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C PROGRAM FFT9(INPUT,OUTPUT,TAPE5=INPUT,TAPE6=OUTPUT) 10
C 20
C ---- PROGRAM DESCRIPTION ---- 30
C PROGRAM FFT9 USES A 4-TH OR 6-TH ORDER 9-POINT DIFFERENCE 40
C FORMULA AND FAST FOURIER TRANSFORM FOR THE NUMERICAL 50
C SOLUTION OF THE ELLIPTIC EQUATION WITH CONSTANT COEFFICIENTS 60
C 70
C (I) CUXX*DDXU + CUYY*DDYU + CU*U = R 80
C 90
C ON A RECTANGULAR REGION 0 .LE. X .LE. SX,0 .LE. Y .LE. SY 100
C AND SUBJECT TO DIRICHLET BOUNDARY CONDITIONS 110
C (II) U = G ON THE BOUNDARY 120
C 130
C NOTE- THE 6-TH ORDER ALGORITHM IS APPLIED ONLY 140
C TO POISSON TYPE OPERATORS 150
C 160
C 170
C ---- INPUT AND OUTPUT TO FFT9 ---- 180
C --PROBLEM DEFINITION-- USER SUPPLIED FORTRAN FUNCTION FOR THE 190
C EVALUATION OF THE RIGHT SIDES (R,G) OF THE DIFFERENTIAL 200
C AND BOUNDARY OPERATORS 210
C 220
C REAL FUNCTION PDERGH(X,Y) 230
C 240
C PDERGH = R 250
C RETURN 260
C END 270
C REAL FUNCTION BCOND(I,X,Y,BVALUS) 280
C TWO DIMENSIONS 290
C VALUES OF BOUNDARY CONDITION ON SIDE I 300
C AT (X,Y) 310
C 320
C I=4 330
C --------------- 340
C I I 350
C I I 360
C I=3 I REGION I I=1 370
C I I 380
C I I 390
C --------------- 400
C I=2 410
C 420
C REAL BVALUS(4) 430
C GO TO(100,101,102,103) , I 440
C 101 BVALUS(4) = G 450
C BCOND = BVALUS(4) 460
C RETURN 470
C 102 BVALUS(4) = G 480
C BCOND = BVALUS(4) 490
C RETURN 500
C 103 BVALUS(4) = G 510
C BCOND = BVALUS(4) 520
C RETURN 530
C 104 BVALUS(4) = G 540
C BCOND = BVALUS(4) 550
C RETURN 560
C END 570
C 580
C USER SUPPLIED SUBROUTINE FOR THE DEFINITION OF 590
C P.D.E CONSTANT COEFFICIENTS 600
C 610
C SUBROUTINE PDE(X,Y,CVALUS) 620
C 630
C REAL CVALUS(7) 640
C CVALUS(1) = CUXX 650
C CVALUS(3) = CUYY 660
C CVALUS(6) = CU 670
C RETURN 680
C END 690
C 700
C 710
C USER SUPPLIED FORTRAN FUNCTION FOR THE TRUE SOLUTION 720
C IF KNOWN 730
C 740
C REAL FUNCTION TRUE(X,Y) 750
C TRUE = ... 760
C RETURN 770
C END 780
C 790
C --REGION AND GRID SPECIFICATIONS-- 800
C 810
C SX,SY - LENGTHS OF SIDES OF RECTANGLE 820
C NGRIDX,NGRIDY - NUMBER OF HORIZONTAL AND 830
C VERTICAL MESH LINES 840
C IQX,IQY - EXPONENTS OF 2 850
C NGRIDX=2**IQX+1,NGRIDY=2**IQY+1 860
C 870
C READ(5,100) SX,SY,NGRIDX,NGRIDY 880
C 100 FORMAT(2F10.0,2I3) 890
C 900
C --OUTPUT CONTROL--USER SUPPLIED DATA 910
C LEVEL - OUTPUT LEVEL DESIRED 920
C NRUNS - NUMBER OF SUCCESIVE RUNS. 930
C IN EACH RUN THE MESH SIZE IS 940
C CUT BY A FACTOR OF 2. 950
C ORDER - RATE OF CONVERGENCE DESIRED 960
C READ(5,102) LEVEL,NRUNS,ORDER 970
C 102 FORMAT(3I2) 980
C 990
C IF LEVEL = 0 PRINT APPROXIMATE SOLUTION AT NODES 1000
C = 1 PRINT MAXIMUM ERROR AND MAXIMUM 1010
C RELATIVE ERROR PROVIDED TRUE SOLUTION 1020
