*DECK POISTG
SUBROUTINE POISTG (NPEROD, N, MPEROD, M, A, B, C, IDIMY, Y,
+ IERROR, W)
C***BEGIN PROLOGUE POISTG
C***PURPOSE Solve a block tridiagonal system of linear equations
C that results from a staggered grid finite difference
C approximation to 2-D elliptic PDE's.
C***LIBRARY SLATEC (FISHPACK)
C***CATEGORY I2B4B
C***TYPE SINGLE PRECISION (POISTG-S)
C***KEYWORDS ELLIPTIC, FISHPACK, HELMHOLTZ, PDE, TRIDIAGONAL
C***AUTHOR Adams, J., (NCAR)
C Swarztrauber, P. N., (NCAR)
C Sweet, R., (NCAR)
C***DESCRIPTION
C
C Subroutine POISTG solves the linear system of equations
C
C A(I)*X(I-1,J) + B(I)*X(I,J) + C(I)*X(I+1,J)
C + X(I,J-1) - 2.*X(I,J) + X(I,J+1) = Y(I,J)
C
C for I=1,2,...,M and J=1,2,...,N.
C
C The indices I+1 and I-1 are evaluated modulo M, i.e.
C X(0,J) = X(M,J) and X(M+1,J) = X(1,J), and X(I,0) may be equal to
C X(I,1) or -X(I,1) and X(I,N+1) may be equal to X(I,N) or -X(I,N)
C depending on an input parameter.
C
C
C * * * * * * * * Parameter Description * * * * * * * * * *
C
C * * * * * * On Input * * * * * *
C
C NPEROD
C Indicates the values which X(I,0) and X(I,N+1) are assumed
C to have.
C = 1 If X(I,0) = -X(I,1) and X(I,N+1) = -X(I,N)
C = 2 If X(I,0) = -X(I,1) and X(I,N+1) = X(I,N)
C = 3 If X(I,0) = X(I,1) and X(I,N+1) = X(I,N)
C = 4 If X(I,0) = X(I,1) and X(I,N+1) = -X(I,N)
C
C N
C The number of unknowns in the J-direction. N must
C be greater than 2.
C
C MPEROD
C = 0 If A(1) and C(M) are not zero
C = 1 If A(1) = C(M) = 0
C
C M
C The number of unknowns in the I-direction. M must
C be greater than 2.
C
C A,B,C
C One-dimensional arrays of length M that specify the coefficients
C in the linear equations given above. If MPEROD = 0 the array
C elements must not depend on the index I, but must be constant.
C Specifically, the subroutine checks the following condition
C
C A(I) = C(1)
C B(I) = B(1)
C C(I) = C(1)
C
C for I = 1, 2, ..., M.
C
C IDIMY
C The row (or first) dimension of the two-dimensional array Y as
C it appears in the program calling POISTG. This parameter is
C used to specify the variable dimension of Y. IDIMY must be at
C least M.
C
C Y
C A two-dimensional array that specifies the values of the
C right side of the linear system of equations given above.
C Y must be dimensioned at least M X N.
C
C W
C A one-dimensional work array that must be provided by the user
C for work space. W may require up to 9M + 4N + M(INT(log2(N)))
C locations. The actual number of locations used is computed by
C POISTG and returned in location W(1).
C
C
C * * * * * * On Output * * * * * *
C
C Y
C Contains the solution X.
C
C IERROR
C An error flag that indicates invalid input parameters. Except
C for number zero, a solution is not attempted.
C = 0 No error
C = 1 If M .LE. 2
C = 2 If N .LE. 2
C = 3 IDIMY .LT. M
C = 4 If NPEROD .LT. 1 or NPEROD .GT. 4
C = 5 If MPEROD .LT. 0 or MPEROD .GT. 1
C = 6 If MPEROD = 0 and
C A(I) .NE. C(1) or B(I) .NE. B(1) or C(I) .NE. C(1)
C for some I = 1, 2, ..., M.
