT(IK) = FOURPI * HHAT(IK) / K |
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30 CONTINUE |
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RETURN |
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END |
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