C IS KNOWN 1030
C = 2 ALSO PRINT TRUE SOLUTION AND 1040
C APPROXIMATE SOLUTION AT THE INTERIOR 1050
C GRID POINTS 1060
C 1070
C IF ORDER = 4 CHOOSE A 4-TH ORDER DIFFERENCE APPR. TO (I) 1080
C = 6 CHOOSE A 6-TH ORDER DIFFERENCE APPR.TO (I) 1090
C --STORAGE--IT IS ASSUMED THAT 3 .LE. IXQ,IQY .LE. 7 . 1100
C IN CASE OF FINER MESH THE DIMENSIONS REQUIRED ARE 1110
C COMPUTED BY FORMULAS GIVEN IN ARRAY LIST DESCRIPTION. 1120
C ---- MAIN VARIABLES OF FFT9 ---- 1130
C 1140
C WORK - WORKING SPACE OF DIMENSION 1150
C ORDER = 6 7+10*MAX(NGRIDX,NGRIDY)+2*NGRIDX*NGRIDY 1160
C Z,Y - ARRAYS USED IN THE FOURIER ANALYSIS-SYNTHESIS 1170
C ORDER = 4 7+10*MAX(NGRIDX,NGRIDY)+NGRIDX*NGRIDY 1180
C DIMENSION OF 'Z,Y' IS NZD=NYD= MAX(NX,NY) 1190
C AKX(I+1)-THE I-TH EIGENVALUE OF THE DIAGONAL BLOCK DIVIDED BY 1200
C THE I-TH EIGENVALUE OF THE OFF DIAGONAL BLOCK 1210
C SQUARED MINUS 2 1220
C DIMENSION OF 'AKX' IS NAKXD = MAX(NX,NY)+2 1230
C 1240
C 1250
C CORE(K) - VALUE OF APPROXIMATE SOLUTION AT NODE K 1260
C DIMENSION OF 'CORE' IS NCORED = NX+2+(NX+1)*(NY+1) 1270
C PTINT( )- VALUES OF G AT HALF LATTICE POINTS 1280
C DIMENSION IS NPINTD = NX*NY 1290
C NOTE- ARRAY 'PTINT' IS NOT USED IN CASE ORDER .EQ. 4 1300
C 1310
C GRIDX,GRIDY - GRID COORDINATES 1320
C DIMENSION OF 'GRIDX,GRIDY' IS 1330
C NGRDXD = NX+1,NGRDYD = NY+1 1340
C 1350
C SX,SY - LENGTHS OF RECTANGULAR REGION 1360
C 1370
C ED(I+1) - THE I-TH EIGENVALUE OF THE OFF DIAGONAL 1380
C BLOCK.DIMENSION OF 'ED' IS NEDD = MAX(NX,NY)+2 1390
C INDEX - INDEX VECTOR USED IN FFT ANALYSIS 1400
C DIMENSION IS INDEXD = MAX(NX,NY) 1410
C SI - SINES VECTOR FOR FFT ANALYSIS AND 1420
C SYNTHESIS. DIMENSION NSID = MAX(NX,NY) 1430
C 1440
C ---- COMMON VARIABLES OF FFT9 ---- 1450
C 1460
C COMMON /MESH/ 1470
C NX,NY - NX = 2**IQX,NY = 2**IQY 1480
C IMIN,IMAX - RANGE OF INTERIOR NODES IN X-DIRECTION 1490
C JMIN,JMAX - RANGE OF INTERIOR NODES IN Y-DIRECTION 1500
C INC - INC = IMAX - IMIN + 3 1510
C IRO - IRO = NX + 3 1520
C IBCX - IBCX = 1 1530
C IQX - EXPONENT OF 2 1540
C IBCY - IBCY = 1 1550
C IQY - EXPONENT OF 2 1560
C HX,HY - MESH SIZE 1570
C HXY2 - HXY2 = (HX/HY)**2 1580
C PI - PI = 3.14... 1590
C POTFAC - NORMALIZATION FACTOR = 2/NX 1600
C COMMON /FDFORM/ 1610
C DLEFT - DIAGONAL ENTRY OF THE OFFDIAGONAL BLOCK 1620
C DIAG,OFFD - DIAGONAL AND OFFDIAGONAL ENTRIES OF THE 1630
C DIAGONAL BLOCK OF THE 9-POINT FORMULA 1640
C RF,HH,CH,RC1,FACTOR - CONSTANTS 1650
C COMMON /FFT/ 1660
C ... - LOCAL VARIABLE 1670
C ---- COMMON VARIABLES DECLARATION ---- 1680
C 1690
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY, 1700
1HX,HY,HXY2,PI,POTFAC,SX,SY 1710
