*DECK POIS3D
SUBROUTINE POIS3D (LPEROD, L, C1, MPEROD, M, C2, NPEROD, N, A, B,
+ C, LDIMF, MDIMF, F, IERROR, W)
C***BEGIN PROLOGUE POIS3D
C***PURPOSE Solve a three-dimensional block tridiagonal linear system
C which arises from a finite difference approximation to a
C three-dimensional Poisson equation using the Fourier
C transform package FFTPAK written by Paul Swarztrauber.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B4B
C***TYPE SINGLE PRECISION (POIS3D-S)
C***KEYWORDS ELLIPTIC PDE, FISHPACK, HELMHOLTZ, POISSON
C***AUTHOR Adams, J., (NCAR)
C Swarztrauber, P. N., (NCAR)
C Sweet, R., (NCAR)
C***DESCRIPTION
C
C Subroutine POIS3D solves the linear system of equations
C
C C1*(X(I-1,J,K)-2.*X(I,J,K)+X(I+1,J,K))
C + C2*(X(I,J-1,K)-2.*X(I,J,K)+X(I,J+1,K))
C + A(K)*X(I,J,K-1)+B(K)*X(I,J,K)+C(K)*X(I,J,K+1) = F(I,J,K)
C
C for I=1,2,...,L , J=1,2,...,M , and K=1,2,...,N .
C
C The indices K-1 and K+1 are evaluated modulo N, i.e.
C X(I,J,0) = X(I,J,N) and X(I,J,N+1) = X(I,J,1). The unknowns
C X(0,J,K), X(L+1,J,K), X(I,0,K), and X(I,M+1,K) are assumed to take
C on certain prescribed values described below.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C
C * * * * * * * * Parameter Description * * * * * * * * * *
C
C
C * * * * * * On Input * * * * * *
C
C LPEROD Indicates the values that X(0,J,K) and X(L+1,J,K) are
C assumed to have.
C
C = 0 If X(0,J,K) = X(L,J,K) and X(L+1,J,K) = X(1,J,K).
C = 1 If X(0,J,K) = X(L+1,J,K) = 0.
C = 2 If X(0,J,K) = 0 and X(L+1,J,K) = X(L-1,J,K).
C = 3 If X(0,J,K) = X(2,J,K) and X(L+1,J,K) = X(L-1,J,K).
C = 4 If X(0,J,K) = X(2,J,K) and X(L+1,J,K) = 0.
C
C L The number of unknowns in the I-direction. L must be at
C least 3.
C
C C1 The real constant that appears in the above equation.
C
C MPEROD Indicates the values that X(I,0,K) and X(I,M+1,K) are
C assumed to have.
C
C = 0 If X(I,0,K) = X(I,M,K) and X(I,M+1,K) = X(I,1,K).
C = 1 If X(I,0,K) = X(I,M+1,K) = 0.
C = 2 If X(I,0,K) = 0 and X(I,M+1,K) = X(I,M-1,K).
C = 3 If X(I,0,K) = X(I,2,K) and X(I,M+1,K) = X(I,M-1,K).
C = 4 If X(I,0,K) = X(I,2,K) and X(I,M+1,K) = 0.
C
C M The number of unknowns in the J-direction. M must be at
C least 3.
C
C C2 The real constant which appears in the above equation.
C
C NPEROD = 0 If A(1) and C(N) are not zero.
C = 1 If A(1) = C(N) = 0.
C
C N The number of unknowns in the K-direction. N must be at
C least 3.
C
C
C A,B,C One-dimensional arrays of length N that specify the
C coefficients in the linear equations given above.
C
C If NPEROD = 0 the array elements must not depend upon the
C index K, but must be constant. Specifically, the
C subroutine checks the following condition
C
C A(K) = C(1)
C C(K) = C(1)
C B(K) = B(1)
C
C for K=1,2,...,N.
C
C LDIMF The row (or first) dimension of the three-dimensional
C array F as it appears in the program calling POIS3D.
C This parameter is used to specify the variable dimension
C of F. LDIMF must be at least L.
C
C MDIMF The column (or second) dimension of the three-dimensional
C array F as it appears in the program calling POIS3D.
C This parameter is used to specify the variable dimension
C of F. MDIMF must be at least M.