C = 7 If MPEROD .EQ. 1 .AND. (A(1).NE.0 .OR. C(M).NE.0)
C
C W
C W(1) contains the required length of W.
C
C *Long Description:
C
C * * * * * * * Program Specifications * * * * * * * * * * * *
C
C Dimension of A(M),B(M),C(M),Y(IDIMY,N),
C Arguments W(see argument list)
C
C Latest June 1, 1977
C Revision
C
C Subprograms POISTG,POSTG2,COSGEN,MERGE,TRIX,TRI3,PIMACH
C Required
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 in 1973
C Revised by Roland Sweet in 1977
C
C
C Space 3297(decimal) = 6341(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 POISTG is roughly proportional
C to M*N*log2(N). Some typical values are listed
C in the table below. More comprehensive timing
C charts may be found in the reference.
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(I) = C(I) = -0.5*B(I) = 1, I=1,2,...,M
C
C and, when MPEROD = 1
C
C A(1) = C(M) = 0
C B(1) = B(M) =-1.
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 POISTG was
C called to produce an approximate solution Z. Then
C the relative error, defined as
C
C E = MAX(ABS(Z(I,J)-X(I,J)))/MAX(ABS(X(I,J)))
C
C where the two maxima are taken over all I=1,2,...,M
C and J=1,2,...,N, was computed. The value of E is
C given in the table below for some typical values of
C M and N.
C
C
C M (=N) MPEROD NPEROD T(MSECS) E
C ------ ------ ------ -------- ------
C
C 31 0-1 1-4 45 9.E-13
C 31 1 1 21 4.E-13
C 31 1 3 41 3.E-13
C 32 0-1 1-4 51 3.E-12
C 32 1 1 32 3.E-13
C 32 1 3 48 1.E-13
C 33 0-1 1-4 42 1.E-12
C 33 1 1 30 4.E-13
C 33 1 3 34 1.E-13
C 63 0-1 1-4 186 3.E-12
C 63 1 1 91 1.E-12
C 63 1 3 173 2.E-13
C 64 0-1 1-4 209 4.E-12
C 64 1 1 128 1.E-12
C 64 1 3 199 6.E-13
C 65 0-1 1-4 143 2.E-13
C 65 1 1 160 1.E-11
C 65 1 3 138 4.E-13
C
C Portability American National Standards Institute FORTRAN.
C The machine dependent constant PI is defined in
C function PIMACH.
C
C Required COS
C Resident
C Routines
C
C Reference Schumann, U. and R. Sweet,'A Direct Method for
C the Solution of Poisson's Equation With Neumann
C Boundary Conditions on a Staggered Grid of
C Arbitrary Size,' J. Comp. Phys. 20(1976),
C pp. 171-182.
C
C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
C***REFERENCES U. Schumann and R. Sweet, A direct method for the
C solution of Poisson's equation with Neumann boundary
C conditions on a staggered grid of arbitrary size,
C Journal of Computational Physics 20, (1976),
C pp. 171-182.