COMMON /FDFORM/ DLEFT,DIAG,OFFD,RF,HH,CH,RC1,FACTOR 1720
COMMON /CPDE/ CUXX,CUYY,CU 1730
COMMON /FFT/ N2,N4,N3,N7,IP,ISL,L1,IBCJ 1740
COMMON WORK(35040) 1750
C 1760
REAL CVALUS(7) 1770
INTEGER ORDER 1780
C 1790
C INITIALIZATIONS 1800
C 1810
IBCX=1 1820
IBCY=1 1830
C 1840
C ***** INPUT ***** 1850
C DEFINE P.D.E COEFFICIENTS 1860
C 1870
CALL PDE (X,Y,CVALUS) 1880
C 1890
CUXX=CVALUS(1) 1900
CUYY=CVALUS(3) 1910
CU=CVALUS(6) 1920
C 1930
C DEFINE GRID SPECIFICATIONS 1940
C 1950
READ (5,102) SX,SY,NGRIDX,NGRIDY 1960
NX=NGRIDX-1 1970
NY=NGRIDY-1 1980
RNX=NX 1990
RNY=NY 2000
RALOG2=1./ALOG(2.) 2010
IQX=ALOG(RNX)*RALOG2 2020
IQY=ALOG(RNY)*RALOG2 2030
C 2040
C DEFINE OUTPUT CONTROL AND ORDER OF FINITE DIFFERENCE 2050
C DESCRITIZATION FORMULA 2060
C 2070
READ (5,103) LEVEL,NRUNS,ORDER 2080
C 2090
C OUTPUT THE INPUT DATA 2100
C 2110
WRITE (6,104) 2120
WRITE (6,105) 2130
WRITE (6,106) CUXX,CUYY,CU 2140
WRITE (6,107) 2150
WRITE (6,108) SX,SY 2160
WRITE (6,109) 2170
WRITE (6,110) ORDER 2180
DO 101 NTIMES=1,NRUNS 2190
C 2200
C ***** DISCRETIZATION ***** 2210
C APPROXIMATE THE DIFFERENCIAL EQUATION WITH 9-POINT DIFF.OPER. 2220
C 2230
NAKXD=MAX0(NX,NY)+2 2240
NEDD=NAKXD 2250
NGRDXD=NX+1 2260
NGRDYD=NY+1 2270
C 2280
IA1=1 2290
IA2=IA1+NAKXD 2300
IA3=IA2+NEDD 2310
IA4=IA3+NGRDXD 2320
IA5=IA4+NGRDYD 2330
CALL DISCRT (ORDER,WORK(IA1),NAKXD,WORK(IA2),NEDD,WORK(IA3),NGR 2340
1 DXD,WORK(IA4),NGRDYD) 2350
C 2360
C GENERATE RIGHT SIDE OF DIFFERENCE EQUATIONS 2370
C 2380
NCORED=NGRDXD*NGRDYD+NAKXD 2390
NPINTD=NCORED 2400
IF (ORDER.EQ.4) NPINTD=1 2410
IA6=IA5+NCORED 2420
IA7=IA6+NPINTD 2430
C 2440
CALL RGHTSD (ORDER,WORK(IA5),NCORED,WORK(IA6),NPINTD,WORK(IA3), 2450
1 NGRDXD,WORK(IA4),NGRDYD) 2460
C 2470
C ***** EQUATION SOLUTION ****** 2480
C 2490
C GENERATE INDECIES AND SINES USED IN THE FOURIER ANALYSIS 2500
C AND SYNTHESIS 2510
C 2520
INDEXD=MAX0(NX,NY) 2530
NSID=INDEXD 2540
IA8=IA7+INDEXD 2550
IA9=IA8+NSID 2560
CALL SETF (IBCX,IQX,WORK(IA7),INDEXD,WORK(IA8),NSID) 2570
C 2580
C 2590
C SOLVE THE BLOCK TRIDIAGONAL SYSTEM OF DIFFERENCE EQUATIONS 2600
C WITH THE FAST FOURIER SERIES METHOD. 2610
C 2620
NZD=NAKXD 2630
IA10=IA9+NZD 2640
NYD=NAKXD 2650
CALL EQSOL (WORK(IA5),NCORED,WORK(IA9),NZD,WORK(IA8),NSID,WORK( 2660
1 IA10),NYD,WORK(IA7),INDEXD,WORK(IA2),NEDD,WORK(IA1),NAKXD) 2670
C 2680
C ***** OUTPUT ****** 2690
C 2700
C 2710
C PRINTS THE COMPUTED SOLUTION AND MAX.ERROR,MAX.RELATIVE 2720
C ERROR IF THE SOLUTION IS KNOWN 2730
C 2740
CALL SUMARY (LEVEL,WORK(IA3),NGRDXD,WORK(IA4),NGRDYD,WORK(IA5), 2750
1 NCORED) 2760
C 2770
C INCREASE EXPONENT OF 2 2780
C 2790
IQX=IQX+1 2800
IQY=IQY+1 2810
NX=2**IQX 2820
NY=2**IQY 2830