C
C F A three-dimensional array that specifies the values of
C the right side of the linear system of equations given
C above. F must be dimensioned at least L x M x N.
C
C W A one-dimensional array that must be provided by the
C user for work space. The length of W must be at least
C 30 + L + M + 2*N + MAX(L,M,N) +
C 7*(INT((L+1)/2) + INT((M+1)/2)).
C
C
C * * * * * * On Output * * * * * *
C
C F Contains the solution X.
C
C IERROR An error flag that indicates invalid input parameters.
C Except for number zero, a solution is not attempted.
C = 0 No error
C = 1 If LPEROD .LT. 0 or .GT. 4
C = 2 If L .LT. 3
C = 3 If MPEROD .LT. 0 or .GT. 4
C = 4 If M .LT. 3
C = 5 If NPEROD .LT. 0 or .GT. 1
C = 6 If N .LT. 3
C = 7 If LDIMF .LT. L
C = 8 If MDIMF .LT. M
C = 9 If A(K) .NE. C(1) or C(K) .NE. C(1) or B(I) .NE.B(1)
C for some K=1,2,...,N.
C = 10 If NPEROD = 1 and A(1) .NE. 0 or C(N) .NE. 0
C
C Since this is the only means of indicating a possibly
C incorrect call to POIS3D, the user should test IERROR
C after the call.
C
C *Long Description:
C
C * * * * * * * Program Specifications * * * * * * * * * * * *
C
C Dimension of A(N),B(N),C(N),F(LDIMF,MDIMF,N),
C Arguments W(see argument list)
C
C Latest December 1, 1978
C Revision
C
C Subprograms POIS3D,POS3D1,TRIDQ,RFFTI,RFFTF,RFFTF1,RFFTB,
C Required RFFTB1,COSTI,COST,SINTI,SINT,COSQI,COSQF,COSQF1
C COSQB,COSQB1,SINQI,SINQF,SINQB,CFFTI,CFFTI1,
C CFFTB,CFFTB1,PASSB2,PASSB3,PASSB4,PASSB,CFFTF,
C CFFTF1,PASSF1,PASSF2,PASSF3,PASSF4,PASSF,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 Roland Sweet at NCAR in July 1977
C
C Algorithm This subroutine solves three-dimensional block
C tridiagonal linear systems arising from finite
C difference approximations to three-dimensional
C Poisson equations using the Fourier transform
C package FFTPAK written by Paul Swarztrauber.
C
C Space 6561(decimal) = 14641(octal) locations on the
C Required NCAR Control Data 7600
C
C Timing and The execution time T on the NCAR Control Data
C Accuracy 7600 for subroutine POIS3D is roughly proportional
C to L*M*N*(log2(L)+log2(M)+5), but also depends on
C input parameters LPEROD and MPEROD. Some typical
C values are listed in the table below when NPEROD=0.
C To measure the accuracy of the algorithm a
C uniform random number generator was used to create
C a solution array X for the system given in the
C 'PURPOSE' with
C
C A(K) = C(K) = -0.5*B(K) = 1, K=1,2,...,N
C
C and, when NPEROD = 1
C
C A(1) = C(N) = 0
C A(N) = C(1) = 2.
C
C The solution X was substituted into the given sys-
C tem and, using double precision, a right side Y was
C computed. Using this array Y subroutine POIS3D was
C called to produce an approximate solution Z. Then
C the relative error, defined as
C
C E = MAX(ABS(Z(I,J,K)-X(I,J,K)))/MAX(ABS(X(I,J,K)))
C
C where the two maxima are taken over I=1,2,...,L,
C J=1,2,...,M and K=1,2,...,N, was computed. The
C value of E is given in the table below for some
C typical values of L,M and N.
C
C
C L(=M=N) LPEROD MPEROD T(MSECS) E
C ------ ------ ------ -------- ------
C
C 16 0 0 272 1.E-13
C 15 1 1 287 4.E-13
C 17 3 3 338 2.E-13
C 32 0 0 1755 2.E-13
C 31 1 1 1894 2.E-12
C 33 3 3 2042 7.E-13
C
C
C Portability American National Standards Institute FORTRAN.
C The machine dependent constant PI is defined in
C function PIMACH.