C***ROUTINES CALLED POSTG2
C***REVISION HISTORY (YYMMDD)
C 801001 DATE WRITTEN
C 861211 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE POISTG
C
C
DIMENSION Y(IDIMY,*)
DIMENSION W(*) ,B(*) ,A(*) ,C(*)
C***FIRST EXECUTABLE STATEMENT POISTG
IERROR = 0
IF (M .LE. 2) IERROR = 1
IF (N .LE. 2) IERROR = 2
IF (IDIMY .LT. M) IERROR = 3
IF (NPEROD.LT.1 .OR. NPEROD.GT.4) IERROR = 4
IF (MPEROD.LT.0 .OR. MPEROD.GT.1) IERROR = 5
IF (MPEROD .EQ. 1) GO TO 103
DO 101 I=1,M
IF (A(I) .NE. C(1)) GO TO 102
IF (C(I) .NE. C(1)) GO TO 102
IF (B(I) .NE. B(1)) GO TO 102
101 CONTINUE
GO TO 104
102 IERROR = 6
RETURN
103 IF (A(1).NE.0. .OR. C(M).NE.0.) IERROR = 7
104 IF (IERROR .NE. 0) RETURN
IWBA = M+1
IWBB = IWBA+M
IWBC = IWBB+M
IWB2 = IWBC+M
IWB3 = IWB2+M
IWW1 = IWB3+M
IWW2 = IWW1+M
IWW3 = IWW2+M
IWD = IWW3+M
IWTCOS = IWD+M
IWP = IWTCOS+4*N
DO 106 I=1,M
K = IWBA+I-1
W(K) = -A(I)
K = IWBC+I-1
W(K) = -C(I)
K = IWBB+I-1
W(K) = 2.-B(I)
DO 105 J=1,N
Y(I,J) = -Y(I,J)
105 CONTINUE
106 CONTINUE
NP = NPEROD
MP = MPEROD+1
GO TO (110,107),MP
107 CONTINUE
GO TO (108,108,108,119),NPEROD
108 CONTINUE
CALL POSTG2 (NP,N,M,W(IWBA),W(IWBB),W(IWBC),IDIMY,Y,W,W(IWB2),
1 W(IWB3),W(IWW1),W(IWW2),W(IWW3),W(IWD),W(IWTCOS),
2 W(IWP))
IPSTOR = W(IWW1)
IREV = 2
IF (NPEROD .EQ. 4) GO TO 120
109 CONTINUE
GO TO (123,129),MP
110 CONTINUE
C
C REORDER UNKNOWNS WHEN MP =0
C
MH = (M+1)/2
MHM1 = MH-1
MODD = 1
IF (MH*2 .EQ. M) MODD = 2
DO 115 J=1,N
DO 111 I=1,MHM1
MHPI = MH+I
MHMI = MH-I
W(I) = Y(MHMI,J)-Y(MHPI,J)
W(MHPI) = Y(MHMI,J)+Y(MHPI,J)
111 CONTINUE
W(MH) = 2.*Y(MH,J)
GO TO (113,112),MODD
112 W(M) = 2.*Y(M,J)
113 CONTINUE
DO 114 I=1,M
Y(I,J) = W(I)
114 CONTINUE
115 CONTINUE
K = IWBC+MHM1-1
I = IWBA+MHM1
W(K) = 0.
W(I) = 0.
W(K+1) = 2.*W(K+1)
GO TO (116,117),MODD
116 CONTINUE
K = IWBB+MHM1-1
W(K) = W(K)-W(I-1)
W(IWBC-1) = W(IWBC-1)+W(IWBB-1)
GO TO 118
117 W(IWBB-1) = W(K+1)
118 CONTINUE
GO TO 107
119 CONTINUE
C
C REVERSE COLUMNS WHEN NPEROD = 4.
C
IREV = 1
NBY2 = N/2
NP = 2
120 DO 122 J=1,NBY2
MSKIP = N+1-J
DO 121 I=1,M
A1 = Y(I,J)
Y(I,J) = Y(I,MSKIP)
Y(I,MSKIP) = A1
121 CONTINUE
122 CONTINUE
GO TO (108,109),IREV
123 CONTINUE
DO 128 J=1,N
DO 124 I=1,MHM1
MHMI = MH-I
MHPI = MH+I
W(MHMI) = .5*(Y(MHPI,J)+Y(I,J))
W(MHPI) = .5*(Y(MHPI,J)-Y(I,J))
124 CONTINUE
W(MH) = .5*Y(MH,J)
GO TO (126,125),MODD
125 W(M) = .5*Y(M,J)
126 CONTINUE
DO 127 I=1,M
Y(I,J) = W(I)
127 CONTINUE
128 CONTINUE
129 CONTINUE
C
C RETURN STORAGE REQUIREMENTS FOR W ARRAY.
C
W(1) = IPSTOR+IWP-1
RETURN
END