101 CONTINUE 2840
STOP 2850
C 2860
102 FORMAT (2F10.0,2I3) 2870
103 FORMAT (3I2) 2880
104 FORMAT (1X,27H******* INPUT DATA ********) 2890
105 FORMAT (1X,37HEQUATION. CUXX*DDXU+CUYY*DDYU+CU*U=R) 2900
106 FORMAT (1X,19HCOEFFICIENTS. CUXX=,F10.3,5HCUYY=,F10.3,3HCU=,F10.3) 2910
107 FORMAT (1X,20HBOUNDARY COND. U = G) 2920
108 FORMAT (1X,24HREGION. 0. .LE. X .LE. ,F8.3,3X,15H0. .LE. Y .LE. , 2930
1F8.3) 2940
109 FORMAT (1X,25H******* SOLUTION ********) 2950
110 FORMAT (1X,15HDISCRETIZATION.,5X,I5,8H- ORDER ,24HDIFFERENCE APPRO 2960
1XIMATION) 2970
C 2980
END 2990
SUBROUTINE RGHTSD (ORDER,CORE,NCORED,PTINT,NPINTD,GRIDX,NGRDXD,GRI 3000
1DY,NGRDYD)
C
C COMPUTES RIGHT SIDE OF THE FINITE DIFFERENCE EQUATIONS
C
C THE ARGUMENTS - ORDER,CORE(NCORED),PTINT(NPINTD),GRIDX(NGRDXD),
C GRIDY(NGRDYD) - DEFINED IN FFT9 MAIN PROGRAM
C
INTEGER ORDER
REAL CORE(NCORED),PTINT(NPINTD),GRIDX(NGRDXD),GRIDY(NGRDYD),BVALUS
1(4)
C
C FFT9 COMMON VARIABLES
C
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
COMMON /FDFORM/ DLEFT,DIAG,OFFD,RF,HH,CH,RC1,FACTOR
C
C INITIALIZATIONS
C
I0=0
J0=0
Z0=0.0
Z1=1.0
C
C FUNCTION EVALUATIONS
C
DO 102 J=J0,NY
W=GRIDY(J+1)
L=IRO+INC*J
DO 101 I=I0,NX
K=L+I
X=GRIDX(I+1)
CORE(K)=PDERGH(X,W)
101 CONTINUE
102 CONTINUE
K=IRO
C
C CORNER INDECIES
C
NR=IRO+NX
NU=IRO+NY*INC
NRU=NU+NX
C
IF (ORDER.EQ.4) GO TO 108
C
C INITIALIZATIONS
C
DO 103 I=I0,NX
CORE(I+1)=CORE(K)
K=K+1
103 CONTINUE
C
C EVALUATE RIGHT SIDE AT THE EXTRA POINTS NEEDED BY SIX
C ORDER FORMULA
C
YMIDL=-HY*.5
DO 105 J=1,NY
L=NX*(J-1)
YMIDL=YMIDL+HY
XMIDL=-HX*.5
DO 104 I=1,NX
XMIDL=XMIDL+HX
K=I+L
PTINT(K)=PDERGH(XMIDL,YMIDL)
104 CONTINUE
105 CONTINUE
C
C COMPUTE THE RIGHT SIDE OF SIX ORDER DIFFERENCE OPERATOR
C
LL=IRO
DO 107 J=JMIN,JMAX
LL=LL+INC
L=LL
CNTRLF=CORE(L)
LDOWN=L-INC
DOWNLF=CORE(LDOWN)
DO 106 I=IMIN,IMAX
LUP=L+INC
LUP2=LUP+2
LUP1=LUP+1
IP1=I+1
IP2=I+2
LP2=L+2
K=L+1
TEMP=DOWNLF+CORE(LUP)+CORE(LUP2)+CORE(IP2)+4.*(CNTRLF+CORE(L
1 UP1)+CORE(LP2)+CORE(IP1))+148.*CORE(K)
C
CNTRLF=CORE(K)
DOWNLF=CORE(IP1)
CORE(I+1)=CORE(K)
IDWN1=I+NX*(J-1)
IDWN2=IDWN1+1
C
IUP1=IDWN1+NX
IUP2=IUP1+1
C
CORE(K)=FACTOR*(TEMP+48.*(PTINT(IDWN1)+PTINT(IDWN2)+PTINT(IU
1 P1)+PTINT(IUP2)))
L=L+1
106 CONTINUE
LP1=L+1
CORE(NX+1)=CORE(LP1)
107 CONTINUE
C
GO TO 112
108 CONTINUE
C
C INITIALIZATION
C
DO 109 I=IMIN,IMAX
K=K+1
CORE(I)=CORE(K)
109 CONTINUE
C
C GENERATE RIGHT SIDE OF DIFFERENCE EQUATIONS
C
L=IRO
DO 111 J=JMIN,JMAX
L=L+INC
K=L
XCENTR=CORE(K)
KRIGHT=K+1
DO 110 I=IMIN,IMAX
K=KRIGHT
KRIGHT=K+1
KUP=K+INC
XLEFT=XCENTR
XRIGHT=CORE(KRIGHT)
XVERT=CORE(I)+CORE(KUP)
XCENTR=CORE(K)
CORE(I)=XCENTR
CORE(K)=FACTOR*(XVERT+HH*(XLEFT+XRIGHT)+CH*XCENTR)
110 CONTINUE