C
C Required COS,SIN,ATAN
C Resident
C Routines
C
C Reference NONE
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C***REFERENCES (NONE)
C***ROUTINES CALLED POS3D1
C***REVISION HISTORY (YYMMDD)
C 801001 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890531 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C***END PROLOGUE POIS3D
DIMENSION A(*) ,B(*) ,C(*) ,
1 F(LDIMF,MDIMF,*) ,W(*) ,SAVE(6)
C***FIRST EXECUTABLE STATEMENT POIS3D
LP = LPEROD+1
MP = MPEROD+1
NP = NPEROD+1
C
C CHECK FOR INVALID INPUT.
C
IERROR = 0
IF (LP.LT.1 .OR. LP.GT.5) IERROR = 1
IF (L .LT. 3) IERROR = 2
IF (MP.LT.1 .OR. MP.GT.5) IERROR = 3
IF (M .LT. 3) IERROR = 4
IF (NP.LT.1 .OR. NP.GT.2) IERROR = 5
IF (N .LT. 3) IERROR = 6
IF (LDIMF .LT. L) IERROR = 7
IF (MDIMF .LT. M) IERROR = 8
IF (NP .NE. 1) GO TO 103
DO 101 K=1,N
IF (A(K) .NE. C(1)) GO TO 102
IF (C(K) .NE. C(1)) GO TO 102
IF (B(K) .NE. B(1)) GO TO 102
101 CONTINUE
GO TO 104
102 IERROR = 9
103 IF (NPEROD.EQ.1 .AND. (A(1).NE.0. .OR. C(N).NE.0.)) IERROR = 10
104 IF (IERROR .NE. 0) GO TO 122
IWYRT = L+1
IWT = IWYRT+M
IWD = IWT+MAX(L,M,N)+1
IWBB = IWD+N
IWX = IWBB+N
IWY = IWX+7*((L+1)/2)+15
GO TO (105,114),NP
C
C REORDER UNKNOWNS WHEN NPEROD = 0.
C
105 NH = (N+1)/2
NHM1 = NH-1
NODD = 1
IF (2*NH .EQ. N) NODD = 2
DO 111 I=1,L
DO 110 J=1,M
DO 106 K=1,NHM1
NHPK = NH+K
NHMK = NH-K
W(K) = F(I,J,NHMK)-F(I,J,NHPK)
W(NHPK) = F(I,J,NHMK)+F(I,J,NHPK)
106 CONTINUE
W(NH) = 2.*F(I,J,NH)
GO TO (108,107),NODD
107 W(N) = 2.*F(I,J,N)
108 DO 109 K=1,N
F(I,J,K) = W(K)
109 CONTINUE
110 CONTINUE
111 CONTINUE
SAVE(1) = C(NHM1)
SAVE(2) = A(NH)
SAVE(3) = C(NH)
SAVE(4) = B(NHM1)
SAVE(5) = B(N)
SAVE(6) = A(N)
C(NHM1) = 0.
A(NH) = 0.
C(NH) = 2.*C(NH)
GO TO (112,113),NODD
112 B(NHM1) = B(NHM1)-A(NH-1)
B(N) = B(N)+A(N)
GO TO 114
113 A(N) = C(NH)
114 CONTINUE
CALL POS3D1 (LP,L,MP,M,N,A,B,C,LDIMF,MDIMF,F,W,W(IWYRT),W(IWT),
1 W(IWD),W(IWX),W(IWY),C1,C2,W(IWBB))
GO TO (115,122),NP
115 DO 121 I=1,L
DO 120 J=1,M
DO 116 K=1,NHM1
NHMK = NH-K
NHPK = NH+K
W(NHMK) = .5*(F(I,J,NHPK)+F(I,J,K))
W(NHPK) = .5*(F(I,J,NHPK)-F(I,J,K))
116 CONTINUE
W(NH) = .5*F(I,J,NH)
GO TO (118,117),NODD
117 W(N) = .5*F(I,J,N)
118 DO 119 K=1,N
F(I,J,K) = W(K)
119 CONTINUE
120 CONTINUE
121 CONTINUE
C(NHM1) = SAVE(1)
A(NH) = SAVE(2)
C(NH) = SAVE(3)
B(NHM1) = SAVE(4)
B(N) = SAVE(5)
A(N) = SAVE(6)
122 CONTINUE
RETURN
END