111 CONTINUE
C
112 CONTINUE
C
C
C BOUNDARY VALUES AT THE CORNERS
C
CORE(IRO)=BCOND(3,0.,0.,BVALUS)
CORE(NR)=BCOND(2,SX,0.,BVALUS)
CORE(NU)=BCOND(4,0.,SY,BVALUS)
CORE(NRU)=BCOND(1,SX,SY,BVALUS)
C
C ENFORCE DIRICHLET BOUNDARY CONDITIONS
C
KLEFT=IRO
KLFTUP=IRO+INC
MRIGHT=NR
MRGTUP=MRIGHT+INC
CORE(KLFTUP)=BCOND(3,0.,HY,BVALUS)
CORE(MRGTUP)=BCOND(1,SX,HY,BVALUS)
DO 113 J=JMIN,JMAX
W=GRIDY(J+2)
KLEFTD=KLEFT
KLEFT=KLFTUP
KLFTUP=KLFTUP+INC
K=KLEFT+1
MRGTD=MRIGHT
MRIGHT=MRGTUP
MRGTUP=MRGTUP+INC
M=MRIGHT-1
CORE(KLFTUP)=BCOND(3,0.,W,BVALUS)
CORE(MRGTUP)=BCOND(1,SX,W,BVALUS)
CORE(K)=CORE(K)-POTFAC*(CORE(KLEFTD)+CORE(KLFTUP)+OFFD*CORE(KLE
1 FT))
CORE(M)=CORE(M)-POTFAC*(CORE(MRGTD)+CORE(MRGTUP)+OFFD*CORE(MRIG
1 HT))
113 CONTINUE
KDOWN=IRO
KRGTD=IRO+1
MUP=NU
MRGTUP=NU+1
CORE(KRGTD)=BCOND(2,HX,0.,BVALUS)
CORE(MRGTUP)=BCOND(4,HX,SY,BVALUS)
DO 114 I=IMIN,IMAX
W=GRIDX(I+2)
KLEFTD=KDOWN
KDOWN=KRGTD
KRGTD=KRGTD+1
MLFTUP=MUP
MUP=MRGTUP
MRGTUP=MRGTUP+1
K=KDOWN+INC
M=MUP-INC
CORE(KRGTD)=BCOND(2,W,0.,BVALUS)
CORE(MRGTUP)=BCOND(4,W,SY,BVALUS)
CORE(K)=CORE(K)-POTFAC*(CORE(KLEFTD)+CORE(KRGTD)+DLEFT*CORE(KDO
1 WN))
CORE(M)=CORE(M)-POTFAC*(CORE(MLFTUP)+CORE(MRGTUP)+DLEFT*CORE(MU
1 P))
114 CONTINUE
C
C CORRECT CORNERS OF RECTANGLE
C
K=IRO+INC+1
KR=K+NX-2
KU=NU-INC+1
KRU=KU+NX-2
CORE(K)=CORE(K)+POTFAC*CORE(IRO)
CORE(KR)=CORE(KR)+POTFAC*CORE(NR)
CORE(KU)=CORE(KU)+POTFAC*CORE(NU)
CORE(KRU)=CORE(KRU)+POTFAC*CORE(NRU)
RETURN
C
END
SUBROUTINE EQSOL (CORE,NCORED,Z,NZD,SI,NSID,Y,NYD,INDEX,INDEXD,ED, 5080
1NEDD,AKX,NAKXD)
C
C SOLVES THE BLOCK TRIDIAGONAL SYSTEM OF DIFFERENCE EQUATIONS
C USING FAST FOURIER SERIES METHOD
C
C THE ARGUMENTS - CORE(NCORED),Z(NZD),SI(NSID),Y(NYD),INDEX(INDEXD),
C ED(NEDD),AKX(NAKXD) - DEFINED IN FFT9 PROGRAM
C
REAL CORE(NCORED),Z(NZD),SI(NSID),Y(NYD),ED(NEDD),AKX(NAKXD)
INTEGER INDEX(INDEXD)
C
C FFT9 COMMON VARIABLES
C
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
COMMON /FDFORM/ DLEFT,DIAG,OFFD,RF,HH,CH,RC1,FACTOR
J1=2
J2=NY-2
C
C MODIFICATION OF EVEN COLUMN VECTORS
C
CALL EVENRD (J1,J2,CORE,NCORED)
C
C PERFORM FOURIER ANALYSIS ON EVEN LINES
C
K=IRO+INC*J1
JUMP=INC+INC
DO 101 J=J1,J2,2
CALL FETCHX (K,Z,NZD,CORE,NCORED)
CALL FOUR (IBCX,IQX,SI,NSID,Z,NZD,Y,NYD,INDEX,INDEXD)
CALL STOREX (K,CORE,NCORED,Y,NYD)
101 K=K+JUMP
C
C DIVISION BY EIGENVALUES OF OFFDIAGONAL MATRIX
C
DO 103 I=IMIN,IMAX
M=IRO+I
DO 102 J=J1,J2,2
K=M+INC*J
CORE(K)=CORE(K)/ED(I+1)
102 CONTINUE
103 CONTINUE
C
C SOLUTION FOR EVEN LINES BY CYCLIC REDUCTION
C
DO 104 K=IMIN,IMAX
A=AKX(K+1)
L=IRO+K
M=JUMP
104 CALL CRED (IBCY,L,M,A,IQY-1,CORE,NCORED)
C
C FOURIER SYNTHESIS ON EVEN LINES
C
K=IRO+INC*J1
DO 105 J=J1,J2,2
CALL FETCHX (K,Z,NZD,CORE,NCORED)
CALL FOUR (IBCX,IQX,SI,NSID,Z,NZD,Y,NYD,INDEX,INDEXD)
CALL STOREX (K,CORE,NCORED,Y,NYD)
105 K=K+JUMP
C
C MODIFICATION OF ODD LINE VECTORS
C
J2=NY-1
F1=1.0/POTFAC
CALL ODDRD (F1,J2,CORE,NCORED)
C
C SOLUTION FOR ODD LINES BY CYCLIC REDUCTION
C
A=-DIAG/OFFD
DO 106 J=1,J2,2
L=IRO+INC*J
106 CALL CRED (IBCX,L,1,A,IQX,CORE,NCORED)
C
RETURN
C
END
SUBROUTINE DISCRT (ORDER,AKX,NAKXD,ED,NEDD,GRIDX,NGRDXD,GRIDY,NGRD 5850
1YD)
C
C SETS CONSTANTS,CALCULATE GRID SPECIFICATIONS,9-POINT
C DIFFERENCE FORMULA AND THE EIGENVALUES OF THE DIAGONAL
C AND OFF DIAGONAL BLOCKS OF THE FINITE DIFFERENCE EQUATIONS
C
C THE ARGUMENTS-ORDER,AKX(NAKXD),ED(NEDD),GRIDX(NGRDXD),
C GRIDY(NGRDYD) ARE DEFINED IN FFT9
C
INTEGER ORDER
REAL AKX(NAKXD),ED(NEDD),GRIDX(NGRDXD),GRIDY(NGRDYD)
C
C FFT9 COMMON VARIABLES
C
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
COMMON /FDFORM/ DLEFT,DIAG,OFFD,RF,HH,CH,RC1,FACTOR
COMMON /CPDE/ C1,C2,C3
C
C GENERATE CONSTANTS
C
PI=3.14159265358979
NX=2**IQX
NY=2**IQY
REV=1./FLOAT(NX)
REVY=1./FLOAT(NY)
POTFAC=2.*REV
HX=SX*REV
HY=SY*REVY
RF=PI*REV
C
C GENERATE GRID SPECIFICATIONS
C
I0=0
DO 101 I=I0,NX
GRIDX(I+1)=FLOAT(I)*HX
101 CONTINUE
DO 102 J=I0,NY
GRIDY(J+1)=FLOAT(J)*HY
102 CONTINUE
IMIN=1
IMAX=NX-1
JMIN=1
JMAX=NY-1
INC=IMAX-IMIN+3
IRO=2+INC
HXY2=(HX/HY)**2
C
C INITIALIZATIONS
C
OFFD=4.
DLEFT=4.
DIAG=-20.
FACTOR=POTFAC*HX**2/60.
C
IF (ORDER.EQ.6) GO TO 103
C
C GENERATE COEFFICIENTS OF 9-POINT FINITE DIFFERENCE STENSIL
C OF FOURTH ORDER
C
RC1=1./C1
C21=RC1*C2
SIGMA=RC1*C3
RGRID=HY/HX
HX2=HX**2
SIGH2=SIGMA*HX2
SIGH12=SIGH2/12.0
RGRSQ=RGRID**2
QUOT=RGRSQ/C21
RR=1.-SIGH12
QQ=C21*(1.-SIGH12*QUOT)/RR
DIV=RGRSQ+QQ
OFFD=12.*QQ*QUOT/DIV-2.0
DLEFT=12.*C21/DIV-2.0
DIAG=(OFFD+2.)*RR*SIGH2-2.*(OFFD+DLEFT)-4.
HH=QQ/C21
CH=(12.*RR-2.)*HH-2.
FACTOR=RC1*POTFAC*RGRSQ*HX2/DIV
103 CONTINUE
C
C CALCULATE EIGENVALUES OF THE OFF-DIAGONAL BLOCKS AND
C DIAGONAL BLOCKS
C
DO 104 I=IMIN,IMAX
II=I+1
DFLI=FLOAT(I)
TWOCOS=2.0*COS(RF*DFLI)
ED(II)=(DLEFT+TWOCOS)**2
RATIO=((DIAG+TWOCOS*OFFD)**2)/ED(II)
AKX(II)=-2.0+RATIO
104 CONTINUE
RETURN
C
END
SUBROUTINE EVENRD (J1,J2,CORE,NCORED) 6800
C
C MODIFIES THE RIGHT SIDE ON EVEN LINE VECTORS WHERE J1 IS
C THE FIRST AND J2 THE LAST EVEN VECTOR.
C THE RIGHT SIDE AND ITS MODIFICATION ARE STORED IN ARRAY CORE
C
C THE ACTUAL ARGUMENT - CORE(NCORED) IS DEFINED IN FFT9
C
REAL CORE(NCORED)
C
C FFT9 COMMON VARIABLES
C
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
COMMON /FDFORM/ DLEFT,DIAG,OFFD,RF,HH,CH,RC1,FACTOR
C
C INITIALIZATIONS
C
I1=IMIN
I2=IMAX-1
L=IRO
JUMP=INC+INC
C
DO 102 J=J1,J2,2
L=L+JUMP
K=L
KRIGHT=K+1
KRDOWN=KRIGHT-INC
KRGTUP=KRIGHT+INC
X2=0.
X3=CORE(KRDOWN)+CORE(KRGTUP)
Y2=0.0
Y3=CORE(KRIGHT)
C
DO 101 I=I1,I2
K=K+1
KRIGHT=KRIGHT+1
KRDOWN=KRDOWN+1
KRGTUP=KRGTUP+1
X1=X2
X2=X3
X3=CORE(KRDOWN)+CORE(KRGTUP)
Y1=Y2
Y2=Y3
Y3=CORE(KRIGHT)
CORE1=X1+X3+DLEFT*X2-OFFD*(Y1+Y3)-DIAG*Y2
CORE(K)=CORE1
101 CONTINUE
K=K+1
CORE1=X2+DLEFT*X3-OFFD*Y2-DIAG*Y3
CORE(K)=CORE1
102 CONTINUE
RETURN
C
END
SUBROUTINE ODDRD (F1,J2,CORE,NCORED) 7350
C
C MODIFIES THE RIGHT SIDE ON ODD-LINE VECTORS WHERE J2 IS THE LAST
C ODD LINE.THE RIGHT SIDE AND ITS MODIFICATION IS STORED IN
C ARRAY CORE(NCORED).
C
C THE ARGUMENT F1 IS A MULTIPLICATION FACTOR
C
REAL CORE(NCORED)
C
C FFT9 COMMON VARIABLES
C
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
COMMON /FDFORM/ DLEFT,DIAG,OFFD,RF,HH,CH,RC1,FACTOR
C
DENOM=1.0/OFFD
DLFT=DLEFT*DENOM
CENTER=F1*DENOM
I2=IMAX-1
L=IRO-INC
JUMP=INC+INC
C
DO 102 J=1,J2,2
L=L+JUMP
K=L
KRIGHT=K+1
KRDOWN=KRIGHT-INC
KRGTUP=KRIGHT+INC
X2=0.0
X3=CORE(KRDOWN)+CORE(KRGTUP)
IF (J.EQ.1) X3=CORE(KRGTUP)
IF (J.EQ.J2) X3=CORE(KRDOWN)
DO 101 I=IMIN,I2
K=K+1
KRIGHT=KRIGHT+1
KRDOWN=KRDOWN+1
KRGTUP=KRGTUP+1
X1=X2
X2=X3
X3=CORE(KRDOWN)+CORE(KRGTUP)
IF (J.EQ.1) X3=CORE(KRGTUP)
IF (J.EQ.J2) X3=CORE(KRDOWN)
CORE1=CENTER*CORE(K)-DENOM*(X1+X3)-DLFT*X2
CORE(K)=CORE1
101 CONTINUE
C
K=K+1
CORE1=CENTER*CORE(K)-DENOM*X2-DLFT*X3
CORE(K)=CORE1
102 CONTINUE
RETURN
C
END
SUBROUTINE CRED (IBC,L,M,A,IP1,CORE,NCORED) 7890
C
C SOLVES TRIANGULAR SYSTEMS BY RECURSIVE CYCLIC REDUCTION
C SEE REFERENCE @3
C
C IBC = 1,IP1-EXPONENT OF 2,A-DIAGONAL ELEMENT,L,M-MESH DEPENDENT
C CONSTANTS USED TO RECOVER RIGHT SIDE FROM ARRAY CORE
C
REAL CORE(NCORED)
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
DIMENSION BB(11)
C
IP=IP1
N2=M
BB(1)=A
B=A
N4=0
N=2**IP
K=L+N*M
IP=IP-1
C
DO 104 N1=1,IP
N4=N4+1
N3=N2
N2=N2+N2
J1=N2+L
J3=K-N3
J2=K-N2
IF (J1.GT.J2) GO TO 102
C
DO 101 J=J1,J2,N2
JMN3=J-N3
JPN3=J+N3
CORE(J)=B*CORE(J)+CORE(JMN3)+CORE(JPN3)
101 CONTINUE
102 B=B*B-2.000
BB(N1+1)=B
IF (B.LE.1.0E14) GO TO 104
C
C SHORT CUT AND SOLVE BY DIVISION IF B
C LARGER THAN DESIRED ACCURACY
C
IF (J1.GT.J2) GO TO 105
C
DO 103 J=J1,J2,N2
103 CORE(J)=-CORE(J)/B
C
IF (IBC.EQ.1) GO TO 105
104 CONTINUE
I=L+N*M/2
CORE(I)=-CORE(I)/B
C
105 DO 108 NN=1,N4
N1=N4-NN
B=BB(N1+1)
J2=K-N3
J1=L+N3
J1PN3=J1+N3
CORE(J1)=(CORE(J1PN3)-CORE(J1))/B
J2MN3=J2-N3
CORE(J2)=(CORE(J2MN3)-CORE(J2))/B
J1=J1+N2
J2=J2-N2
IF (J1.GT.J2) GO TO 107
C
DO 106 J=J1,J2,N2
JMN3=J-N3
JPN3=J+N3
CORE(J)=(CORE(JMN3)+CORE(JPN3)-CORE(J))/B
106 CONTINUE
107 N2=N3
108 N3=N3/2
RETURN
C
END
SUBROUTINE STOREX (K,CORE,NCORED,Y,NYD) 8650
C
C TRANSFERS DATA FROM ARRAY Y TO ARRAY CORE
C AFTER FOURIER ANALYSIS
C
C THE ARGUMENTS - K IS THE NODE,CORE(NCORED),Y(NYD) DEFINED IN FFT9
C
REAL CORE(NCORED),Y(NYD)
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
DO 101 I=IMIN,IMAX
KPI=K+I
IP1=I+1
CORE(KPI)=Y(IP1)
101 CONTINUE
RETURN
C
END
SUBROUTINE FETCHX (K,Z,NZD,CORE,NCORED) 8830
C
C TRANSFERS DATA FROM ARRAY CORE TO ARRAY Z PRIOR TO
C FOURIER ANALYSIS
C
C THE ARGUMENTS - K NODE,Z(NZD),CORE(NCORED) ARE DEFINED IN FFT9
C
REAL Z(NZD),CORE(NCORED)
COMMON /MESH/ NX,NY,IMIN,IMAX,JMIN,JMAX,INC,IRO,IBCX,IQX,IBCY,IQY,
1HX,HY,HXY2,PI,POTFAC,SX,SY
DO 101 I=IMIN,IMAX
KPI=K+I
IP1=I+1
Z(IP1)=CORE(KPI)
101 CONTINUE
RETURN
C
END
SUBROUTINE KFOLD (INDEX,INDEXD,SI,NSID,Y,NYD,Z,NZD) 9010
C
C EVALUATES THE SUMMATIONS AND DOES ALL THE MULTIPLICATIONS
C BY A RECURSIVE TECHNIQUE (SEE REFERENCE @3 )
C
C THE ARGUMENTS-INDEX(INDEXD),SI(NSID),Y(NYD),Z(NZD) DEFINED IN FFT9
C
REAL SI(NSID),Y(NYD),Z(NZD)
INTEGER INDEX(INDEXD)
C
C FFT9 COMMON VARIABLES
C
COMMON /FFT/ N2,N4,N3,N7,IP,ISL,L1,IBCJ
C
JS1=N2
I=1
J5=ISL+N2
IS1=ISL
IC1=L1
JS1=JS1/2
C
C GO TO FIRST TIME IS LAST TIME
C
IF (JS1.EQ.1) GO TO 106
SN=SI(1)
IS1=IS1+JS1
IC1=IC1+JS1
J3=IS1+JS1
C
101 ISO=IS1-JS1
ICO=IC1-JS1
ODD1=SN*(Z(IC1)-Z(IS1))
ODD2=SN*(Z(IC1)+Z(IS1))
Z(IC1)=Z(ICO)-ODD1
Z(ICO)=Z(ICO)+ODD1
Z(IS1)=-Z(ISO)+ODD2
Z(ISO)=Z(ISO)+ODD2
IS1=IS1+1
IC1=IC1+1
IF (IS1.NE.J3) GO TO 101
C
I=I+1
102 IS1=ISL
IC1=L1
JS1=JS1/2
C
C GO TO LAST TIME WITH K IN PAIRS
C
IF (JS1.EQ.1) GO TO 107
C
C TAKE K IN PAIRS INTERCHANGING SN AND CS
C
103 SN=SI(I)
I=I+1
CS=SI(I)
IS1=IS1+JS1
IC1=IC1+JS1
J3=IS1+JS1
C
104 ISO=IS1-JS1
ICO=IC1-JS1
ODD1=CS*Z(IC1)-SN*Z(IS1)
ODD2=SN*Z(IC1)+CS*Z(IS1)
Z(IC1)=Z(ICO)-ODD1
Z(ICO)=Z(ICO)+ODD1
Z(IS1)=-Z(ISO)+ODD2
Z(ISO)=Z(ISO)+ODD2
IS1=IS1+1
IC1=IC1+1
IF (IS1.NE.J3) GO TO 104
C
IS1=IS1+JS1
IC1=IC1+JS1
J3=IS1+JS1
C
105 ISO=IS1-JS1
ICO=IC1-JS1
ODD1=SN*Z(IC1)-CS*Z(IS1)
ODD2=CS*Z(IC1)+SN*Z(IS1)
Z(IC1)=Z(ICO)-ODD1
Z(ICO)=Z(ICO)+ODD1
Z(IS1)=-Z(ISO)+ODD2
Z(ISO)=Z(ISO)+ODD2
IS1=IS1+1
IC1=IC1+1
IF (IS1.NE.J3) GO TO 105
I=I+1
IF (IS1.EQ.J5) GO TO 102
GO TO 103
C
C LAST TIME IS FIRST TIME
C
106 K1=INDEX(I)
SN=SI(I)
ISO=IS1
IS1=IS1+JS1
ICO=IC1
IC1=IC1+JS1
ODD1=SN*(Z(IC1)-Z(IS1))
Y(K1+1)=Z(ICO)+ODD1